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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a or H 1 ).
Collect data in a way designed to test the hypothesis.
Perform an appropriate statistical test .
Decide whether to reject or fail to reject your null hypothesis.
Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

an estimate of the difference in average height between the two groups.
a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

Normal distribution
Descriptive statistics
Measures of central tendency
Correlation coefficient

Methodology

Cluster sampling
Stratified sampling
Types of interviews
Cohort study
Thematic analysis

Research bias

Implicit bias
Cognitive bias
Survivorship bias
Availability heuristic
Nonresponse bias
Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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9.1: Introduction to Hypothesis Testing

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Kyle Siegrist
University of Alabama in Huntsville via Random Services

Basic Theory

Preliminaries.

As usual, our starting point is a random experiment with an underlying sample space and a probability measure $\P$. In the basic statistical model, we have an observable random variable $\bs{X}$ taking values in a set $S$. In general, $\bs{X}$ can have quite a complicated structure. For example, if the experiment is to sample $n$ objects from a population and record various measurements of interest, then \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where $X_i$ is the vector of measurements for the $i$th object. The most important special case occurs when $(X_1, X_2, \ldots, X_n)$ are independent and identically distributed. In this case, we have a random sample of size $n$ from the common distribution.

The purpose of this section is to define and discuss the basic concepts of statistical hypothesis testing . Collectively, these concepts are sometimes referred to as the Neyman-Pearson framework, in honor of Jerzy Neyman and Egon Pearson, who first formalized them.

A statistical hypothesis is a statement about the distribution of $\bs{X}$. Equivalently, a statistical hypothesis specifies a set of possible distributions of $\bs{X}$: the set of distributions for which the statement is true. A hypothesis that specifies a single distribution for $\bs{X}$ is called simple ; a hypothesis that specifies more than one distribution for $\bs{X}$ is called composite .

In hypothesis testing , the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis . The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$.

An hypothesis test is a statistical decision ; the conclusion will either be to reject the null hypothesis in favor of the alternative, or to fail to reject the null hypothesis. The decision that we make must, of course, be based on the observed value $\bs{x}$ of the data vector $\bs{X}$. Thus, we will find an appropriate subset $R$ of the sample space $S$ and reject $H_0$ if and only if $\bs{x} \in R$. The set $R$ is known as the rejection region or the critical region . Note the asymmetry between the null and alternative hypotheses. This asymmetry is due to the fact that we assume the null hypothesis, in a sense, and then see if there is sufficient evidence in $\bs{x}$ to overturn this assumption in favor of the alternative.

An hypothesis test is a statistical analogy to proof by contradiction, in a sense. Suppose for a moment that $H_1$ is a statement in a mathematical theory and that $H_0$ is its negation. One way that we can prove $H_1$ is to assume $H_0$ and work our way logically to a contradiction. In an hypothesis test, we don't prove anything of course, but there are similarities. We assume $H_0$ and then see if the data $\bs{x}$ are sufficiently at odds with that assumption that we feel justified in rejecting $H_0$ in favor of $H_1$.

Often, the critical region is defined in terms of a statistic $w(\bs{X})$, known as a test statistic , where $w$ is a function from $S$ into another set $T$. We find an appropriate rejection region $R_T \subseteq T$ and reject $H_0$ when the observed value $w(\bs{x}) \in R_T$. Thus, the rejection region in $S$ is then $R = w^{-1}(R_T) = \left\{\bs{x} \in S: w(\bs{x}) \in R_T\right\}$. As usual, the use of a statistic often allows significant data reduction when the dimension of the test statistic is much smaller than the dimension of the data vector.

The ultimate decision may be correct or may be in error. There are two types of errors, depending on which of the hypotheses is actually true.

Types of errors:

A type 1 error is rejecting the null hypothesis $H_0$ when $H_0$ is true.
A type 2 error is failing to reject the null hypothesis $H_0$ when the alternative hypothesis $H_1$ is true.

Similarly, there are two ways to make a correct decision: we could reject $H_0$ when $H_1$ is true or we could fail to reject $H_0$ when $H_0$ is true. The possibilities are summarized in the following table:

Of course, when we observe $\bs{X} = \bs{x}$ and make our decision, either we will have made the correct decision or we will have committed an error, and usually we will never know which of these events has occurred. Prior to gathering the data, however, we can consider the probabilities of the various errors.

If $H_0$ is true (that is, the distribution of $\bs{X}$ is specified by $H_0$), then $\P(\bs{X} \in R)$ is the probability of a type 1 error for this distribution. If $H_0$ is composite, then $H_0$ specifies a variety of different distributions for $\bs{X}$ and thus there is a set of type 1 error probabilities.

The maximum probability of a type 1 error, over the set of distributions specified by $ H_0 $, is the significance level of the test or the size of the critical region.

The significance level is often denoted by $\alpha$. Usually, the rejection region is constructed so that the significance level is a prescribed, small value (typically 0.1, 0.05, 0.01).

If $H_1$ is true (that is, the distribution of $\bs{X}$ is specified by $H_1$), then $\P(\bs{X} \notin R)$ is the probability of a type 2 error for this distribution. Again, if $H_1$ is composite then $H_1$ specifies a variety of different distributions for $\bs{X}$, and thus there will be a set of type 2 error probabilities. Generally, there is a tradeoff between the type 1 and type 2 error probabilities. If we reduce the probability of a type 1 error, by making the rejection region $R$ smaller, we necessarily increase the probability of a type 2 error because the complementary region $S \setminus R$ is larger.

The extreme cases can give us some insight. First consider the decision rule in which we never reject $H_0$, regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = \emptyset$. A type 1 error is impossible, so the significance level is 0. On the other hand, the probability of a type 2 error is 1 for any distribution defined by $H_1$. At the other extreme, consider the decision rule in which we always rejects $H_0$ regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = S$. A type 2 error is impossible, but now the probability of a type 1 error is 1 for any distribution defined by $H_0$. In between these two worthless tests are meaningful tests that take the evidence $\bs{x}$ into account.

If $H_1$ is true, so that the distribution of $\bs{X}$ is specified by $H_1$, then $\P(\bs{X} \in R)$, the probability of rejecting $H_0$ is the power of the test for that distribution.

Thus the power of the test for a distribution specified by $ H_1 $ is the probability of making the correct decision.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with region $R_1$ is uniformly more powerful than the test with region $R_2$ if \[ \P(\bs{X} \in R_1) \ge \P(\bs{X} \in R_2) \text{ for every distribution of } \bs{X} \text{ specified by } H_1 \]

Naturally, in this case, we would prefer the first test. Often, however, two tests will not be uniformly ordered; one test will be more powerful for some distributions specified by $H_1$ while the other test will be more powerful for other distributions specified by $H_1$.

If a test has significance level $\alpha$ and is uniformly more powerful than any other test with significance level $\alpha$, then the test is said to be a uniformly most powerful test at level $\alpha$.

Clearly a uniformly most powerful test is the best we can do.

$P$-value

In most cases, we have a general procedure that allows us to construct a test (that is, a rejection region $R_\alpha$) for any given significance level $\alpha \in (0, 1)$. Typically, $R_\alpha$ decreases (in the subset sense) as $\alpha$ decreases.

The $P$-value of the observed value $\bs{x}$ of $\bs{X}$, denoted $P(\bs{x})$, is defined to be the smallest $\alpha$ for which $\bs{x} \in R_\alpha$; that is, the smallest significance level for which $H_0$ is rejected, given $\bs{X} = \bs{x}$.

Knowing $P(\bs{x})$ allows us to test $H_0$ at any significance level for the given data $\bs{x}$: If $P(\bs{x}) \le \alpha$ then we would reject $H_0$ at significance level $\alpha$; if $P(\bs{x}) \gt \alpha$ then we fail to reject $H_0$ at significance level $\alpha$. Note that $P(\bs{X})$ is a statistic . Informally, $P(\bs{x})$ can often be thought of as the probability of an outcome as or more extreme than the observed value $\bs{x}$, where extreme is interpreted relative to the null hypothesis $H_0$.

Analogy with Justice Systems

There is a helpful analogy between statistical hypothesis testing and the criminal justice system in the US and various other countries. Consider a person charged with a crime. The presumed null hypothesis is that the person is innocent of the crime; the conjectured alternative hypothesis is that the person is guilty of the crime. The test of the hypotheses is a trial with evidence presented by both sides playing the role of the data. After considering the evidence, the jury delivers the decision as either not guilty or guilty . Note that innocent is not a possible verdict of the jury, because it is not the point of the trial to prove the person innocent. Rather, the point of the trial is to see whether there is sufficient evidence to overturn the null hypothesis that the person is innocent in favor of the alternative hypothesis of that the person is guilty. A type 1 error is convicting a person who is innocent; a type 2 error is acquitting a person who is guilty. Generally, a type 1 error is considered the more serious of the two possible errors, so in an attempt to hold the chance of a type 1 error to a very low level, the standard for conviction in serious criminal cases is beyond a reasonable doubt .

Tests of an Unknown Parameter

Hypothesis testing is a very general concept, but an important special class occurs when the distribution of the data variable $\bs{X}$ depends on a parameter $\theta$ taking values in a parameter space $\Theta$. The parameter may be vector-valued, so that $\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_n)$ and $\Theta \subseteq \R^k$ for some $k \in \N_+$. The hypotheses generally take the form \[ H_0: \theta \in \Theta_0 \text{ versus } H_1: \theta \notin \Theta_0 \] where $\Theta_0$ is a prescribed subset of the parameter space $\Theta$. In this setting, the probabilities of making an error or a correct decision depend on the true value of $\theta$. If $R$ is the rejection region, then the power function $ Q $ is given by \[ Q(\theta) = \P_\theta(\bs{X} \in R), \quad \theta \in \Theta \] The power function gives a lot of information about the test.

The power function satisfies the following properties:

$Q(\theta)$ is the probability of a type 1 error when $\theta \in \Theta_0$.
$\max\left\{Q(\theta): \theta \in \Theta_0\right\}$ is the significance level of the test.
$1 - Q(\theta)$ is the probability of a type 2 error when $\theta \notin \Theta_0$.
$Q(\theta)$ is the power of the test when $\theta \notin \Theta_0$.

If we have two tests, we can compare them by means of their power functions.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with rejection region $R_1$ is uniformly more powerful than the test with rejection region $R_2$ if $ Q_1(\theta) \ge Q_2(\theta)$ for all $ \theta \notin \Theta_0 $.

Most hypothesis tests of an unknown real parameter $\theta$ fall into three special cases:

Suppose that $ \theta $ is a real parameter and $ \theta_0 \in \Theta $ a specified value. The tests below are respectively the two-sided test , the left-tailed test , and the right-tailed test .

$H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$
$H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$

Thus the tests are named after the conjectured alternative. Of course, there may be other unknown parameters besides $\theta$ (known as nuisance parameters ).

Equivalence Between Hypothesis Test and Confidence Sets

There is an equivalence between hypothesis tests and confidence sets for a parameter $\theta$.

Suppose that $C(\bs{x})$ is a $1 - \alpha$ level confidence set for $\theta$. The following test has significance level $\alpha$ for the hypothesis $ H_0: \theta = \theta_0 $ versus $ H_1: \theta \ne \theta_0 $: Reject $H_0$ if and only if $\theta_0 \notin C(\bs{x})$

By definition, $\P[\theta \in C(\bs{X})] = 1 - \alpha$. Hence if $H_0$ is true so that $\theta = \theta_0$, then the probability of a type 1 error is $P[\theta \notin C(\bs{X})] = \alpha$.

Equivalently, we fail to reject $H_0$ at significance level $\alpha$ if and only if $\theta_0$ is in the corresponding $1 - \alpha$ level confidence set. In particular, this equivalence applies to interval estimates of a real parameter $\theta$ and the common tests for $\theta$ given above .

In each case below, the confidence interval has confidence level $1 - \alpha$ and the test has significance level $\alpha$.

Suppose that $\left[L(\bs{X}, U(\bs{X})\right]$ is a two-sided confidence interval for $\theta$. Reject $H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$ or $\theta_0 \gt U(\bs{X})$.
Suppose that $L(\bs{X})$ is a confidence lower bound for $\theta$. Reject $H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$.
Suppose that $U(\bs{X})$ is a confidence upper bound for $\theta$. Reject $H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$ if and only if $\theta_0 \gt U(\bs{X})$.

Pivot Variables and Test Statistics

Recall that confidence sets of an unknown parameter $\theta$ are often constructed through a pivot variable , that is, a random variable $W(\bs{X}, \theta)$ that depends on the data vector $\bs{X}$ and the parameter $\theta$, but whose distribution does not depend on $\theta$ and is known. In this case, a natural test statistic for the basic tests given above is $W(\bs{X}, \theta_0)$.

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S.3.1 hypothesis testing (critical value approach).

The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the " critical value ." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are:

Specify the null and alternative hypotheses.
Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. To conduct the hypothesis test for the population mean μ , we use the t -statistic $t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}$ which follows a t -distribution with n - 1 degrees of freedom.
Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error — which is denoted $\alpha$ (greek letter "alpha") and is called the " significance level of the test " — is small (typically 0.01, 0.05, or 0.10).
Compare the test statistic to the critical value. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less extreme than the critical value, do not reject the null hypothesis.

Example S.3.1.1

Mean gpa section .

In our example concerning the mean grade point average, suppose we take a random sample of n = 15 students majoring in mathematics. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right-Tailed

The critical value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the t -value, denoted t $\alpha$ , n - 1 , such that the probability to the right of it is $\alpha$. It can be shown using either statistical software or a t -table that the critical value t 0.05,14 is 1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3 if the test statistic t * is greater than 1.7613. Visually, the rejection region is shaded red in the graph.

t distribution graph for a t value of 1.76131

Left-Tailed

The critical value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the t -value, denoted -t ( $\alpha$ , n - 1) , such that the probability to the left of it is $\alpha$. It can be shown using either statistical software or a t -table that the critical value -t 0.05,14 is -1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3 if the test statistic t * is less than -1.7613. Visually, the rejection region is shaded red in the graph.

There are two critical values for the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 — one for the left-tail denoted -t ( $\alpha$ / 2, n - 1) and one for the right-tail denoted t ( $\alpha$ / 2, n - 1) . The value - t ( $\alpha$ /2, n - 1) is the t -value such that the probability to the left of it is $\alpha$/2, and the value t ( $\alpha$ /2, n - 1) is the t -value such that the probability to the right of it is $\alpha$/2. It can be shown using either statistical software or a t -table that the critical value -t 0.025,14 is -2.1448 and the critical value t 0.025,14 is 2.1448. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3 if the test statistic t * is less than -2.1448 or greater than 2.1448. Visually, the rejection region is shaded red in the graph.

t distribution graph for a two tailed test of 0.05 level of significance

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

What is Hypothesis Testing?
Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
Example : Testing a new drug.
Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

Symbolized by the Greek letter alpha (α).
Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

Symbolized by the Greek letter beta (β).
Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

Set up Hypotheses : Before starting, you make a prediction:
Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

The average healing time in the Drug Group is 2 hours.
The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

Correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, skewness and kurtosis – peaks and tails, understanding data through skewness and kurtosis”, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

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Statistical Analysis: Developing and Testing Hypotheses

Statistical hypothesis testing is sometimes known as confirmatory data analysis. It is a way of drawing inferences from data. In the process, you develop a hypothesis or theory about what you might see in your research. You then test that hypothesis against the data that you collect.

Hypothesis testing is generally used when you want to compare two groups, or compare a group against an idealised position.

Before You Start: Developing A Research Hypothesis

Before you can do any kind of research in social science fields such as management, you need a research question or hypothesis. Research is generally designed to either answer a research question or consider a research hypothesis . These two are closely linked, and generally one or the other is used, rather than both.

A research question is the question that your research sets out to answer . For example:

Do men and women like ice cream equally?

Do men and women like the same flavours of ice cream?

What are the main problems in the market for ice cream?

How can the market for ice cream be segmented and targeted?

Research hypotheses are statements of what you believe you will find in your research.

These are then tested statistically during the research to see if your belief is correct. Examples include:

Men and women like ice cream to different extents.

Men and women like different flavours of ice cream.

Men are more likely than women to like mint ice cream.

Women are more likely than men to like chocolate ice cream.

Both men and women prefer strawberry to vanilla ice cream.

Relationships vs Differences

Research hypotheses can be expressed in terms of differences between groups, or relationships between variables. However, these are two sides of the same coin: almost any hypothesis could be set out in either way.

For example:

There is a relationship between gender and liking ice cream OR

Men are more likely to like ice cream than women.

Testing Research Hypotheses

The purpose of statistical hypothesis testing is to use a sample to draw inferences about a population.

Testing research hypotheses requires a number of steps:

Step 1. Define your research hypothesis

The first step in any hypothesis testing is to identify your hypothesis, which you will then go on to test. How you define your hypothesis may affect the type of statistical testing that you do, so it is important to be clear about it. In particular, consider whether you are going to hypothesise simply that there is a relationship or speculate about the direction of the relationship.

Using the examples above:

There is a relationship between gender and liking ice cream is a non-directional hypothesis. You have simply specified that there is a relationship, not whether men or women like ice cream more.

However, men are more likely to like ice cream than women is directional : you have specified which gender is more likely to like ice cream.

Generally, it is better not to specify direction unless you are moderately sure about it.

Step 2. Define the null hypothesis

The null hypothesis is basically a statement of what you are hoping to disprove: the opposite of your ‘guess’ about the relationship. For example, in the hypotheses above, the null hypothesis would be:

Men and women like ice cream equally, or

There is no relationship between gender and ice cream.

This also defines your ‘alternative hypothesis’ which is your ‘test hypothesis’ ( men like ice cream more than women ). Your null hypothesis is generally that there is no difference, because this is the simplest position.

The purpose of hypothesis testing is to disprove the null hypothesis. If you cannot disprove the null hypothesis, you have to assume it is correct.

Step 3. Develop a summary measure that describes your variable of interest for each group you wish to compare

Our page on Simple Statistical Analysis describes several summary measures, including two of the most common, mean and median.

The next step in your hypothesis testing is to develop a summary measure for each of your groups. For example, to test the gender differences in liking for ice cream, you might ask people how much they liked ice cream on a scale of 1 to 5. Alternatively, you might have data about the number of times that ice creams are consumed each week in the summer months.

You then need to produce a summary measure for each group, usually mean and standard deviation. These may be similar for each group, or quite different.

Step 4. Choose a reference distribution and calculate a test statistic

To decide whether there is a genuine difference between the two groups, you have to use a reference distribution against which to measure the values from the two groups.

The most common source of reference distributions is a standard distribution such as the normal distribution or t - distribution. These two are the same, except that the standard deviation of the t -distribution is estimated from the sample, and that of the normal distribution is known. There is more about this in our page on Statistical Distributions .

You then compare the summary data from the two groups by using them to calculate a test statistic. There is a standard formula for every test statistic and reference distribution. The test and reference distribution depend on your data and the purpose of your testing (see below).

The test that you use to compare your groups will depend on how many groups you have, the type of data that you have collected, and how reliable your data are. In general, you would use different tests for comparing two groups than you would for comparing three or more.

Our page Surveys and Survey Design explains that there are two types of answer scale, continuous and categorical. Age, for example, is a continuous scale, although it can also be grouped into categories. You may also find it helpful to read our page on Types of Data .

Gender is a category scale.

For a continuous scale, you can use the mean values of the two groups that you are comparing.

For a category scale, you need to use the median values.

Source: Easterby-Smith, Thorpe and Jackson, Management Research 4th Edition

One- or Two-Tailed Test

The other thing that you have to decide is whether you use what is known as a ‘one-tailed’ or ‘two-tailed’ test.

This allows you to compare differences between groups in either one or both directions.

In practice, this boils down to whether your research hypothesis is expressed as ‘x is likely to be more than y’, or ‘x is likely to be different from y’. If you are confident of the direction of the distance (that is, you are sure that the only options are that ‘x is likely to be more than y’ or ‘x and y are the same’), then your test will be one-tailed. If not, it will be two-tailed .

If there is any doubt, it is better to use a two-tailed test.

You should only use a one-tailed test when you are certain about the direction of the difference, and it doesn’t matter if you are wrong.

The graph under Step 5 shows a two-tailed test.

If you are not very confident about the quality of the data collected, for example because the inputting was done quickly and cheaply, or because the data have not been checked, then you may prefer to use the median even if the data are continuous to avoid any problems with outliers. This makes the tests more robust, and the results more reliable.

Our page on correlations suggests that you may also want to plot a scattergraph before undertaking any further analysis. This will also help you to identify any outliers or potential problems with the data.

Calculating the Test Statistic

For each type of test, there is a standard formula for the test statistic. For example, for the t -test, it is:

(M1-M2)/SE(diff)

M1 is the mean of the first group

M2 is the mean of the second group

SE(diff) is the standard error of the difference, which is calculated from the standard deviation and the sample size of each group.

The formula for calculating the standard error of the difference between means is:

sd 2 = the square of the standard deviation of the source population (i.e., the variance);
n a = the size of sample A; and
n b = the size of sample B.

Step 5. Identify Acceptance and Rejection Regions

The final part of the test is to see if your test statistic is significant—in other words, whether you are going to accept or reject your null hypothesis. You need to consider first what level of significance is required. This tells you the probability that you have achieved your result by chance.

Significance (or p-value) is usually required to be either 5% or 1%, meaning that you are 95% or 99% confident that your result was not achieved by chance.

NOTE: the significance level is sometimes expressed as p < 0.05 or p < 0.01.

For more about significance, you may like to read our page on Significance and Confidence Intervals .

The graph below shows a reference distribution (this one could be either the normal or the t- distribution) with the acceptance and rejection regions marked. It also shows the critical values. µ is the mean. For more about this, you may like to read our page on Statistical Distributions .

Reference distribution showing acceptance and rejection regions, critical values and mean.

The critical values are identified from published statistical tables for your reference distribution, which are available for different levels of significance.

If your test statistic falls within either of the two rejection regions (that is, it is greater than the higher critical value, or less than the lower one), you will reject the null hypothesis. You can therefore accept your alternative hypothesis.

Step 6. Draw Conclusions and Inferences

The final step is to draw conclusions.

If your test statistic fell within the rejection region, and you have rejected the null hypothesis, you can therefore conclude that there is a gender difference in liking for ice cream, using the example above.

Types of Error

There are four possible outcomes from statistical testing (see table):

The groups are different, and you conclude that they are different (correct result)

The groups are different, but you conclude that they are not (Type II error)

The groups are the same, but you conclude that they are different (Type I error)

The groups are the same, and you conclude that they are the same (correct result).

Type I errors are generally considered more important than Type II, because they have the potential to change the status quo.

For example, if you wrongly conclude that a new medical treatment is effective, doctors are likely to move to providing that treatment. Patients may receive the treatment instead of an alternative that could have fewer side effects, and pharmaceutical companies may stop looking for an alternative treatment.

Data Handling and Algebra - The Skills You Need Guide to Numeracy

How Do You Formulate (Important) Hypotheses?

Open Access
First Online: 03 December 2022

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James Hiebert 6 ,
Jinfa Cai 7 ,
Stephen Hwang 7 ,
Anne K Morris 6 &
Charles Hohensee 6

Part of the book series: Research in Mathematics Education ((RME))

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Building on the ideas in Chap. 1, we describe formulating, testing, and revising hypotheses as a continuing cycle of clarifying what you want to study, making predictions about what you might find together with developing your reasons for these predictions, imagining tests of these predictions, revising your predictions and rationales, and so on. Many resources feed this process, including reading what others have found about similar phenomena, talking with colleagues, conducting pilot studies, and writing drafts as you revise your thinking. Although you might think you cannot predict what you will find, it is always possible—with enough reading and conversations and pilot studies—to make some good guesses. And, once you guess what you will find and write out the reasons for these guesses you are on your way to scientific inquiry. As you refine your hypotheses, you can assess their research importance by asking how connected they are to problems your research community really wants to solve.

You have full access to this open access chapter, Download chapter PDF

Part I. Getting Started

We want to begin by addressing a question you might have had as you read the title of this chapter. You are likely to hear, or read in other sources, that the research process begins by asking research questions . For reasons we gave in Chap. 1 , and more we will describe in this and later chapters, we emphasize formulating, testing, and revising hypotheses. However, it is important to know that asking and answering research questions involve many of the same activities, so we are not describing a completely different process.

We acknowledge that many researchers do not actually begin by formulating hypotheses. In other words, researchers rarely get a researchable idea by writing out a well-formulated hypothesis. Instead, their initial ideas for what they study come from a variety of sources. Then, after they have the idea for a study, they do lots of background reading and thinking and talking before they are ready to formulate a hypothesis. So, for readers who are at the very beginning and do not yet have an idea for a study, let’s back up. Where do research ideas come from?

There are no formulas or algorithms that spawn a researchable idea. But as you begin the process, you can ask yourself some questions. Your answers to these questions can help you move forward.

What are you curious about? What are you passionate about? What have you wondered about as an educator? These are questions that look inward, questions about yourself.

What do you think are the most pressing educational problems? Which problems are you in the best position to address? What change(s) do you think would help all students learn more productively? These are questions that look outward, questions about phenomena you have observed.

What are the main areas of research in the field? What are the big questions that are being asked? These are questions about the general landscape of the field.

What have you read about in the research literature that caught your attention? What have you read that prompted you to think about extending the profession’s knowledge about this? What have you read that made you ask, “I wonder why this is true?” These are questions about how you can build on what is known in the field.

What are some research questions or testable hypotheses that have been identified by other researchers for future research? This, too, is a question about how you can build on what is known in the field. Taking up such questions or hypotheses can help by providing some existing scaffolding that others have constructed.

What research is being done by your immediate colleagues or your advisor that is of interest to you? These are questions about topics for which you will likely receive local support.

Exercise 2.1

Brainstorm some answers for each set of questions. Record them. Then step back and look at the places of intersection. Did you have similar answers across several questions? Write out, as clearly as you can, the topic that captures your primary interest, at least at this point. We will give you a chance to update your responses as you study this book.

Part II. Paths from a General Interest to an Informed Hypothesis

There are many different paths you might take from conceiving an idea for a study, maybe even a vague idea, to formulating a prediction that leads to an informed hypothesis that can be tested. We will explore some of the paths we recommend.

We will assume you have completed Exercise 2.1 in Part I and have some written answers to the six questions that preceded it as well as a statement that describes your topic of interest. This very first statement could take several different forms: a description of a problem you want to study, a question you want to address, or a hypothesis you want to test. We recommend that you begin with one of these three forms, the one that makes most sense to you. There is an advantage to using all three and flexibly choosing the one that is most meaningful at the time and for a particular study. You can then move from one to the other as you think more about your research study and you develop your initial idea. To get a sense of how the process might unfold, consider the following alternative paths.

Beginning with a Prediction If You Have One

Sometimes, when you notice an educational problem or have a question about an educational situation or phenomenon, you quickly have an idea that might help solve the problem or answer the question. Here are three examples.

You are a teacher, and you noticed a problem with the way the textbook presented two related concepts in two consecutive lessons. Almost as soon as you noticed the problem, it occurred to you that the two lessons could be taught more effectively in the reverse order. You predicted better outcomes if the order was reversed, and you even had a preliminary rationale for why this would be true.

You are a graduate student and you read that students often misunderstand a particular aspect of graphing linear functions. You predicted that, by listening to small groups of students working together, you could hear new details that would help you understand this misconception.

You are a curriculum supervisor and you observed sixth-grade classrooms where students were learning about decimal fractions. After talking with several experienced teachers, you predicted that beginning with percentages might be a good way to introduce students to decimal fractions.

We begin with the path of making predictions because we see the other two paths as leading into this one at some point in the process (see Fig. 2.1 ). Starting with this path does not mean you did not sense a problem you wanted to solve or a question you wanted to answer.

The process flow diagram of initiation of hypothesis. It starts with a problem situation and leads to a prediction following the question to the hypothesis.

Three Pathways to Formulating Informed Hypotheses

Notice that your predictions can come from a variety of sources—your own experience, reading, and talking with colleagues. Most likely, as you write out your predictions you also think about the educational problem for which your prediction is a potential solution. Writing a clear description of the problem will be useful as you proceed. Notice also that it is easy to change each of your predictions into a question. When you formulate a prediction, you are actually answering a question, even though the question might be implicit. Making that implicit question explicit can generate a first draft of the research question that accompanies your prediction. For example, suppose you are the curriculum supervisor who predicts that teaching percentages first would be a good way to introduce decimal fractions. In an obvious shift in form, you could ask, “In what ways would teaching percentages benefit students’ initial learning of decimal fractions?”

The picture has a difference between a question and a prediction: a question simply asks what you will find whereas a prediction also says what you expect to find; written.

There are advantages to starting with the prediction form if you can make an educated guess about what you will find. Making a prediction forces you to think now about several things you will need to think about at some point anyway. It is better to think about them earlier rather than later. If you state your prediction clearly and explicitly, you can begin to ask yourself three questions about your prediction: Why do I expect to observe what I am predicting? Why did I make that prediction? (These two questions essentially ask what your rationale is for your prediction.) And, how can I test to see if it’s right? This is where the benefits of making predictions begin.

Asking yourself why you predicted what you did, and then asking yourself why you answered the first “why” question as you did, can be a powerful chain of thought that lays the groundwork for an increasingly accurate prediction and an increasingly well-reasoned rationale. For example, suppose you are the curriculum supervisor above who predicted that beginning by teaching percentages would be a good way to introduce students to decimal fractions. Why did you make this prediction? Maybe because students are familiar with percentages in everyday life so they could use what they know to anchor their thinking about hundredths. Why would that be helpful? Because if students could connect hundredths in percentage form with hundredths in decimal fraction form, they could bring their meaning of percentages into decimal fractions. But how would that help? If students understood that a decimal fraction like 0.35 meant 35 of 100, then they could use their understanding of hundredths to explore the meaning of tenths, thousandths, and so on. Why would that be useful? By continuing to ask yourself why you gave the previous answer, you can begin building your rationale and, as you build your rationale, you will find yourself revisiting your prediction, often making it more precise and explicit. If you were the curriculum supervisor and continued the reasoning in the previous sentences, you might elaborate your prediction by specifying the way in which percentages should be taught in order to have a positive effect on particular aspects of students’ understanding of decimal fractions.

Developing a Rationale for Your Predictions

Keeping your initial predictions in mind, you can read what others already know about the phenomenon. Your reading can now become targeted with a clear purpose.

By reading and talking with colleagues, you can develop more complete reasons for your predictions. It is likely that you will also decide to revise your predictions based on what you learn from your reading. As you develop sound reasons for your predictions, you are creating your rationales, and your predictions together with your rationales become your hypotheses. The more you learn about what is already known about your research topic, the more refined will be your predictions and the clearer and more complete your rationales. We will use the term more informed hypotheses to describe this evolution of your hypotheses.

The picture says you develop sound reasons for your predictions, you are creating your rationales, and your predictions together with your rationales become your hypotheses.

Developing more informed hypotheses is a good thing because it means: (1) you understand the reasons for your predictions; (2) you will be able to imagine how you can test your hypotheses; (3) you can more easily convince your colleagues that they are important hypotheses—they are hypotheses worth testing; and (4) at the end of your study, you will be able to more easily interpret the results of your test and to revise your hypotheses to demonstrate what you have learned by conducting the study.

Imagining Testing Your Hypotheses

Because we have tied together predictions and rationales to constitute hypotheses, testing hypotheses means testing predictions and rationales. Testing predictions means comparing empirical observations, or findings, with the predictions. Testing rationales means using these comparisons to evaluate the adequacy or soundness of the rationales.

Imagining how you might test your hypotheses does not mean working out the details for exactly how you would test them. Rather, it means thinking ahead about how you could do this. Recall the descriptor of scientific inquiry: “experience carefully planned in advance” (Fisher, 1935). Asking whether predictions are testable and whether rationales can be evaluated is simply planning in advance.

You might read that testing hypotheses means simply assessing whether predictions are correct or incorrect. In our view, it is more useful to think of testing as a means of gathering enough information to compare your findings with your predictions, revise your rationales, and propose more accurate predictions. So, asking yourself whether hypotheses can be tested means asking whether information could be collected to assess the accuracy of your predictions and whether the information will show you how to revise your rationales to sharpen your predictions.

Cycles of Building Rationales and Planning to Test Your Predictions

Scientific reasoning is a dialogue between the possible and the actual, an interplay between hypotheses and the logical expectations they give rise to: there is a restless to-and-fro motion of thought, the formulation and rectification of hypotheses (Medawar, 1982 , p.72).

As you ask yourself about how you could test your predictions, you will inevitably revise your rationales and sharpen your predictions. Your hypotheses will become more informed, more targeted, and more explicit. They will make clearer to you and others what, exactly, you plan to study.

When will you know that your hypotheses are clear and precise enough? Because of the way we define hypotheses, this question asks about both rationales and predictions. If a rationale you are building lets you make a number of quite different predictions that are equally plausible rather than a single, primary prediction, then your hypothesis needs further refinement by building a more complete and precise rationale. Also, if you cannot briefly describe to your colleagues a believable way to test your prediction, then you need to phrase it more clearly and precisely.

Each time you strengthen your rationales, you might need to adjust your predictions. And, each time you clarify your predictions, you might need to adjust your rationales. The cycle of going back and forth to keep your predictions and rationales tightly aligned has many payoffs down the road. Every decision you make from this point on will be in the interests of providing a transparent and convincing test of your hypotheses and explaining how the results of your test dictate specific revisions to your hypotheses. As you make these decisions (described in the succeeding chapters), you will probably return to clarify your hypotheses even further. But, you will be in a much better position, at each point, if you begin with well-informed hypotheses.

Beginning by Asking Questions to Clarify Your Interests

Instead of starting with predictions, a second path you might take devotes more time at the beginning to asking questions as you zero in on what you want to study. Some researchers suggest you start this way (e.g., Gournelos et al., 2019 ). Specifically, with this second path, the first statement you write to express your research interest would be a question. For example, you might ask, “Why do ninth-grade students change the way they think about linear equations after studying quadratic equations?” or “How do first graders solve simple arithmetic problems before they have been taught to add and subtract?”

The first phrasing of your question might be quite general or vague. As you think about your question and what you really want to know, you are likely to ask follow-up questions. These questions will almost always be more specific than your first question. The questions will also express more clearly what you want to know. So, the question “How do first graders solve simple arithmetic problems before they have been taught to add and subtract” might evolve into “Before first graders have been taught to solve arithmetic problems, what strategies do they use to solve arithmetic problems with sums and products below 20?” As you read and learn about what others already know about your questions, you will continually revise your questions toward clearer and more explicit and more precise versions that zero in on what you really want to know. The question above might become, “Before they are taught to solve arithmetic problems, what strategies do beginning first graders use to solve arithmetic problems with sums and products below 20 if they are read story problems and given physical counters to help them keep track of the quantities?”

Imagining Answers to Your Questions

If you monitor your own thinking as you ask questions, you are likely to begin forming some guesses about answers, even to the early versions of the questions. What do students learn about quadratic functions that influences changes in their proportional reasoning when dealing with linear functions? It could be that if you analyze the moments during instruction on quadratic equations that are extensions of the proportional reasoning involved in solving linear equations, there are times when students receive further experience reasoning proportionally. You might predict that these are the experiences that have a “backward transfer” effect (Hohensee, 2014 ).

These initial guesses about answers to your questions are your first predictions. The first predicted answers are likely to be hunches or fuzzy, vague guesses. This simply means you do not know very much yet about the question you are asking. Your first predictions, no matter how unfocused or tentative, represent the most you know at the time about the question you are asking. They help you gauge where you are in your thinking.

Shifting to the Hypothesis Formulation and Testing Path

Research questions can play an important role in the research process. They provide a succinct way of capturing your research interests and communicating them to others. When colleagues want to know about your work, they will often ask “What are your research questions?” It is good to have a ready answer.

However, research questions have limitations. They do not capture the three images of scientific inquiry presented in Chap. 1 . Due, in part, to this less expansive depiction of the process, research questions do not take you very far. They do not provide a guide that leads you through the phases of conducting a study.

Consequently, when you can imagine an answer to your research question, we recommend that you move onto the hypothesis formulation and testing path. Imagining an answer to your question means you can make plausible predictions. You can now begin clarifying the reasons for your predictions and transform your early predictions into hypotheses (predictions along with rationales). We recommend you do this as soon as you have guesses about the answers to your questions because formulating, testing, and revising hypotheses offers a tool that puts you squarely on the path of scientific inquiry. It is a tool that can guide you through the entire process of conducting a research study.

This does not mean you are finished asking questions. Predictions are often created as answers to questions. So, we encourage you to continue asking questions to clarify what you want to know. But your target shifts from only asking questions to also proposing predictions for the answers and developing reasons the answers will be accurate predictions. It is by predicting answers, and explaining why you made those predictions, that you become engaged in scientific inquiry.

Cycles of Refining Questions and Predicting Answers

An example might provide a sense of how this process plays out. Suppose you are reading about Vygotsky’s ( 1987 ) zone of proximal development (ZPD), and you realize this concept might help you understand why your high school students had trouble learning exponential functions. Maybe they were outside this zone when you tried to teach exponential functions. In order to recognize students who would benefit from instruction, you might ask, “How can I identify students who are within the ZPD around exponential functions?” What would you predict? Maybe students in this ZPD are those who already had knowledge of related functions. You could write out some reasons for this prediction, like “students who understand linear and quadratic functions are more likely to extend their knowledge to exponential functions.” But what kind of data would you need to test this? What would count as “understanding”? Are linear and quadratic the functions you should assess? Even if they are, how could you tell whether students who scored well on tests of linear and quadratic functions were within the ZPD of exponential functions? How, in the end, would you measure what it means to be in this ZPD? So, asking a series of reasonable questions raised some red flags about the way your initial question was phrased, and you decide to revise it.

You set the stage for revising your question by defining ZPD as the zone within which students can solve an exponential function problem by making only one additional conceptual connection between what they already know and exponential functions. Your revised question is, “Based on students’ knowledge of linear and quadratic functions, which students are within the ZPD of exponential functions?” This time you know what kind of data you need: the number of conceptual connections students need to bridge from their knowledge of related functions to exponential functions. How can you collect these data? Would you need to see into the minds of the students? Or, are there ways to test the number of conceptual connections someone makes to move from one topic to another? Do methods exist for gathering these data? You decide this is not realistic, so you now have a choice: revise the question further or move your research in a different direction.

Notice that we do not use the term research question for all these early versions of questions that begin clarifying for yourself what you want to study. These early versions are too vague and general to be called research questions. In this book, we save the term research question for a question that comes near the end of the work and captures exactly what you want to study . By the time you are ready to specify a research question, you will be thinking about your study in terms of hypotheses and tests. When your hypotheses are in final form and include clear predictions about what you will find, it will be easy to state the research questions that accompany your predictions.

To reiterate one of the key points of this chapter: hypotheses carry much more information than research questions. Using our definition, hypotheses include predictions about what the answer might be to the question plus reasons for why you think so. Unlike research questions, hypotheses capture all three images of scientific inquiry presented in Chap. 1 (planning, observing and explaining, and revising one’s thinking). Your hypotheses represent the most you know, at the moment, about your research topic. The same cannot be said for research questions.

Beginning with a Research Problem

When you wrote answers to the six questions at the end of Part I of this chapter, you might have identified a research interest by stating it as a problem. This is the third path you might take to begin your research. Perhaps your description of your problem might look something like this: “When I tried to teach my middle school students by presenting them with a challenging problem without showing them how to solve similar problems, they didn’t exert much effort trying to find a solution but instead waited for me to show them how to solve the problem.” You do not have a specific question in mind, and you do not have an idea for why the problem exists, so you do not have a prediction about how to solve it. Writing a statement of this problem as clearly as possible could be the first step in your research journey.

As you think more about this problem, it will feel natural to ask questions about it. For example, why did some students show more initiative than others? What could I have done to get them started? How could I have encouraged the students to keep trying without giving away the solution? You are now on the path of asking questions—not research questions yet, but questions that are helping you focus your interest.

As you continue to think about these questions, reflect on your own experience, and read what others know about this problem, you will likely develop some guesses about the answers to the questions. They might be somewhat vague answers, and you might not have lots of confidence they are correct, but they are guesses that you can turn into predictions. Now you are on the hypothesis-formulation-and-testing path. This means you are on the path of asking yourself why you believe the predictions are correct, developing rationales for the predictions, asking what kinds of empirical observations would test your predictions, and refining your rationales and predictions as you read the literature and talk with colleagues.

A simple diagram that summarizes the three paths we have described is shown in Fig. 2.1 . Each row of arrows represents one pathway for formulating an informed hypothesis. The dotted arrows in the first two rows represent parts of the pathways that a researcher may have implicitly travelled through already (without an intent to form a prediction) but that ultimately inform the researcher’s development of a question or prediction.

Part III. One Researcher’s Experience Launching a Scientific Inquiry

Martha was in her third year of her doctoral program and beginning to identify a topic for her dissertation. Based on (a) her experience as a high school mathematics teacher and a curriculum supervisor, (b) the reading she has done to this point, and (c) her conversations with her colleagues, she has developed an interest in what kinds of professional development experiences (let’s call them learning opportunities [LOs] for teachers) are most effective. Where does she go from here?

Exercise 2.2

Before you continue reading, please write down some suggestions for Martha about where she should start.

A natural thing for Martha to do at this point is to ask herself some additional questions, questions that specify further what she wants to learn: What kinds of LOs do most teachers experience? How do these experiences change teachers’ practices and beliefs? Are some LOs more effective than others? What makes them more effective?

To focus her questions and decide what she really wants to know, she continues reading but now targets her reading toward everything she can find that suggests possible answers to these questions. She also talks with her colleagues to get more ideas about possible answers to these or related questions. Over several weeks or months, she finds herself being drawn to questions about what makes LOs effective, especially for helping teachers teach more conceptually. She zeroes in on the question, “What makes LOs for teachers effective for improving their teaching for conceptual understanding?”

This question is more focused than her first questions, but it is still too general for Martha to define a research study. How does she know it is too general? She uses two criteria. First, she notices that the predictions she makes about the answers to the question are all over the place; they are not constrained by the reasons she has assembled for her predictions. One prediction is that LOs are more effective when they help teachers learn content. Martha makes this guess because previous research suggests that effective LOs for teachers include attention to content. But this rationale allows lots of different predictions. For example, LOs are more effective when they focus on the content teachers will teach; LOs are more effective when they focus on content beyond what teachers will teach so teachers see how their instruction fits with what their students will encounter later; and LOs are more effective when they are tailored to the level of content knowledge participants have when they begin the LOs. The rationale she can provide at this point does not point to a particular prediction.

A second measure Martha uses to decide her question is too general is that the predictions she can make regarding the answers seem very difficult to test. How could she test, for example, whether LOs should focus on content beyond what teachers will teach? What does “content beyond what teachers teach” mean? How could you tell whether teachers use their new knowledge of later content to inform their teaching?

Before anticipating what Martha’s next question might be, it is important to pause and recognize how predicting the answers to her questions moved Martha into a new phase in the research process. As she makes predictions, works out the reasons for them, and imagines how she might test them, she is immersed in scientific inquiry. This intellectual work is the main engine that drives the research process. Also notice that revisions in the questions asked, the predictions made, and the rationales built represent the updated thinking (Chap. 1 ) that occurs as Martha continues to define her study.

Based on all these considerations and her continued reading, Martha revises the question again. The question now reads, “Do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?” Although she feels like the question is more specific, she realizes that the answer to the question is either “yes” or “no.” This, by itself, is a red flag. Answers of “yes” or “no” would not contribute much to understanding the relationships between these LOs for teachers and changes in their teaching. Recall from Chap. 1 that understanding how things work, explaining why things work, is the goal of scientific inquiry.

Martha continues by trying to understand why she believes the answer is “yes.” When she tries to write out reasons for predicting “yes,” she realizes that her prediction depends on a variety of factors. If teachers already have deep knowledge of the content, the LOs might not affect them as much as other teachers. If the LOs do not help teachers develop their own conceptual understanding, they are not likely to change their teaching. By trying to build the rationale for her prediction—thus formulating a hypothesis—Martha realizes that the question still is not precise and clear enough.

Martha uses what she learned when developing the rationale and rephrases the question as follows: “ Under what conditions do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?” Through several additional cycles of thinking through the rationale for her predictions and how she might test them, Martha specifies her question even further: “Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from LOs that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?”

Each version of Martha’s question has become more specific. This has occurred as she has (a) identified a starting condition for the teachers—they lack conceptual knowledge of linear functions, (b) specified the mathematics content as linear functions, and (c) included a condition or purpose of the LO—it is aimed at conceptual learning.

Because of the way Martha’s question is now phrased, her predictions will require thinking about the conditions that could influence what teachers learn from the LOs and how this learning could affect their teaching. She might predict that if teachers engaged in LOs that extended over multiple sessions, they would develop deeper understanding which would, in turn, prompt changes in their teaching. Or she might predict that if the LOs included examples of how their conceptual learning could translate into different instructional activities for their students, teachers would be more likely to change their teaching. Reasons for these predictions would likely come from research about the effects of professional development on teachers’ practice.

As Martha thinks about testing her predictions, she realizes it will probably be easier to measure the conditions under which teachers are learning than the changes in the conceptual emphasis in their instruction. She makes a note to continue searching the literature for ways to measure the “conceptualness” of teaching.

As she refines her predictions and expresses her reasons for the predictions, she formulates a hypothesis (in this case several hypotheses) that will guide her research. As she makes predictions and develops the rationales for these predictions, she will probably continue revising her question. She might decide, for example, that she is not interested in studying the condition of different numbers of LO sessions and so decides to remove this condition from consideration by including in her question something like “. . . over five 2-hour sessions . . .”

At this point, Martha has developed a research question, articulated a number of predictions, and developed rationales for them. Her current question is: “Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from five 2-hour LO sessions that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?” Her hypothesis is:

Prediction: Participating teachers will show changes in their teaching with a greater emphasis on conceptual understanding, with larger changes on linear function topics directly addressed in the LOs than on other topics.

Brief Description of Rationale: (1) Past research has shown correlations between teachers’ specific mathematics knowledge of a topic and the quality of their teaching of that topic. This does not mean an increase in knowledge causes higher quality teaching but it allows for that possibility. (2) Transfer is usually difficult for teachers, but the examples developed during the LO sessions will help them use what they learned to teach for conceptual understanding. This is because the examples developed during the LO sessions are much like those that will be used by the teachers. So larger changes will be found when teachers are teaching the linear function topics addressed in the LOs.

Notice it is more straightforward to imagine how Martha could test this prediction because it is more precise than previous predictions. Notice also that by asking how to test a particular prediction, Martha will be faced with a decision about whether testing this prediction will tell her something she wants to learn. If not, she can return to the research question and consider how to specify it further and, perhaps, constrain further the conditions that could affect the data.

As Martha formulates her hypotheses and goes through multiple cycles of refining her question(s), articulating her predictions, and developing her rationales, she is constantly building the theoretical framework for her study. Because the theoretical framework is the topic for Chap. 3 , we will pause here and pick up Martha’s story in the next chapter. Spoiler alert: Martha’s experience contains some surprising twists and turns.

Before leaving Martha, however, we point out two aspects of the process in which she has been engaged. First, it can be useful to think about the process as identifying (1) the variables targeted in her predictions, (2) the mechanisms she believes explain the relationships among the variables, and (3) the definitions of all the terms that are special to her educational problem. By variables, we mean things that can be measured and, when measured, can take on different values. In Martha’s case, the variables are the conceptualness of teaching and the content topics addressed in the LOs. The mechanisms are cognitive processes that enable teachers to see the relevance of what they learn in PD to their own teaching and that enable the transfer of learning from one setting to another. Definitions are the precise descriptions of how the important ideas relevant to the research are conceptualized. In Martha’s case, definitions must be provided for terms like conceptual understanding, linear functions, LOs, each of the topics related to linear functions, instructional setting, and knowledge transfer.

A second aspect of the process is a practice that Martha acquired as part of her graduate program, a practice that can go unnoticed. Martha writes out, in full sentences, her thinking as she wrestles with her research question, her predictions of the answers, and the rationales for her predictions. Writing is a tool for organizing thinking and we recommend you use it throughout the scientific inquiry process. We say more about this at the end of the chapter.

Here are the questions Martha wrote as she developed a clearer sense of what question she wanted to answer and what answer she predicted. The list shows the increasing refinement that occurred as she continued to read, think, talk, and write.

Early questions: What kinds of LOs do most teachers experience? How do these experiences change teachers’ practices and beliefs? Are some LOs more effective than others? What makes them more effective?

First focused question: What makes LOs for teachers effective for improving their teaching for conceptual understanding?

Question after trying to predict the answer and imagining how to test the prediction: Do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?

Question after developing an initial rationale for her prediction: Under what conditions do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?

Question after developing a more precise prediction and richer rationale: Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from five 2-hour LO sessions that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?

Part IV. An Illustrative Dialogue

The story of Martha described the major steps she took to refine her thinking. However, there is a lot of work that went on behind the scenes that wasn’t part of the story. For example, Martha had conversations with fellow students and professors that sharpened her thinking. What do these conversations look like? Because they are such an important part of the inquiry process, it will be helpful to “listen in” on the kinds of conversations that students might have with their advisors.

Here is a dialogue between a beginning student, Sam (S), and their advisor, Dr. Avery (A). They are meeting to discuss data Sam collected for a course project. The dialogue below is happening very early on in Sam’s conceptualization of the study, prior even to systematic reading of the literature.

Thanks for meeting with me today. As you know, I was able to collect some data for a course project a few weeks ago, but I’m having trouble analyzing the data, so I need your help. Let me try to explain the problem. As you know, I wanted to understand what middle-school teachers do to promote girls’ achievement in a mathematics class. I conducted four observations in each of three teachers’ classrooms. I also interviewed each teacher once about the four lessons I observed, and I interviewed two girls from each of the teachers’ classes. Obviously, I have a ton of data. But when I look at all these data, I don’t really know what I learned about my topic. When I was observing the teachers, I thought I might have observed some ways the teachers were promoting girls’ achievement, but then I wasn’t sure how to interpret my data. I didn’t know if the things I was observing were actually promoting girls’ achievement.

What were some of your observations?

Well, in a couple of my classroom observations, teachers called on girls to give an answer, even when the girls didn’t have their hands up. I thought that this might be a way that teachers were promoting the girls’ achievement. But then the girls didn’t say anything about that when I interviewed them and also the teachers didn’t do it in every class. So, it’s hard to know what effect, if any, this might have had on their learning or their motivation to learn. I didn’t want to ask the girls during the interview specifically about the teacher calling on them, and without the girls bringing it up themselves, I didn’t know if it had any effect.

Well, why didn’t you want to ask the girls about being called on?

Because I wanted to leave it as open as possible; I didn’t want to influence what they were going to say. I didn’t want to put words in their mouths. I wanted to know what they thought the teacher was doing that promoted their mathematical achievement and so I only asked the girls general questions, like “Do you think the teacher does things to promote girls’ mathematical achievement?” and “Can you describe specific experiences you have had that you believe do and do not promote your mathematical achievement?”

So then, how did they answer those general questions?

Well, with very general answers, such as that the teacher knows their names, offers review sessions, grades their homework fairly, gives them opportunities to earn extra credit, lets them ask questions, and always answers their questions. Nothing specific that helps me know what teaching actions specifically target girls’ mathematics achievement.

OK. Any ideas about what you might do next?

Well, I remember that when I was planning this data collection for my course, you suggested I might want to be more targeted and specific about what I was looking for. I can see now that more targeted questions would have made my data more interpretable in terms of connecting teaching actions to the mathematical achievement of girls. But I just didn’t want to influence what the girls would say.

Yes, I remember when you were planning your course project, you wanted to keep it open. You didn’t want to miss out on discovering something new and interesting. What do you think now about this issue?

Well, I still don’t want to put words in their mouths. I want to know what they think. But I see that if I ask really open questions, I have no guarantee they will talk about what I want them to talk about. I guess I still like the idea of an open study, but I see that it’s a risky approach. Leaving the questions too open meant I didn’t constrain their responses and there were too many ways they could interpret and answer the questions. And there are too many ways I could interpret their responses.

By this point in the dialogue, Sam has realized that open data (i.e., data not testing a specific prediction) is difficult to interpret. In the next part, Dr. Avery explains why collecting open data was not helping Sam achieve goals for her study that had motivated collecting open data in the first place.

Yes, I totally agree. Even for an experienced researcher, it can be difficult to make sense of this kind of open, messy data. However, if you design a study with a more specific focus, you can create questions for participants that are more targeted because you will be interested in their answers to these specific questions. Let’s reflect back on your data collection. What can you learn from it for the future?

When I think about it now, I realize that I didn’t think about the distinction between all the different constructs at play in my study, and I didn’t choose which one I was focusing on. One construct was the teaching moves that teachers think could be promoting achievement. Another is what teachers deliberately do to promote girls’ mathematics achievement, if anything. Another was the teaching moves that actually do support girls’ mathematics achievement. Another was what teachers were doing that supported girls’ mathematics achievement versus the mathematics achievement of all students. Another was students’ perception of what their teacher was doing to promote girls’ mathematics achievement. I now see that any one of these constructs could have been the focus of a study and that I didn’t really decide which of these was the focus of my course project prior to collecting data.

So, since you told me that the topic of this course project is probably what you’ll eventually want to study for your dissertation, which of these constructs are you most interested in?

I think I’m more interested in the teacher moves that teachers deliberately do to promote girls’ achievement. But I’m still worried about asking teachers directly and getting too specific about what they do because I don’t want to bias what they will say. And I chose qualitative methods and an exploratory design because I thought it would allow for a more open approach, an approach that helps me see what’s going on and that doesn’t bias or predetermine the results.

Well, it seems to me you are conflating three issues. One issue is how to conduct an unbiased study. Another issue is how specific to make your study. And the third issue is whether or not to choose an exploratory or qualitative study design. Those three issues are not the same. For example, designing a study that’s more open or more exploratory is not how researchers make studies fair and unbiased. In fact, it would be quite easy to create an open study that is biased. For example, you could ask very open questions and then interpret the responses in a way that unintentionally, and even unknowingly, aligns with what you were hoping the findings would say. Actually, you could argue that by adding more specificity and narrowing your focus, you’re creating constraints that prevent bias. The same goes for an exploratory or qualitative study; they can be biased or unbiased. So, let’s talk about what is meant by getting more specific. Within your new focus on what teachers deliberately do, there are many things that would be interesting to look at, such as teacher moves that address math anxiety, moves that allow girls to answer questions more frequently, moves that are specifically fitted to student thinking about specific mathematical content, and so on. What are one or two things that are most interesting to you? One way to answer this question is by thinking back to where your interest in this topic began.

In the preceding part of the dialogue, Dr. Avery explained how the goals Sam had for their study were not being met with open data. In the next part, Sam begins to articulate a prediction, which Sam and Dr. Avery then sharpen.

Actually, I became interested in this topic because of an experience I had in college when I was in a class of mostly girls. During whole class discussions, we were supposed to critically evaluate each other’s mathematical thinking, but we were too polite to do that. Instead, we just praised each other’s work. But it was so different in our small groups. It seemed easier to critique each other’s thinking and to push each other to better solutions in small groups. I began wondering how to get girls to be more critical of each other’s thinking in a whole class discussion in order to push everyone’s thinking.

Okay, this is great information. Why not use this idea to zoom-in on a more manageable and interpretable study? You could look specifically at how teachers support girls in critically evaluating each other’s thinking during whole class discussions. That would be a much more targeted and specific topic. Do you have predictions about what teachers could do in that situation, keeping in mind that you are looking specifically at girls’ mathematical achievement, not students in general?

Well, what I noticed was that small groups provided more social and emotional support for girls, whereas the whole class discussion did not provide that same support. The girls felt more comfortable critiquing each other’s thinking in small groups. So, I guess I predict that when the social and emotional supports that are present in small groups are extended to the whole class discussion, girls would be more willing to evaluate each other’s mathematical thinking critically during whole class discussion . I guess ultimately, I’d like to know how the whole class discussion could be used to enhance, rather than undermine, the social and emotional support that is present in the small groups.

Okay, then where would you start? Would you start with a study of what the teachers say they will do during whole class discussion and then observe if that happens during whole class discussion?

But part of my prediction also involves the small groups. So, I’d also like to include small groups in my study if possible. If I focus on whole groups, I won’t be exploring what I am interested in. My interest is broader than just the whole class discussion.

That makes sense, but there are many different things you could look at as part of your prediction, more than you can do in one study. For instance, if your prediction is that when the social and emotional supports that are present in small groups are extended to whole class discussions, girls would be more willing to evaluate each other’s mathematical thinking critically during whole class discussions , then you could ask the following questions: What are the social and emotional supports that are present in small groups?; In which small groups do they exist?; Is it groups that are made up only of girls?; Does every small group do this, and for groups that do this, when do these supports get created?; What kinds of small group activities that teachers ask them to work on are associated with these supports?; Do the same social and emotional supports that apply to small groups even apply to whole group discussion?

All your questions make me realize that my prediction about extending social and emotional supports to whole class discussions first requires me to have a better understanding of the social and emotional supports that exist in small groups. In fact, I first need to find out whether those supports commonly exist in small groups or is that just my experience working in small groups. So, I think I will first have to figure out what small groups do to support each other and then, in a later study, I could ask a teacher to implement those supports during whole class discussions and find out how you can do that. Yeah, now I’m seeing that.

The previous part of the dialogue illustrates how continuing to ask questions about one’s initial prediction is a good way to make it more and more precise (and researchable). In the next part, we see how developing a precise prediction has the added benefit of setting the researcher up for future studies.

Yes, I agree that for your first study, you should probably look at small groups. In other words, you should focus on only a part of your prediction for now, namely the part that says there are social and emotional supports in small groups that support girls in critiquing each other’s thinking . That begins to sharpen the focus of your prediction, but you’ll want to continue to refine it. For example, right now, the question that this prediction leads to is a question with a yes or no answer, but what you’ve said so far suggests to me that you are looking for more than that.

Yes, I want to know more than just whether there are supports. I’d like to know what kinds. That’s why I wanted to do a qualitative study.

Okay, this aligns more with my thinking about research as being prediction driven. It’s about collecting data that would help you revise your existing predictions into better ones. What I mean is that you would focus on collecting data that would allow you to refine your prediction, make it more nuanced, and go beyond what is already known. Does that make sense, and if so, what would that look like for your prediction?

Oh yes, I like that. I guess that would mean that, based on the data I collect for this next study, I could develop a more refined prediction that, for example, more specifically identifies and differentiates between different kinds of social and emotional supports that are present in small groups, or maybe that identifies the kinds of small groups that they occur in, or that predicts when and how frequently or infrequently they occur, or about the features of the small group tasks in which they occur, etc. I now realize that, although I chose qualitative research to make my study be more open, really the reason qualitative research fits my purposes is because it will allow me to explore fine-grained aspects of social and emotional supports that may exist for girls in small groups.

Yes, exactly! And then, based on the data you collect, you can include in your revised prediction those new fine-grained aspects. Furthermore, you will have a story to tell about your study in your written report, namely the story about your evolving prediction. In other words, your written report can largely tell how you filled out and refined your prediction as you learned more from carrying out the study. And even though you might not use them right away, you are also going to be able to develop new predictions that you would not have even thought of about social and emotional supports in small groups and your aim of extending them to whole-class discussions, had you not done this study. That will set you up to follow up on those new predictions in future studies. For example, you might have more refined ideas after you collect the data about the goals for critiquing student thinking in small groups versus the goals for critiquing student thinking during whole class discussion. You might even begin to think that some of the social and emotional supports you observe are not even replicable or even applicable to or appropriate for whole-class discussions, because the supports play different roles in different contexts. So, to summarize what I’m saying, what you look at in this study, even though it will be very focused, sets you up for a research program that will allow you to more fully investigate your broader interest in this topic, where each new study builds on your prior body of work. That’s why it is so important to be explicit about the best place to start this research, so that you can build on it.

I see what you are saying. We started this conversation talking about my course project data. What I think I should have done was figure out explicitly what I needed to learn with that study with the intention of then taking what I learned and using it as the basis for the next study. I didn’t do that, and so I didn’t collect data that pushed forward my thinking in ways that would guide my next study. It would be as if I was starting over with my next study.

Sam and Dr. Avery have just explored how specifying a prediction reveals additional complexities that could become fodder for developing a systematic research program. Next, we watch Sam beginning to recognize the level of specificity required for a prediction to be testable.

One thing that would have really helped would have been if you had had a specific prediction going into your data collection for your course project.

Well, I didn’t really have much of an explicit prediction in mind when I designed my methods.

Think back, you must have had some kind of prediction, even if it was implicit.

Well, yes, I guess I was predicting that teachers would enact moves that supported girls’ mathematical achievement. And I observed classrooms to identify those teacher moves, I interviewed teachers to ask them about the moves I observed, and I interviewed students to see if they mentioned those moves as promoting their mathematical achievement. The goal of my course project was to identify teacher moves that support girls’ mathematical achievement. And my specific research question was: What teacher moves support girls’ mathematical achievement?

So, really you were asking the teacher and students to show and tell you what those moves are and the effects of those moves, as a result putting the onus on your participants to provide the answers to your research question for you. I have an idea, let’s try a thought experiment. You come up with data collection methods for testing the prediction that there are social and emotional supports in small groups that support girls in critiquing each other’s thinking that still puts the onus on the participants. And then I’ll see if I can think of data collection methods that would not put the onus on the participants.

Hmm, well. .. I guess I could simply interview girls who participated in small groups and ask them “are there social and emotional supports that you use in small groups that support your group in critiquing each other’s thinking and if so, what are they?” In that case, I would be putting the onus on them to be aware of the social dynamics of small groups and to have thought about these constructs as much as I have. Okay now can you continue the thought experiment? What might the data collection methods look like if I didn’t put the onus on the participants?

First, I would pick a setting in which it was only girls at this point to reduce the number of variables. Then, personally I would want to observe a lot of groups of girls interacting in groups around tasks. I would be looking for instances when the conversation about students’ ideas was shut down and instances when the conversation about students’ ideas involved critiquing of ideas and building on each other’s thinking. I would also look at what happened just before and during those instances, such as: did the student continue to talk after their thinking was critiqued, did other students do anything to encourage the student to build on their own thinking (i.e., constructive criticism) or how did they support or shut down continued participation. In fact, now that I think about it, “critiquing each other’s thinking” can be defined in a number of different ways. I could mean just commenting on someone’s thinking, judging correctness and incorrectness, constructive criticism that moves the thinking forward, etc. If you put the onus on the participants to answer your research question, you are stuck with their definition, and they won’t have thought about this very much, if at all.

I think that what you are also saying is that my definitions would affect my data collection. If I think that critiquing each other’s thinking means that the group moves their thinking forward toward more valid and complete mathematical solutions, then I’m going to focus on different moves than if I define it another way, such as just making a comment on each other’s thinking and making each other feel comfortable enough to keep participating. In fact, am I going to look at individual instances of critiquing or look at entire sequences in which the critiquing leads to a goal? This seems like a unit of analysis question, and I would need to develop a more nuanced prediction that would make explicit what that unit of analysis is.

I agree, your definition of “critiquing each other’s thinking” could entirely change what you are predicting. One prediction could be based on defining critiquing as a one-shot event in which someone makes one comment on another person’s thinking. In this case the prediction would be that there are social and emotional supports in small groups that support girls in making an evaluative comment on another student’s thinking. Another prediction could be based on defining critiquing as a back-and-forth process in which the thinking gets built on and refined. In that case, the prediction would be something like that there are social and emotional supports in small groups that support girls in critiquing each other’s thinking in ways that do not shut down the conversation but that lead to sustained conversations that move each other toward more valid and complete solutions.

Well, I think I am more interested in the second prediction because it is more compatible with my long-term interests, which are that I’m interested in extending small group supports to whole class discussions. The second prediction is more appropriate for eventually looking at girls in whole class discussion. During whole class discussion, the teacher tries to get a sustained conversation going that moves the students’ thinking forward. So, if I learn about small group supports that lead to sustained conversations that move each other toward more valid and complete solutions , those supports might transfer to whole class discussions.

In the previous part of the dialogue, Dr. Avery and Sam showed how narrowing down a prediction to one that is testable requires making numerous important decisions, including how to define the constructs referred to in the prediction. In the final part of the dialogue, Dr. Avery and Sam begin to outline the reading Sam will have to do to develop a rationale for the specific prediction.

Do you see how your prediction and definitions are getting more and more specific? You now need to read extensively to further refine your prediction.

Well, I should probably read about micro dynamics of small group interactions, anything about interactions in small groups, and what is already known about small group interactions that support sustained conversations that move students’ thinking toward more valid and complete solutions. I guess I could also look at research on whole-class discussion methods that support sustained conversations that move the class to more mathematically valid and complete solutions, because it might give me ideas for what to look for in the small groups. I might also need to focus on research about how learners develop understandings about a particular subject matter so that I know what “more valid and complete solutions” look like. I also need to read about social and emotional supports but focus on how they support students cognitively, rather than in other ways.

Sounds good, let’s get together after you have processed some of this literature and we can talk about refining your prediction based on what you read and also the methods that will best suit testing that prediction.

Great! Thanks for meeting with me. I feel like I have a much better set of tools that push my own thinking forward and allow me to target something specific that will lead to more interpretable data.

Part V. Is It Always Possible to Formulate Hypotheses?

In Chap. 1 , we noted you are likely to read that research does not require formulating hypotheses. Some sources describe doing research without making predictions and developing rationales for these predictions. Some researchers say you cannot always make predictions—you do not know enough about the situation. In fact, some argue for the value of not making predictions (e.g., Glaser & Holton, 2004 ; Merton, 1968 ; Nemirovsky, 2011 ). These are important points of view, so we will devote this section to discussing them.

Can You Always Predict What You Will Find?

One reason some researchers say you do not need to make predictions is that it can be difficult to imagine what you will find. This argument comes up most often for descriptive studies. Suppose you want to describe the nature of a situation you do not know much about. Can you still make a prediction about what you will find? We believe that, although you do not know exactly what you will find, you probably have a hunch or, at a minimum, a very fuzzy idea. It would be unusual to ask a question about a situation you want to know about without at least a fuzzy inkling of what you might find. The original question just would not occur to you. We acknowledge you might have only a vague idea of what you will find and you might not have much confidence in your prediction. However, we expect if you monitor your own thinking you will discover you have developed a suspicion along the way, regardless how vague the suspicion might be. Through the cyclic process we discussed above, that suspicion or hunch gradually evolves and turns into a prediction.

The Benefits of Making Predictions Even When They Are Wrong: An Example from the 1970s

One of us was a graduate student at the University of Wisconsin in the late 1970s, assigned as a research assistant to a project that was investigating young children’s thinking about simple arithmetic. A new curriculum was being written, and the developers wanted to know how to introduce the earliest concepts and skills to kindergarten and first-grade children. The directors of the project did not know what to expect because, at the time, there was little research on five- and six-year-olds’ pre-instruction strategies for adding and subtracting.

After consulting what literature was available, talking with teachers, analyzing the nature of different types of addition and subtraction problems, and debating with each other, the research team formulated some hypotheses about children’s performance. Following the usual assumptions at the time and recognizing the new curriculum would introduce the concepts, the researchers predicted that, before instruction, most children would not be able to solve the problems. Based on the rationale that some young children did not yet recognize the simple form for written problems (e.g., 5 + 3 = ___), the researchers predicted that the best chance for success would be to read problems as stories (e.g., Jesse had 5 apples and then found 3 more. How many does she have now?). They reasoned that, even though children would have difficulty on all the problems, some story problems would be easier because the semantic structure is easier to follow. For example, they predicted the above story about adding 3 apples to 5 would be easier than a problem like, “Jesse had some apples in the refrigerator. She put in 2 more and now has 6. How many were in the refrigerator at the beginning?” Based on the rationale that children would need to count to solve the problems and that it can be difficult to keep track of the numbers, they predicted children would be more successful if they were given counters. Finally, accepting the common reasoning that larger numbers are more difficult than smaller numbers, they predicted children would be more successful if all the numbers in a problem were below 10.

Although these predictions were not very precise and the rationales were not strongly convincing, these hypotheses prompted the researchers to design the study to test their predictions. This meant they would collect data by presenting a variety of problems under a variety of conditions. Because the goal was to describe children’s thinking, problems were presented to students in individual interviews. Problems with different semantic structures were included, counters were available for some problems but not others, and some problems had sums to 9 whereas others had sums to 20 or more.

The punchline of this story is that gathering data under these conditions, prompted by the predictions, made all the difference in what the researchers learned. Contrary to predictions, children could solve addition and subtraction problems before instruction. Counters were important because almost all the solution strategies were based on counting which meant that memory was an issue because many strategies require counting in two ways simultaneously. For example, subtracting 4 from 7 was usually solved by counting down from 7 while counting up from 1 to 4 to keep track of counting down. Because children acted out the stories with their counters, the semantic structure of the story was also important. Stories that were easier to read and write were also easier to solve.

To make a very long story very short, other researchers were, at about the same time, reporting similar results about children’s pre-instruction arithmetic capabilities. A clear pattern emerged regarding the relative difficulty of different problem types (semantic structures) and the strategies children used to solve each type. As the data were replicated, the researchers recognized that kindergarten and first-grade teachers could make good use of this information when they introduced simple arithmetic. This is how Cognitively Guided Instruction (CGI) was born (Carpenter et al., 1989 ; Fennema et al., 1996 ).

To reiterate, the point of this example is that the study conducted to describe children’s thinking would have looked quite different if the researchers had made no predictions. They would have had no reason to choose the particular problems and present them under different conditions. The fact that some of the predictions were completely wrong is not the point. The predictions created the conditions under which the predictions were tested which, in turn, created learning opportunities for the researchers that would not have existed without the predictions. The lesson is that even research that aims to simply describe a phenomenon can benefit from hypotheses. As signaled in Chap. 1 , this also serves as another example of “failing productively.”

Suggestions for What to Do When You Do Not Have Predictions

There likely are exceptions to our claim about being able to make a prediction about what you will find. For example, there could be rare cases where researchers truly have no idea what they will find and can come up with no predictions and even no hunches. And, no research has been reported on related phenomena that would offer some guidance. If you find yourself in this position, we suggest one of three approaches: revise your question, conduct a pilot study, or choose another question.

Because there are many advantages to making predictions explicit and then writing out the reasons for these predictions, one approach is to adjust your question just enough to allow you to make a prediction. Perhaps you can build on descriptions that other researchers have provided for related situations and consider how you can extend this work. Building on previous descriptions will enable you to make predictions about the situation you want to describe.

A second approach is to conduct a small pilot study or, better, a series of small pilot studies to develop some preliminary ideas of what you might find. If you can identify a small sample of participants who are similar to those in your study, you can try out at least some of your research plans to help make and refine your predictions. As we detail later, you can also use pilot studies to check whether key aspects of your methods (e.g., tasks, interview questions, data collection methods) work as you expect.

A third approach is to return to your list of interests and choose one that has been studied previously. Sometimes this is the wisest choice. It is very difficult for beginning researchers to conduct research in brand-new areas where no hunches or predictions are possible. In addition, the contributions of this research can be limited. Recall the earlier story about one of us “failing productively” by completing a dissertation in a somewhat new area. If, after an exhaustive search, you find that no one has investigated the phenomenon in which you are interested or even related phenomena, it can be best to move in a different direction. You will read recommendations in other sources to find a “gap” in the research and develop a study to “fill the gap.” This can be helpful advice if the gap is very small. However, if the gap is large, too large to predict what you might find, the study will present severe challenges. It will be more productive to extend work that has already been done than to launch into an entirely new area.

Should You Always Try to Predict What You Will Find?

In short, our answer to the question in the heading is “yes.” But this calls for further explanation.

Suppose you want to observe a second-grade classroom in order to investigate how students talk about adding and subtracting whole numbers. You might think, “I don’t want to bias my thinking; I want to be completely open to what I see in the classroom.” Sam shared a similar point of view at the beginning of the dialogue: “I wanted to leave it as open as possible; I didn’t want to influence what they were going to say.” Some researchers say that beginning your research study by making predictions is inappropriate precisely because it will bias your observations and results. The argument is that by bringing a set of preconceptions, you will confirm what you expected to find and be blind to other observations and outcomes. The following quote illustrates this view: “The first step in gaining theoretical sensitivity is to enter the research setting with as few predetermined ideas as possible—especially logically deducted, a priori hypotheses. In this posture, the analyst is able to remain sensitive to the data by being able to record events and detect happenings without first having them filtered through and squared with pre-existing hypotheses and biases” (Glaser, 1978, pp. 2–3).

We take a different point of view. In fact, we believe there are several compelling reasons for making your predictions explicit.

Making Your Predictions Explicit Increases Your Chances of Productive Observations

Because your predictions are an extension of what is already known, they prepare you to identify more nuanced relationships that can advance our understanding of a phenomenon. For example, rather than simply noticing, in a general sense, that students talking about addition and subtraction leads them to better understandings, you might, based on your prediction, make the specific observation that talking about addition and subtraction in a particular way helps students to think more deeply about a particular concept related to addition and subtraction. Going into a study without predictions can bring less sensitivity rather than more to the study of a phenomenon. Drawing on knowledge about related phenomena by reading the literature and conducting pilot studies allows you to be much more sensitive and your observations to be more productive.

Making Your Predictions Explicit Allows You to Guard Against Biases

Some genres and methods of educational research are, in fact, rooted in philosophical traditions (e.g., Husserl, 1929/ 1973 ) that explicitly call for researchers to temporarily “bracket” or set aside existing theory as well as their prior knowledge and experience to better enter into the experience of the participants in the research. However, this does not mean ignoring one’s own knowledge and experience or turning a blind eye to what has been learned by others. Much more than the simplistic image of emptying one’s mind of preconceptions and implicit biases (arguably an impossible feat to begin with), the goal is to be as reflective as possible about one’s prior knowledge and conceptions and as transparent as possible about how they may guide observations and shape interpretations (Levitt et al., 2018 ).

We believe it is better to be honest about the predictions you are almost sure to have because then you can deliberately plan to minimize the chances they will influence what you find and how you interpret your results. For starters, it is important to recognize that acknowledging you have some guesses about what you will find does not make them more influential. Because you are likely to have them anyway, we recommend being explicit about what they are. It is easier to deal with biases that are explicit than those that lurk in the background and are not acknowledged.

What do we mean by “deal with biases”? Some journals require you to include a statement about your “positionality” with respect to the participants in your study and the observations you are making to gather data. Formulating clear hypotheses is, in our view, a direct response to this request. The reasons for your predictions are your explicit statements about your positionality. Often there are methodological strategies you can use to protect the study from undue influences of bias. In other words, making your vague predictions explicit can help you design your study so you minimize the bias of your findings.

Making Your Predictions Explicit Can Help You See What You Did Not Predict

Making your predictions explicit does not need to blind you to what is different than expected. It does not need to force you to see only what you want to see. Instead, it can actually increase your sensitivity to noticing features of the situation that are surprising, features you did not predict. Results can stand out when you did not expect to see them.

In contrast, not bringing your biases to consciousness might subtly shift your attention away from these unexpected results in ways that you are not aware of. This path can lead to claiming no biases and no unexpected findings without being conscious of them. You cannot observe everything, and some things inevitably will be overlooked. If you have predicted what you will see, you can design your study so that the unexpected results become more salient rather than less.

Returning to the example of observing a second-grade classroom, we note that the field already knows a great deal about how students talk about addition and subtraction. Being cognizant of what others have observed allows you to enter the classroom with some clear predictions about what will happen. The rationales for these predictions are based on all the related knowledge you have before stepping into the classroom, and the predictions and rationales help you to better deal with what you see. This is partly because you are likely to be surprised by the things you did not anticipate. There is almost always something that will surprise you because your predictions will almost always be incomplete or too general. This sensitivity to the unanticipated—the sense of surprise that sparks your curiosity—is an indication of your openness to the phenomenon you are studying.

Making Your Predictions Explicit Allows You to Plan in Advance

Recall from Chap. 1 the descriptor of scientific inquiry: “Experience carefully planned in advance.” If you make no predictions about what might happen, it is very difficult, if not impossible, to plan your study in advance. Again, you cannot observe everything, so you must make decisions about what you will observe. What kind of data will you plan to collect? Why would you collect these data instead of others? If you have no idea what to expect, on what basis will you make these consequential decisions? Even if your predictions are vague and your rationales for the predictions are a bit shaky, at least they provide a direction for your plan. They allow you to explain why you are planning this study and collecting these data. They allow you to “carefully plan in advance.”

Making Your Predictions Explicit Allows You to Put Your Rationales in Harm’s Way

Rationales are developed to justify the predictions. Rationales represent your best reasoning about the research problem you are studying. How can you tell whether your reasoning is sound? You can try it out with colleagues. However, the best way to test it is to put it in “harm’s way” (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003 p. 10). And the best approach to putting your reasoning in harm’s way is to test the predictions it generates. Regardless if you are conducting a qualitative or quantitative study, rationales can be improved only if they generate testable predictions. This is possible only if predictions are explicit and precise. As we described earlier, rationales are evaluated for their soundness and refined in light of the specific differences between predictions and empirical observations.

Making Your Predictions Explicit Forces You to Organize and Extend Your (and the Field’s) Thinking

By writing out your predictions (even hunches or fuzzy guesses) and by reflecting on why you have these predictions and making these reasons explicit for yourself, you are advancing your thinking about the questions you really want to answer. This means you are making progress toward formulating your research questions and your final hypotheses. Making more progress in your own thinking before you conduct your study increases the chances your study will be of higher quality and will be exactly the study you intended. Making predictions, developing rationales, and imagining tests are tools you can use to push your thinking forward before you even collect data.

Suppose you wonder how preservice teachers in your university’s teacher preparation program will solve particular kinds of math problems. You are interested in this question because you have noticed several PSTs solve them in unexpected ways. As you ask the question you want to answer, you make predictions about what you expect to see. When you reflect on why you made these predictions, you realize that some PSTs might use particular solution strategies because they were taught to use some of them in an earlier course, and they might believe you expect them to solve the problems in these ways. By being explicit about why you are making particular predictions, you realize that you might be answering a different question than you intend (“How much do PSTs remember from previous courses?” or even “To what extent do PSTs believe different instructors have similar expectations?”). Now you can either change your question or change the design of your study (i.e., the sample of students you will use) or both. You are advancing your thinking by being explicit about your predictions and why you are making them.

The Costs of Not Making Predictions

Avoiding making predictions, for whatever reason, comes with significant costs. It prevents you from learning very much about your research topic. It would require not reading related research, not talking with your colleagues, and not conducting pilot studies because, if you do, you are likely to find a prediction creeping into your thinking. Not doing these things would forego the benefits of advancing your thinking before you collect data. It would amount to conducting the study with as little forethought as possible.

Part VI. How Do You Formulate Important Hypotheses?

We provided a partial answer in Chap. 1 to the question of a hypothesis’ importance when we encouraged considering the ultimate goal to which a study’s findings might contribute. You might want to reread Part III of Chap. 1 where we offered our opinions about the purposes of doing research. We also recommend reading the March 2019 editorial in the Journal for Research in Mathematics Education (Cai et al., 2019b ) in which we address what constitutes important educational research.

As we argued in Chap. 1 and in the March 2019 editorial, a worthy ultimate goal for educational research is to improve the learning opportunities for all students. However, arguments can be made for other ultimate goals as well. To gauge the importance of your hypotheses, think about how clearly you can connect them to a goal the educational community considers important. In addition, given the descriptors of scientific inquiry proposed in Chap. 1 , think about how testing your hypotheses will help you (and the community) understand what you are studying. Will you have a better explanation for the phenomenon after your study than before?

Although we address the question of importance again, and in more detail, in Chap. 5 , it is useful to know here that you can determine the significance or importance of your hypotheses when you formulate them. The importance need not depend on the data you collect or the results you report. The importance can come from the fact that, based on the results of your study, you will be able to offer revised hypotheses that help the field better understand an important issue. In large part, it is these revised hypotheses rather than the data that determine a study’s importance.

A critical caveat to this discussion is that few hypotheses are self-evidently important. They are important only if you make the case for their importance. Even if you follow closely the guidelines we suggest for formulating an important hypothesis, you must develop an argument that convinces others. This argument will be presented in the research paper you write.

The picture has a few hypotheses that are self-evidently important. They are important only if you make the case for their importance; written.

Consider Martha’s hypothesis presented earlier. When we left Martha, she predicted that “Participating teachers will show changes in their teaching with a greater emphasis on conceptual understanding with larger changes on linear function topics directly addressed in the LOs than on other topics.” For researchers and educators not intimately familiar with this area of research, it is not apparent why someone should spend a year or more conducting a dissertation to test this prediction. Her rationale, summarized earlier, begins to describe why this could be an important hypothesis. But it is by writing a clear argument that explains her rationale to readers that she will convince them of its importance.

How Martha fills in her rationale so she can create a clear written argument for its importance is taken up in Chap. 3 . As we indicated, Martha’s work in this regard led her to make some interesting decisions, in part due to her own assessment of what was important.

Part VII. Beginning to Write the Research Paper for Your Study

It is common to think that researchers conduct a study and then, after the data are collected and analyzed, begin writing the paper about the study. We recommend an alternative, especially for beginning researchers. We believe it is better to write drafts of the paper at the same time you are planning and conducting your study. The paper will gradually evolve as you work through successive phases of the scientific inquiry process. Consequently, we will call this paper your evolving research paper .

The picture has, we believe it is better to write drafts of the paper at the same time you are planning and conducting your study; written.

You will use your evolving research paper to communicate your study, but you can also use writing as a tool for thinking and organizing your thinking while planning and conducting the study. Used as a tool for thinking, you can write drafts of your ideas to check on the clarity of your thinking, and then you can step back and reflect on how to clarify it further. Be sure to avoid jargon and general terms that are not well defined. Ask yourself whether someone not in your field, maybe a sibling, a parent, or a friend, would be able to understand what you mean. You are likely to write multiple drafts with lots of scribbling, crossing out, and revising.

Used as a tool for communicating, writing the best version of what you know before moving to the next phase will help you record your decisions and the reasons for them before you forget important details. This best-version-for-now paper also provides the basis for your thinking about the next phase of your scientific inquiry.

At this point in the process, you will be writing your (research) questions, the answers you predict, and the rationales for your predictions. The predictions you make should be direct answers to your research questions and should flow logically from (or be directly supported by) the rationales you present. In addition, you will have a written statement of the study’s purpose or, said another way, an argument for the importance of the hypotheses you will be testing. It is in the early sections of your paper that you will convince your audience about the importance of your hypotheses.

In our experience, presenting research questions is a more common form of stating the goal of a research study than presenting well-formulated hypotheses. Authors sometimes present a hypothesis, often as a simple prediction of what they might find. The hypothesis is then forgotten and not used to guide the analysis or interpretations of the findings. In other words, authors seldom use hypotheses to do the kind of work we describe. This means that many research articles you read will not treat hypotheses as we suggest. We believe these are missed opportunities to present research in a more compelling and informative way. We intend to provide enough guidance in the remaining chapters for you to feel comfortable organizing your evolving research paper around formulating, testing, and revising hypotheses.

While we were editing one of the leading research journals in mathematics education ( JRME ), we conducted a study of reviewers’ critiques of papers submitted to the journal. Two of the five most common concerns were: (1) the research questions were unclear, and (2) the answers to the questions did not make a substantial contribution to the field. These are likely to be major concerns for the reviewers of all research journals. We hope the knowledge and skills you have acquired working through this chapter will allow you to write the opening to your evolving research paper in a way that addresses these concerns. Much of the chapter should help make your research questions clear, and the prior section on formulating “important hypotheses” will help you convey the contribution of your study.

Exercise 2.3

Look back at your answers to the sets of questions before part II of this chapter.

Think about how you would argue for the importance of your current interest.

Write your interest in the form of (1) a research problem, (2) a research question, and (3) a prediction with the beginnings of a rationale. You will update these as you read the remaining chapters.

Part VIII. The Heart of Scientific Inquiry

In this chapter, we have described the process of formulating hypotheses. This process is at the heart of scientific inquiry. It is where doing research begins. Conducting research always involves formulating, testing, and revising hypotheses. This is true regardless of your research questions and whether you are using qualitative, quantitative, or mixed methods. Without engaging in this process in a deliberate, intense, relentless way, your study will reveal less than it could. By engaging in this process, you are maximizing what you, and others, can learn from conducting your study.

In the next chapter, we build on the ideas we have developed in the first two chapters to describe the purpose and nature of theoretical frameworks . The term theoretical framework, along with closely related terms like conceptual framework, can be somewhat mysterious for beginning researchers and can seem like a requirement for writing a paper rather than an aid for conducting research. We will show how theoretical frameworks grow from formulating hypotheses—from developing rationales for the predicted answers to your research questions. We will propose some practical suggestions for building theoretical frameworks and show how useful they can be. In addition, we will continue Martha’s story from the point at which we paused earlier—developing her theoretical framework.

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Hiebert, J., Cai, J., Hwang, S., Morris, A.K., Hohensee, C. (2023). How Do You Formulate (Important) Hypotheses?. In: Doing Research: A New Researcher’s Guide. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-19078-0_2

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Statistics Made Easy

Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

Published by Zach

How to Write a Hypothesis: A Step-by-Step Guide

Introduction

An overview of the research hypothesis, different types of hypotheses, variables in a hypothesis, how to formulate an effective research hypothesis, designing a study around your hypothesis.

The scientific method can derive and test predictions as hypotheses. Empirical research can then provide support (or lack thereof) for the hypotheses. Even failure to find support for a hypothesis still represents a valuable contribution to scientific knowledge. Let's look more closely at the idea of the hypothesis and the role it plays in research.

As much as the term exists in everyday language, there is a detailed development that informs the word "hypothesis" when applied to research. A good research hypothesis is informed by prior research and guides research design and data analysis , so it is important to understand how a hypothesis is defined and understood by researchers.

What is the simple definition of a hypothesis?

A hypothesis is a testable prediction about an outcome between two or more variables . It functions as a navigational tool in the research process, directing what you aim to predict and how.

What is the hypothesis for in research?

In research, a hypothesis serves as the cornerstone for your empirical study. It not only lays out what you aim to investigate but also provides a structured approach for your data collection and analysis.

Essentially, it bridges the gap between the theoretical and the empirical, guiding your investigation throughout its course.

What is an example of a hypothesis?

If you are studying the relationship between physical exercise and mental health, a suitable hypothesis could be: "Regular physical exercise leads to improved mental well-being among adults."

This statement constitutes a specific and testable hypothesis that directly relates to the variables you are investigating.

What makes a good hypothesis?

A good hypothesis possesses several key characteristics. Firstly, it must be testable, allowing you to analyze data through empirical means, such as observation or experimentation, to assess if there is significant support for the hypothesis. Secondly, a hypothesis should be specific and unambiguous, giving a clear understanding of the expected relationship between variables. Lastly, it should be grounded in existing research or theoretical frameworks , ensuring its relevance and applicability.

Understanding the types of hypotheses can greatly enhance how you construct and work with hypotheses. While all hypotheses serve the essential function of guiding your study, there are varying purposes among the types of hypotheses. In addition, all hypotheses stand in contrast to the null hypothesis, or the assumption that there is no significant relationship between the variables .

Here, we explore various kinds of hypotheses to provide you with the tools needed to craft effective hypotheses for your specific research needs. Bear in mind that many of these hypothesis types may overlap with one another, and the specific type that is typically used will likely depend on the area of research and methodology you are following.

Null hypothesis

The null hypothesis is a statement that there is no effect or relationship between the variables being studied. In statistical terms, it serves as the default assumption that any observed differences are due to random chance.

For example, if you're studying the effect of a drug on blood pressure, the null hypothesis might state that the drug has no effect.

Alternative hypothesis

Contrary to the null hypothesis, the alternative hypothesis suggests that there is a significant relationship or effect between variables.

Using the drug example, the alternative hypothesis would posit that the drug does indeed affect blood pressure. This is what researchers aim to prove.

Simple hypothesis

A simple hypothesis makes a prediction about the relationship between two variables, and only two variables.

For example, "Increased study time results in better exam scores." Here, "study time" and "exam scores" are the only variables involved.

Complex hypothesis

A complex hypothesis, as the name suggests, involves more than two variables. For instance, "Increased study time and access to resources result in better exam scores." Here, "study time," "access to resources," and "exam scores" are all variables.

This hypothesis refers to multiple potential mediating variables. Other hypotheses could also include predictions about variables that moderate the relationship between the independent variable and dependent variable .

Directional hypothesis

A directional hypothesis specifies the direction of the expected relationship between variables. For example, "Eating more fruits and vegetables leads to a decrease in heart disease."

Here, the direction of heart disease is explicitly predicted to decrease, due to effects from eating more fruits and vegetables. All hypotheses typically specify the expected direction of the relationship between the independent and dependent variable, such that researchers can test if this prediction holds in their data analysis .

Statistical hypothesis

A statistical hypothesis is one that is testable through statistical methods, providing a numerical value that can be analyzed. This is commonly seen in quantitative research .

For example, "There is a statistically significant difference in test scores between students who study for one hour and those who study for two."

Empirical hypothesis

An empirical hypothesis is derived from observations and is tested through empirical methods, often through experimentation or survey data . Empirical hypotheses may also be assessed with statistical analyses.

For example, "Regular exercise is correlated with a lower incidence of depression," could be tested through surveys that measure exercise frequency and depression levels.

Causal hypothesis

A causal hypothesis proposes that one variable causes a change in another. This type of hypothesis is often tested through controlled experiments.

For example, "Smoking causes lung cancer," assumes a direct causal relationship.

Associative hypothesis

Unlike causal hypotheses, associative hypotheses suggest a relationship between variables but do not imply causation.

For instance, "People who smoke are more likely to get lung cancer," notes an association but doesn't claim that smoking causes lung cancer directly.

Relational hypothesis

A relational hypothesis explores the relationship between two or more variables but doesn't specify the nature of the relationship.

For example, "There is a relationship between diet and heart health," leaves the nature of the relationship (causal, associative, etc.) open to interpretation.

Logical hypothesis

A logical hypothesis is based on sound reasoning and logical principles. It's often used in theoretical research to explore abstract concepts, rather than being based on empirical data.

For example, "If all men are mortal and Socrates is a man, then Socrates is mortal," employs logical reasoning to make its point.

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In any research hypothesis, variables play a critical role. These are the elements or factors that the researcher manipulates, controls, or measures. Understanding variables is essential for crafting a clear, testable hypothesis and for the stages of research that follow, such as data collection and analysis.

In the realm of hypotheses, there are generally two types of variables to consider: independent and dependent. Independent variables are what you, as the researcher, manipulate or change in your study. It's considered the cause in the relationship you're investigating. For instance, in a study examining the impact of sleep duration on academic performance, the independent variable would be the amount of sleep participants get.

Conversely, the dependent variable is the outcome you measure to gauge the effect of your manipulation. It's the effect in the cause-and-effect relationship. The dependent variable thus refers to the main outcome of interest in your study. In the same sleep study example, the academic performance, perhaps measured by exam scores or GPA, would be the dependent variable.

Beyond these two primary types, you might also encounter control variables. These are variables that could potentially influence the outcome and are therefore kept constant to isolate the relationship between the independent and dependent variables . For example, in the sleep and academic performance study, control variables could include age, diet, or even the subject of study.

By clearly identifying and understanding the roles of these variables in your hypothesis, you set the stage for a methodologically sound research project. It helps you develop focused research questions, design appropriate experiments or observations, and carry out meaningful data analysis . It's a step that lays the groundwork for the success of your entire study.

Crafting a strong, testable hypothesis is crucial for the success of any research project. It sets the stage for everything from your study design to data collection and analysis . Below are some key considerations to keep in mind when formulating your hypothesis:

Be specific : A vague hypothesis can lead to ambiguous results and interpretations . Clearly define your variables and the expected relationship between them.
Ensure testability : A good hypothesis should be testable through empirical means, whether by observation , experimentation, or other forms of data analysis.
Ground in literature : Before creating your hypothesis, consult existing research and theories. This not only helps you identify gaps in current knowledge but also gives you valuable context and credibility for crafting your hypothesis.
Use simple language : While your hypothesis should be conceptually sound, it doesn't have to be complicated. Aim for clarity and simplicity in your wording.
State direction, if applicable : If your hypothesis involves a directional outcome (e.g., "increase" or "decrease"), make sure to specify this. You also need to think about how you will measure whether or not the outcome moved in the direction you predicted.
Keep it focused : One of the common pitfalls in hypothesis formulation is trying to answer too many questions at once. Keep your hypothesis focused on a specific issue or relationship.
Account for control variables : Identify any variables that could potentially impact the outcome and consider how you will control for them in your study.
Be ethical : Make sure your hypothesis and the methods for testing it comply with ethical standards , particularly if your research involves human or animal subjects.

Designing your study involves multiple key phases that help ensure the rigor and validity of your research. Here we discuss these crucial components in more detail.

Literature review

Starting with a comprehensive literature review is essential. This step allows you to understand the existing body of knowledge related to your hypothesis and helps you identify gaps that your research could fill. Your research should aim to contribute some novel understanding to existing literature, and your hypotheses can reflect this. A literature review also provides valuable insights into how similar research projects were executed, thereby helping you fine-tune your own approach.

Research methods

Choosing the right research methods is critical. Whether it's a survey, an experiment, or observational study, the methodology should be the most appropriate for testing your hypothesis. Your choice of methods will also depend on whether your research is quantitative, qualitative, or mixed-methods. Make sure the chosen methods align well with the variables you are studying and the type of data you need.

Preliminary research

Before diving into a full-scale study, it’s often beneficial to conduct preliminary research or a pilot study . This allows you to test your research methods on a smaller scale, refine your tools, and identify any potential issues. For instance, a pilot survey can help you determine if your questions are clear and if the survey effectively captures the data you need. This step can save you both time and resources in the long run.

Data analysis

Finally, planning your data analysis in advance is crucial for a successful study. Decide which statistical or analytical tools are most suited for your data type and research questions . For quantitative research, you might opt for t-tests, ANOVA, or regression analyses. For qualitative research , thematic analysis or grounded theory may be more appropriate. This phase is integral for interpreting your results and drawing meaningful conclusions in relation to your research question.

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University of Oregon Libraries
Research Guides

How to Write a Literature Review

5. Critically Analyze and Evaluate
Literature Reviews: A Recap
Reading Journal Articles
Does it Describe a Literature Review?
1. Identify the Question
2. Review Discipline Styles
Searching Article Databases
Finding Full-Text of an Article
Citation Chaining
When to Stop Searching
4. Manage Your References

Critically analyze and evaluate

Tip: read and annotate pdfs.

6. Synthesize
7. Write a Literature Review

Ask yourself questions like these about each book or article you include:

What is the research question?
What is the primary methodology used?
How was the data gathered?
How is the data presented?
What are the main conclusions?
Are these conclusions reasonable?
What theories are used to support the researcher's conclusions?

Take notes on the articles as you read them and identify any themes or concepts that may apply to your research question.

This sample template (below) may also be useful for critically reading and organizing your articles. Or you can use this online form and email yourself a copy .

Sample Template for Critical Analysis of the Literature

Opening an article in PDF format in Acrobat Reader will allow you to use "sticky notes" and "highlighting" to make notes on the article without printing it out. Make sure to save the edited file so you don't lose your notes!

Some Citation Managers like Mendeley also have highlighting and annotation features.Here's a screen capture of a pdf in Mendeley with highlighting, notes, and various colors:

Screen capture of Mendeley desktop showing note, highlight, and color tools. Tips include adding notes and highlighting, and using different colors for other purposes like quotations

Screen capture from a UO Librarian's Mendeley Desktop app

Learn more about citation management software in the previous step: 4. Manage Your References
<< Previous: 4. Manage Your References
Next: 6. Synthesize >>
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J Korean Med Sci
v.37(16); 2022 Apr 25

A Practical Guide to Writing Quantitative and Qualitative Research Questions and Hypotheses in Scholarly Articles

Edward barroga.

1 Department of General Education, Graduate School of Nursing Science, St. Luke’s International University, Tokyo, Japan.

Glafera Janet Matanguihan

2 Department of Biological Sciences, Messiah University, Mechanicsburg, PA, USA.

The development of research questions and the subsequent hypotheses are prerequisites to defining the main research purpose and specific objectives of a study. Consequently, these objectives determine the study design and research outcome. The development of research questions is a process based on knowledge of current trends, cutting-edge studies, and technological advances in the research field. Excellent research questions are focused and require a comprehensive literature search and in-depth understanding of the problem being investigated. Initially, research questions may be written as descriptive questions which could be developed into inferential questions. These questions must be specific and concise to provide a clear foundation for developing hypotheses. Hypotheses are more formal predictions about the research outcomes. These specify the possible results that may or may not be expected regarding the relationship between groups. Thus, research questions and hypotheses clarify the main purpose and specific objectives of the study, which in turn dictate the design of the study, its direction, and outcome. Studies developed from good research questions and hypotheses will have trustworthy outcomes with wide-ranging social and health implications.

INTRODUCTION

Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses. 1 , 2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results. 3 , 4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the inception of novel studies and the ethical testing of ideas. 5 , 6

It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or qualitative research, particularly when conceptualizing research questions and hypotheses. 4

There is a continuing need to support researchers in the creation of innovative research questions and hypotheses, as well as for journal articles that carefully review these elements. 1 When research questions and hypotheses are not carefully thought of, unethical studies and poor outcomes usually ensue. Carefully formulated research questions and hypotheses define well-founded objectives, which in turn determine the appropriate design, course, and outcome of the study. This article then aims to discuss in detail the various aspects of crafting research questions and hypotheses, with the goal of guiding researchers as they develop their own. Examples from the authors and peer-reviewed scientific articles in the healthcare field are provided to illustrate key points.

DEFINITIONS AND RELATIONSHIP OF RESEARCH QUESTIONS AND HYPOTHESES

A research question is what a study aims to answer after data analysis and interpretation. The answer is written in length in the discussion section of the paper. Thus, the research question gives a preview of the different parts and variables of the study meant to address the problem posed in the research question. 1 An excellent research question clarifies the research writing while facilitating understanding of the research topic, objective, scope, and limitations of the study. 5

On the other hand, a research hypothesis is an educated statement of an expected outcome. This statement is based on background research and current knowledge. 8 , 9 The research hypothesis makes a specific prediction about a new phenomenon 10 or a formal statement on the expected relationship between an independent variable and a dependent variable. 3 , 11 It provides a tentative answer to the research question to be tested or explored. 4

Hypotheses employ reasoning to predict a theory-based outcome. 10 These can also be developed from theories by focusing on components of theories that have not yet been observed. 10 The validity of hypotheses is often based on the testability of the prediction made in a reproducible experiment. 8

Conversely, hypotheses can also be rephrased as research questions. Several hypotheses based on existing theories and knowledge may be needed to answer a research question. Developing ethical research questions and hypotheses creates a research design that has logical relationships among variables. These relationships serve as a solid foundation for the conduct of the study. 4 , 11 Haphazardly constructed research questions can result in poorly formulated hypotheses and improper study designs, leading to unreliable results. Thus, the formulations of relevant research questions and verifiable hypotheses are crucial when beginning research. 12

CHARACTERISTICS OF GOOD RESEARCH QUESTIONS AND HYPOTHESES

Excellent research questions are specific and focused. These integrate collective data and observations to confirm or refute the subsequent hypotheses. Well-constructed hypotheses are based on previous reports and verify the research context. These are realistic, in-depth, sufficiently complex, and reproducible. More importantly, these hypotheses can be addressed and tested. 13

There are several characteristics of well-developed hypotheses. Good hypotheses are 1) empirically testable 7 , 10 , 11 , 13 ; 2) backed by preliminary evidence 9 ; 3) testable by ethical research 7 , 9 ; 4) based on original ideas 9 ; 5) have evidenced-based logical reasoning 10 ; and 6) can be predicted. 11 Good hypotheses can infer ethical and positive implications, indicating the presence of a relationship or effect relevant to the research theme. 7 , 11 These are initially developed from a general theory and branch into specific hypotheses by deductive reasoning. In the absence of a theory to base the hypotheses, inductive reasoning based on specific observations or findings form more general hypotheses. 10

TYPES OF RESEARCH QUESTIONS AND HYPOTHESES

Research questions and hypotheses are developed according to the type of research, which can be broadly classified into quantitative and qualitative research. We provide a summary of the types of research questions and hypotheses under quantitative and qualitative research categories in Table 1 .

Research questions in quantitative research

In quantitative research, research questions inquire about the relationships among variables being investigated and are usually framed at the start of the study. These are precise and typically linked to the subject population, dependent and independent variables, and research design. 1 Research questions may also attempt to describe the behavior of a population in relation to one or more variables, or describe the characteristics of variables to be measured ( descriptive research questions ). 1 , 5 , 14 These questions may also aim to discover differences between groups within the context of an outcome variable ( comparative research questions ), 1 , 5 , 14 or elucidate trends and interactions among variables ( relationship research questions ). 1 , 5 We provide examples of descriptive, comparative, and relationship research questions in quantitative research in Table 2 .

Hypotheses in quantitative research

In quantitative research, hypotheses predict the expected relationships among variables. 15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable ( simple hypothesis ) or 2) between two or more independent and dependent variables ( complex hypothesis ). 4 , 11 Hypotheses may also specify the expected direction to be followed and imply an intellectual commitment to a particular outcome ( directional hypothesis ) 4 . On the other hand, hypotheses may not predict the exact direction and are used in the absence of a theory, or when findings contradict previous studies ( non-directional hypothesis ). 4 In addition, hypotheses can 1) define interdependency between variables ( associative hypothesis ), 4 2) propose an effect on the dependent variable from manipulation of the independent variable ( causal hypothesis ), 4 3) state a negative relationship between two variables ( null hypothesis ), 4 , 11 , 15 4) replace the working hypothesis if rejected ( alternative hypothesis ), 15 explain the relationship of phenomena to possibly generate a theory ( working hypothesis ), 11 5) involve quantifiable variables that can be tested statistically ( statistical hypothesis ), 11 6) or express a relationship whose interlinks can be verified logically ( logical hypothesis ). 11 We provide examples of simple, complex, directional, non-directional, associative, causal, null, alternative, working, statistical, and logical hypotheses in quantitative research, as well as the definition of quantitative hypothesis-testing research in Table 3 .

Research questions in qualitative research

Unlike research questions in quantitative research, research questions in qualitative research are usually continuously reviewed and reformulated. The central question and associated subquestions are stated more than the hypotheses. 15 The central question broadly explores a complex set of factors surrounding the central phenomenon, aiming to present the varied perspectives of participants. 15

There are varied goals for which qualitative research questions are developed. These questions can function in several ways, such as to 1) identify and describe existing conditions ( contextual research question s); 2) describe a phenomenon ( descriptive research questions ); 3) assess the effectiveness of existing methods, protocols, theories, or procedures ( evaluation research questions ); 4) examine a phenomenon or analyze the reasons or relationships between subjects or phenomena ( explanatory research questions ); or 5) focus on unknown aspects of a particular topic ( exploratory research questions ). 5 In addition, some qualitative research questions provide new ideas for the development of theories and actions ( generative research questions ) or advance specific ideologies of a position ( ideological research questions ). 1 Other qualitative research questions may build on a body of existing literature and become working guidelines ( ethnographic research questions ). Research questions may also be broadly stated without specific reference to the existing literature or a typology of questions ( phenomenological research questions ), may be directed towards generating a theory of some process ( grounded theory questions ), or may address a description of the case and the emerging themes ( qualitative case study questions ). 15 We provide examples of contextual, descriptive, evaluation, explanatory, exploratory, generative, ideological, ethnographic, phenomenological, grounded theory, and qualitative case study research questions in qualitative research in Table 4 , and the definition of qualitative hypothesis-generating research in Table 5 .

Qualitative studies usually pose at least one central research question and several subquestions starting with How or What . These research questions use exploratory verbs such as explore or describe . These also focus on one central phenomenon of interest, and may mention the participants and research site. 15

Hypotheses in qualitative research

Hypotheses in qualitative research are stated in the form of a clear statement concerning the problem to be investigated. Unlike in quantitative research where hypotheses are usually developed to be tested, qualitative research can lead to both hypothesis-testing and hypothesis-generating outcomes. 2 When studies require both quantitative and qualitative research questions, this suggests an integrative process between both research methods wherein a single mixed-methods research question can be developed. 1

FRAMEWORKS FOR DEVELOPING RESEARCH QUESTIONS AND HYPOTHESES

Research questions followed by hypotheses should be developed before the start of the study. 1 , 12 , 14 It is crucial to develop feasible research questions on a topic that is interesting to both the researcher and the scientific community. This can be achieved by a meticulous review of previous and current studies to establish a novel topic. Specific areas are subsequently focused on to generate ethical research questions. The relevance of the research questions is evaluated in terms of clarity of the resulting data, specificity of the methodology, objectivity of the outcome, depth of the research, and impact of the study. 1 , 5 These aspects constitute the FINER criteria (i.e., Feasible, Interesting, Novel, Ethical, and Relevant). 1 Clarity and effectiveness are achieved if research questions meet the FINER criteria. In addition to the FINER criteria, Ratan et al. described focus, complexity, novelty, feasibility, and measurability for evaluating the effectiveness of research questions. 14

The PICOT and PEO frameworks are also used when developing research questions. 1 The following elements are addressed in these frameworks, PICOT: P-population/patients/problem, I-intervention or indicator being studied, C-comparison group, O-outcome of interest, and T-timeframe of the study; PEO: P-population being studied, E-exposure to preexisting conditions, and O-outcome of interest. 1 Research questions are also considered good if these meet the “FINERMAPS” framework: Feasible, Interesting, Novel, Ethical, Relevant, Manageable, Appropriate, Potential value/publishable, and Systematic. 14

As we indicated earlier, research questions and hypotheses that are not carefully formulated result in unethical studies or poor outcomes. To illustrate this, we provide some examples of ambiguous research question and hypotheses that result in unclear and weak research objectives in quantitative research ( Table 6 ) 16 and qualitative research ( Table 7 ) 17 , and how to transform these ambiguous research question(s) and hypothesis(es) into clear and good statements.

a These statements were composed for comparison and illustrative purposes only.

b These statements are direct quotes from Higashihara and Horiuchi. 16

a This statement is a direct quote from Shimoda et al. 17

The other statements were composed for comparison and illustrative purposes only.

CONSTRUCTING RESEARCH QUESTIONS AND HYPOTHESES

To construct effective research questions and hypotheses, it is very important to 1) clarify the background and 2) identify the research problem at the outset of the research, within a specific timeframe. 9 Then, 3) review or conduct preliminary research to collect all available knowledge about the possible research questions by studying theories and previous studies. 18 Afterwards, 4) construct research questions to investigate the research problem. Identify variables to be accessed from the research questions 4 and make operational definitions of constructs from the research problem and questions. Thereafter, 5) construct specific deductive or inductive predictions in the form of hypotheses. 4 Finally, 6) state the study aims . This general flow for constructing effective research questions and hypotheses prior to conducting research is shown in Fig. 1 .

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Research questions are used more frequently in qualitative research than objectives or hypotheses. 3 These questions seek to discover, understand, explore or describe experiences by asking “What” or “How.” The questions are open-ended to elicit a description rather than to relate variables or compare groups. The questions are continually reviewed, reformulated, and changed during the qualitative study. 3 Research questions are also used more frequently in survey projects than hypotheses in experiments in quantitative research to compare variables and their relationships.

Hypotheses are constructed based on the variables identified and as an if-then statement, following the template, ‘If a specific action is taken, then a certain outcome is expected.’ At this stage, some ideas regarding expectations from the research to be conducted must be drawn. 18 Then, the variables to be manipulated (independent) and influenced (dependent) are defined. 4 Thereafter, the hypothesis is stated and refined, and reproducible data tailored to the hypothesis are identified, collected, and analyzed. 4 The hypotheses must be testable and specific, 18 and should describe the variables and their relationships, the specific group being studied, and the predicted research outcome. 18 Hypotheses construction involves a testable proposition to be deduced from theory, and independent and dependent variables to be separated and measured separately. 3 Therefore, good hypotheses must be based on good research questions constructed at the start of a study or trial. 12

In summary, research questions are constructed after establishing the background of the study. Hypotheses are then developed based on the research questions. Thus, it is crucial to have excellent research questions to generate superior hypotheses. In turn, these would determine the research objectives and the design of the study, and ultimately, the outcome of the research. 12 Algorithms for building research questions and hypotheses are shown in Fig. 2 for quantitative research and in Fig. 3 for qualitative research.

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EXAMPLES OF RESEARCH QUESTIONS FROM PUBLISHED ARTICLES

EXAMPLE 1. Descriptive research question (quantitative research)
- Presents research variables to be assessed (distinct phenotypes and subphenotypes)
“BACKGROUND: Since COVID-19 was identified, its clinical and biological heterogeneity has been recognized. Identifying COVID-19 phenotypes might help guide basic, clinical, and translational research efforts.
RESEARCH QUESTION: Does the clinical spectrum of patients with COVID-19 contain distinct phenotypes and subphenotypes? ” 19
EXAMPLE 2. Relationship research question (quantitative research)
- Shows interactions between dependent variable (static postural control) and independent variable (peripheral visual field loss)
“Background: Integration of visual, vestibular, and proprioceptive sensations contributes to postural control. People with peripheral visual field loss have serious postural instability. However, the directional specificity of postural stability and sensory reweighting caused by gradual peripheral visual field loss remain unclear.
Research question: What are the effects of peripheral visual field loss on static postural control ?” 20
EXAMPLE 3. Comparative research question (quantitative research)
- Clarifies the difference among groups with an outcome variable (patients enrolled in COMPERA with moderate PH or severe PH in COPD) and another group without the outcome variable (patients with idiopathic pulmonary arterial hypertension (IPAH))
“BACKGROUND: Pulmonary hypertension (PH) in COPD is a poorly investigated clinical condition.
RESEARCH QUESTION: Which factors determine the outcome of PH in COPD?
STUDY DESIGN AND METHODS: We analyzed the characteristics and outcome of patients enrolled in the Comparative, Prospective Registry of Newly Initiated Therapies for Pulmonary Hypertension (COMPERA) with moderate or severe PH in COPD as defined during the 6th PH World Symposium who received medical therapy for PH and compared them with patients with idiopathic pulmonary arterial hypertension (IPAH) .” 21
EXAMPLE 4. Exploratory research question (qualitative research)
- Explores areas that have not been fully investigated (perspectives of families and children who receive care in clinic-based child obesity treatment) to have a deeper understanding of the research problem
“Problem: Interventions for children with obesity lead to only modest improvements in BMI and long-term outcomes, and data are limited on the perspectives of families of children with obesity in clinic-based treatment. This scoping review seeks to answer the question: What is known about the perspectives of families and children who receive care in clinic-based child obesity treatment? This review aims to explore the scope of perspectives reported by families of children with obesity who have received individualized outpatient clinic-based obesity treatment.” 22
EXAMPLE 5. Relationship research question (quantitative research)
- Defines interactions between dependent variable (use of ankle strategies) and independent variable (changes in muscle tone)
“Background: To maintain an upright standing posture against external disturbances, the human body mainly employs two types of postural control strategies: “ankle strategy” and “hip strategy.” While it has been reported that the magnitude of the disturbance alters the use of postural control strategies, it has not been elucidated how the level of muscle tone, one of the crucial parameters of bodily function, determines the use of each strategy. We have previously confirmed using forward dynamics simulations of human musculoskeletal models that an increased muscle tone promotes the use of ankle strategies. The objective of the present study was to experimentally evaluate a hypothesis: an increased muscle tone promotes the use of ankle strategies. Research question: Do changes in the muscle tone affect the use of ankle strategies ?” 23

EXAMPLES OF HYPOTHESES IN PUBLISHED ARTICLES

EXAMPLE 1. Working hypothesis (quantitative research)
- A hypothesis that is initially accepted for further research to produce a feasible theory
“As fever may have benefit in shortening the duration of viral illness, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response when taken during the early stages of COVID-19 illness .” 24
“In conclusion, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response . The difference in perceived safety of these agents in COVID-19 illness could be related to the more potent efficacy to reduce fever with ibuprofen compared to acetaminophen. Compelling data on the benefit of fever warrant further research and review to determine when to treat or withhold ibuprofen for early stage fever for COVID-19 and other related viral illnesses .” 24
EXAMPLE 2. Exploratory hypothesis (qualitative research)
- Explores particular areas deeper to clarify subjective experience and develop a formal hypothesis potentially testable in a future quantitative approach
“We hypothesized that when thinking about a past experience of help-seeking, a self distancing prompt would cause increased help-seeking intentions and more favorable help-seeking outcome expectations .” 25
“Conclusion
Although a priori hypotheses were not supported, further research is warranted as results indicate the potential for using self-distancing approaches to increasing help-seeking among some people with depressive symptomatology.” 25
EXAMPLE 3. Hypothesis-generating research to establish a framework for hypothesis testing (qualitative research)
“We hypothesize that compassionate care is beneficial for patients (better outcomes), healthcare systems and payers (lower costs), and healthcare providers (lower burnout). ” 26
Compassionomics is the branch of knowledge and scientific study of the effects of compassionate healthcare. Our main hypotheses are that compassionate healthcare is beneficial for (1) patients, by improving clinical outcomes, (2) healthcare systems and payers, by supporting financial sustainability, and (3) HCPs, by lowering burnout and promoting resilience and well-being. The purpose of this paper is to establish a scientific framework for testing the hypotheses above . If these hypotheses are confirmed through rigorous research, compassionomics will belong in the science of evidence-based medicine, with major implications for all healthcare domains.” 26
EXAMPLE 4. Statistical hypothesis (quantitative research)
- An assumption is made about the relationship among several population characteristics ( gender differences in sociodemographic and clinical characteristics of adults with ADHD ). Validity is tested by statistical experiment or analysis ( chi-square test, Students t-test, and logistic regression analysis)
“Our research investigated gender differences in sociodemographic and clinical characteristics of adults with ADHD in a Japanese clinical sample. Due to unique Japanese cultural ideals and expectations of women's behavior that are in opposition to ADHD symptoms, we hypothesized that women with ADHD experience more difficulties and present more dysfunctions than men . We tested the following hypotheses: first, women with ADHD have more comorbidities than men with ADHD; second, women with ADHD experience more social hardships than men, such as having less full-time employment and being more likely to be divorced.” 27
“Statistical Analysis
( text omitted ) Between-gender comparisons were made using the chi-squared test for categorical variables and Students t-test for continuous variables…( text omitted ). A logistic regression analysis was performed for employment status, marital status, and comorbidity to evaluate the independent effects of gender on these dependent variables.” 27

EXAMPLES OF HYPOTHESIS AS WRITTEN IN PUBLISHED ARTICLES IN RELATION TO OTHER PARTS

EXAMPLE 1. Background, hypotheses, and aims are provided
“Pregnant women need skilled care during pregnancy and childbirth, but that skilled care is often delayed in some countries …( text omitted ). The focused antenatal care (FANC) model of WHO recommends that nurses provide information or counseling to all pregnant women …( text omitted ). Job aids are visual support materials that provide the right kind of information using graphics and words in a simple and yet effective manner. When nurses are not highly trained or have many work details to attend to, these job aids can serve as a content reminder for the nurses and can be used for educating their patients (Jennings, Yebadokpo, Affo, & Agbogbe, 2010) ( text omitted ). Importantly, additional evidence is needed to confirm how job aids can further improve the quality of ANC counseling by health workers in maternal care …( text omitted )” 28
“ This has led us to hypothesize that the quality of ANC counseling would be better if supported by job aids. Consequently, a better quality of ANC counseling is expected to produce higher levels of awareness concerning the danger signs of pregnancy and a more favorable impression of the caring behavior of nurses .” 28
“This study aimed to examine the differences in the responses of pregnant women to a job aid-supported intervention during ANC visit in terms of 1) their understanding of the danger signs of pregnancy and 2) their impression of the caring behaviors of nurses to pregnant women in rural Tanzania.” 28
EXAMPLE 2. Background, hypotheses, and aims are provided
“We conducted a two-arm randomized controlled trial (RCT) to evaluate and compare changes in salivary cortisol and oxytocin levels of first-time pregnant women between experimental and control groups. The women in the experimental group touched and held an infant for 30 min (experimental intervention protocol), whereas those in the control group watched a DVD movie of an infant (control intervention protocol). The primary outcome was salivary cortisol level and the secondary outcome was salivary oxytocin level.” 29
“ We hypothesize that at 30 min after touching and holding an infant, the salivary cortisol level will significantly decrease and the salivary oxytocin level will increase in the experimental group compared with the control group .” 29
EXAMPLE 3. Background, aim, and hypothesis are provided
“In countries where the maternal mortality ratio remains high, antenatal education to increase Birth Preparedness and Complication Readiness (BPCR) is considered one of the top priorities [1]. BPCR includes birth plans during the antenatal period, such as the birthplace, birth attendant, transportation, health facility for complications, expenses, and birth materials, as well as family coordination to achieve such birth plans. In Tanzania, although increasing, only about half of all pregnant women attend an antenatal clinic more than four times [4]. Moreover, the information provided during antenatal care (ANC) is insufficient. In the resource-poor settings, antenatal group education is a potential approach because of the limited time for individual counseling at antenatal clinics.” 30
“This study aimed to evaluate an antenatal group education program among pregnant women and their families with respect to birth-preparedness and maternal and infant outcomes in rural villages of Tanzania.” 30
“ The study hypothesis was if Tanzanian pregnant women and their families received a family-oriented antenatal group education, they would (1) have a higher level of BPCR, (2) attend antenatal clinic four or more times, (3) give birth in a health facility, (4) have less complications of women at birth, and (5) have less complications and deaths of infants than those who did not receive the education .” 30

Research questions and hypotheses are crucial components to any type of research, whether quantitative or qualitative. These questions should be developed at the very beginning of the study. Excellent research questions lead to superior hypotheses, which, like a compass, set the direction of research, and can often determine the successful conduct of the study. Many research studies have floundered because the development of research questions and subsequent hypotheses was not given the thought and meticulous attention needed. The development of research questions and hypotheses is an iterative process based on extensive knowledge of the literature and insightful grasp of the knowledge gap. Focused, concise, and specific research questions provide a strong foundation for constructing hypotheses which serve as formal predictions about the research outcomes. Research questions and hypotheses are crucial elements of research that should not be overlooked. They should be carefully thought of and constructed when planning research. This avoids unethical studies and poor outcomes by defining well-founded objectives that determine the design, course, and outcome of the study.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

Conceptualization: Barroga E, Matanguihan GJ.
Methodology: Barroga E, Matanguihan GJ.
Writing - original draft: Barroga E, Matanguihan GJ.
Writing - review & editing: Barroga E, Matanguihan GJ.

Data Science Society

Data.Platform

Statistical Hypothesis Testing: How to Calculate Critical Values

Testing statistical hypotheses is one of the most important parts of data analysis. It lets the researcher and analyst conclude the whole community from a small sample. In this case, critical values are useful because they help figure out if the results are worthy of attention.

The goal of this article is to define a critical value calculator, talk about why it’s important in statistical hypothesis testing, and show how to use one.

What is statistical hypothesis testing?

Statistical hypothesis testing is a methodical way to draw conclusions about a whole community from a small group of samples. In this step, observed data is compared to an expected or null hypothesis to see if there is a difference that is caused by an effect, chance, or just a human mistake. Hypotheses are put to the test in economics, social studies, and science in order to come to reasonable conclusions.

What are critical values?

In this case, critical values are limits or borders that are used during hypothesis testing to see how important a test statistic is. In hypothesis testing, the critical value is compared to a test statistic that measures the difference between the data that was noticed and the value that was thought to be true. A critical value calculator is used to evaluate if there is sufficient information in the observed results that would make it possible to invalidate the zero hypothesis.

How to calculate critical values

Step 1: identify the test statistic.

Before you can figure out the key values, you need to choose the right test statistic for your hypothesis test. The “test statistic” is a number that shows that the data are different from the “null value.” This is a list of test statistics. Which one to use depends on the data or hypothesis being tested.

Examples of these statistics are the Z-score, T-statistics, F-statistics, and Chi-squared statistics. Here’s a brief overview of when each test statistic is typically used:

Z-score: If you have data that has a normal distribution, you can find out what the group mean and standard deviation are.

T-statistic: The t-statistic is used to test hypotheses when the sample size is small, or the community standard deviation is unknown.

F-statistic: In ANOVA tests, F-statistics are used to find changes between the variances of different groups or treatments.

The chi-squared measure is used for tests that use categorical data, such as the goodness of fit test or the test for independence in a contingency table.

Once you’ve found the best statistic for a hypothesis test, move on to the next step.

Step 2: Determine the degrees of freedom

Degrees of freedom (df) are one of the important things that are used to figure out critical numbers. Freedom of degrees refers to the number of separate factors that are linked to your dataset. The number of degrees of freedom changes based on the test measure that is used.

For example, to find the critical numbers for a T-statistic, one is usually taken away from n to get an idea of the degrees of freedom. An F-statistic in ANOVA, on the other hand, has two sets of degrees of freedom: one for the numerator (which is the difference between groups) and one for the denominator (which is the difference within groups).

Because of this, you need to figure out the right number of degrees of freedom for your analysis and not use the wrong numbers because they lead to wrong results. If you need to find the right degree of freedom values for your test statistic, look at the appropriate statistical tables or sources.

Step 3: One needs to find the critical value in a critical value table

A critical value table is an important part of any hypothesis test. For each degree of freedom and significant level, the table shows the test statistic values that go with them. This critical number sets a limit on how often the null should be rejected.

One example is a two-tailed Z-test with a significance level of 0.05 (alpha = 0.05). If you know the number of degrees of freedom, you can find the critical value that is equal to alpha/2 (0.025) in the

Also, the T-table shows the important number for alpha/2 and your degrees of freedom for the T-distribution with degrees of freedom.

Step 4: Do you believe that the test statistic is bigger than the critical value?

After that, we will compare this test statistic with the critical number we chose from the table. So, you will reject the null hypothesis if your test result is more extreme than what is needed for a significance level (the tail of the distribution above the critical value). This shows that the data that was seen is very different, which means it probably wasn’t just a matter of chance. On the other hand, you can’t reject the null hypothesis if your test statistic doesn’t fall in the rejection area. In this case, the data that was noticed is not enough to show that the value that was hypothesized might be wrong.

In the field of statistical hypothesis testing, researchers and other analysts need to know what key values are and how to find them. So, critical values are a common way to figure out how important the results of tests are. When researchers check to see if the test statistic is greater than or similar to the critical value, they can tell if their data supports the null hypothesis or not.

Always use the right critical value tables, and keep in mind that degrees of freedom are a big part of making sure that statistical analysis is correct and thorough. Using statistical software can also help cut down on mistakes and make the math part of this process easier.

Hypothesis testing is built on important values that help people come to conclusions, make decisions, make progress in science, and learn more. Critical value calculation is a skill that everyone who works with statistics needs to have.

Developing a Hypothesis

Rajiv S. Jhangiani; I-Chant A. Chiang; Carrie Cuttler; and Dana C. Leighton

Learning Objectives

Distinguish between a theory and a hypothesis.
Discover how theories are used to generate hypotheses and how the results of studies can be used to further inform theories.
Understand the characteristics of a good hypothesis.

Theories and Hypotheses

Before describing how to develop a hypothesis, it is important to distinguish between a theory and a hypothesis. A theory is a coherent explanation or interpretation of one or more phenomena. Although theories can take a variety of forms, one thing they have in common is that they go beyond the phenomena they explain by including variables, structures, processes, functions, or organizing principles that have not been observed directly. Consider, for example, Zajonc’s theory of social facilitation and social inhibition (1965) [1] . He proposed that being watched by others while performing a task creates a general state of physiological arousal, which increases the likelihood of the dominant (most likely) response. So for highly practiced tasks, being watched increases the tendency to make correct responses, but for relatively unpracticed tasks, being watched increases the tendency to make incorrect responses. Notice that this theory—which has come to be called drive theory—provides an explanation of both social facilitation and social inhibition that goes beyond the phenomena themselves by including concepts such as “arousal” and “dominant response,” along with processes such as the effect of arousal on the dominant response.

Outside of science, referring to an idea as a theory often implies that it is untested—perhaps no more than a wild guess. In science, however, the term theory has no such implication. A theory is simply an explanation or interpretation of a set of phenomena. It can be untested, but it can also be extensively tested, well supported, and accepted as an accurate description of the world by the scientific community. The theory of evolution by natural selection, for example, is a theory because it is an explanation of the diversity of life on earth—not because it is untested or unsupported by scientific research. On the contrary, the evidence for this theory is overwhelmingly positive and nearly all scientists accept its basic assumptions as accurate. Similarly, the “germ theory” of disease is a theory because it is an explanation of the origin of various diseases, not because there is any doubt that many diseases are caused by microorganisms that infect the body.

A hypothesis , on the other hand, is a specific prediction about a new phenomenon that should be observed if a particular theory is accurate. It is an explanation that relies on just a few key concepts. Hypotheses are often specific predictions about what will happen in a particular study. They are developed by considering existing evidence and using reasoning to infer what will happen in the specific context of interest. Hypotheses are often but not always derived from theories. So a hypothesis is often a prediction based on a theory but some hypotheses are a-theoretical and only after a set of observations have been made, is a theory developed. This is because theories are broad in nature and they explain larger bodies of data. So if our research question is really original then we may need to collect some data and make some observations before we can develop a broader theory.

Theories and hypotheses always have this if-then relationship. “ If drive theory is correct, then cockroaches should run through a straight runway faster, and a branching runway more slowly, when other cockroaches are present.” Although hypotheses are usually expressed as statements, they can always be rephrased as questions. “Do cockroaches run through a straight runway faster when other cockroaches are present?” Thus deriving hypotheses from theories is an excellent way of generating interesting research questions.

But how do researchers derive hypotheses from theories? One way is to generate a research question using the techniques discussed in this chapter and then ask whether any theory implies an answer to that question. For example, you might wonder whether expressive writing about positive experiences improves health as much as expressive writing about traumatic experiences. Although this question is an interesting one on its own, you might then ask whether the habituation theory—the idea that expressive writing causes people to habituate to negative thoughts and feelings—implies an answer. In this case, it seems clear that if the habituation theory is correct, then expressive writing about positive experiences should not be effective because it would not cause people to habituate to negative thoughts and feelings. A second way to derive hypotheses from theories is to focus on some component of the theory that has not yet been directly observed. For example, a researcher could focus on the process of habituation—perhaps hypothesizing that people should show fewer signs of emotional distress with each new writing session.

Among the very best hypotheses are those that distinguish between competing theories. For example, Norbert Schwarz and his colleagues considered two theories of how people make judgments about themselves, such as how assertive they are (Schwarz et al., 1991) [2] . Both theories held that such judgments are based on relevant examples that people bring to mind. However, one theory was that people base their judgments on the number of examples they bring to mind and the other was that people base their judgments on how easily they bring those examples to mind. To test these theories, the researchers asked people to recall either six times when they were assertive (which is easy for most people) or 12 times (which is difficult for most people). Then they asked them to judge their own assertiveness. Note that the number-of-examples theory implies that people who recalled 12 examples should judge themselves to be more assertive because they recalled more examples, but the ease-of-examples theory implies that participants who recalled six examples should judge themselves as more assertive because recalling the examples was easier. Thus the two theories made opposite predictions so that only one of the predictions could be confirmed. The surprising result was that participants who recalled fewer examples judged themselves to be more assertive—providing particularly convincing evidence in favor of the ease-of-retrieval theory over the number-of-examples theory.

Theory Testing

The primary way that scientific researchers use theories is sometimes called the hypothetico-deductive method (although this term is much more likely to be used by philosophers of science than by scientists themselves). Researchers begin with a set of phenomena and either construct a theory to explain or interpret them or choose an existing theory to work with. They then make a prediction about some new phenomenon that should be observed if the theory is correct. Again, this prediction is called a hypothesis. The researchers then conduct an empirical study to test the hypothesis. Finally, they reevaluate the theory in light of the new results and revise it if necessary. This process is usually conceptualized as a cycle because the researchers can then derive a new hypothesis from the revised theory, conduct a new empirical study to test the hypothesis, and so on. As Figure 2.3 shows, this approach meshes nicely with the model of scientific research in psychology presented earlier in the textbook—creating a more detailed model of “theoretically motivated” or “theory-driven” research.

As an example, let us consider Zajonc’s research on social facilitation and inhibition. He started with a somewhat contradictory pattern of results from the research literature. He then constructed his drive theory, according to which being watched by others while performing a task causes physiological arousal, which increases an organism’s tendency to make the dominant response. This theory predicts social facilitation for well-learned tasks and social inhibition for poorly learned tasks. He now had a theory that organized previous results in a meaningful way—but he still needed to test it. He hypothesized that if his theory was correct, he should observe that the presence of others improves performance in a simple laboratory task but inhibits performance in a difficult version of the very same laboratory task. To test this hypothesis, one of the studies he conducted used cockroaches as subjects (Zajonc, Heingartner, & Herman, 1969) [3] . The cockroaches ran either down a straight runway (an easy task for a cockroach) or through a cross-shaped maze (a difficult task for a cockroach) to escape into a dark chamber when a light was shined on them. They did this either while alone or in the presence of other cockroaches in clear plastic “audience boxes.” Zajonc found that cockroaches in the straight runway reached their goal more quickly in the presence of other cockroaches, but cockroaches in the cross-shaped maze reached their goal more slowly when they were in the presence of other cockroaches. Thus he confirmed his hypothesis and provided support for his drive theory. (Zajonc also showed that drive theory existed in humans [Zajonc & Sales, 1966] [4] in many other studies afterward).

Incorporating Theory into Your Research

When you write your research report or plan your presentation, be aware that there are two basic ways that researchers usually include theory. The first is to raise a research question, answer that question by conducting a new study, and then offer one or more theories (usually more) to explain or interpret the results. This format works well for applied research questions and for research questions that existing theories do not address. The second way is to describe one or more existing theories, derive a hypothesis from one of those theories, test the hypothesis in a new study, and finally reevaluate the theory. This format works well when there is an existing theory that addresses the research question—especially if the resulting hypothesis is surprising or conflicts with a hypothesis derived from a different theory.

To use theories in your research will not only give you guidance in coming up with experiment ideas and possible projects, but it lends legitimacy to your work. Psychologists have been interested in a variety of human behaviors and have developed many theories along the way. Using established theories will help you break new ground as a researcher, not limit you from developing your own ideas.

Characteristics of a Good Hypothesis

There are three general characteristics of a good hypothesis. First, a good hypothesis must be testable and falsifiable . We must be able to test the hypothesis using the methods of science and if you’ll recall Popper’s falsifiability criterion, it must be possible to gather evidence that will disconfirm the hypothesis if it is indeed false. Second, a good hypothesis must be logical. As described above, hypotheses are more than just a random guess. Hypotheses should be informed by previous theories or observations and logical reasoning. Typically, we begin with a broad and general theory and use deductive reasoning to generate a more specific hypothesis to test based on that theory. Occasionally, however, when there is no theory to inform our hypothesis, we use inductive reasoning which involves using specific observations or research findings to form a more general hypothesis. Finally, the hypothesis should be positive. That is, the hypothesis should make a positive statement about the existence of a relationship or effect, rather than a statement that a relationship or effect does not exist. As scientists, we don’t set out to show that relationships do not exist or that effects do not occur so our hypotheses should not be worded in a way to suggest that an effect or relationship does not exist. The nature of science is to assume that something does not exist and then seek to find evidence to prove this wrong, to show that it really does exist. That may seem backward to you but that is the nature of the scientific method. The underlying reason for this is beyond the scope of this chapter but it has to do with statistical theory.

Zajonc, R. B. (1965). Social facilitation. Science, 149 , 269–274 ↵
Schwarz, N., Bless, H., Strack, F., Klumpp, G., Rittenauer-Schatka, H., & Simons, A. (1991). Ease of retrieval as information: Another look at the availability heuristic. Journal of Personality and Social Psychology, 61 , 195–202. ↵
Zajonc, R. B., Heingartner, A., & Herman, E. M. (1969). Social enhancement and impairment of performance in the cockroach. Journal of Personality and Social Psychology, 13 , 83–92. ↵
Zajonc, R.B. & Sales, S.M. (1966). Social facilitation of dominant and subordinate responses. Journal of Experimental Social Psychology, 2 , 160-168. ↵

A coherent explanation or interpretation of one or more phenomena.

A specific prediction about a new phenomenon that should be observed if a particular theory is accurate.

A cyclical process of theory development, starting with an observed phenomenon, then developing or using a theory to make a specific prediction of what should happen if that theory is correct, testing that prediction, refining the theory in light of the findings, and using that refined theory to develop new hypotheses, and so on.

The ability to test the hypothesis using the methods of science and the possibility to gather evidence that will disconfirm the hypothesis if it is indeed false.

Developing a Hypothesis Copyright © 2022 by Rajiv S. Jhangiani; I-Chant A. Chiang; Carrie Cuttler; and Dana C. Leighton is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Write a Critical Review of a Scientific Journal Article

1. identify how and why the research was carried out, 2. establish the research context, 3. evaluate the research, 4. establish the significance of the research.

Writing Your Critique

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Read the article(s) carefully and use the questions below to help you identify how and why the research was carried out. Look at the following sections:

Introduction

What was the objective of the study?
What methods were used to accomplish this purpose (e.g., systematic recording of observations, analysis and evaluation of published research, assessment of theory, etc.)?
What techniques were used and how was each technique performed?
What kind of data can be obtained using each technique?
How are such data interpreted?
What kind of information is produced by using the technique?
What objective evidence was obtained from the authors’ eﬀorts (observations, measurements, etc.)?
What were the results of the study?
How was each technique used to obtain each result?
What statistical tests were used to evaluate the signiﬁcance of the conclusions based on numeric or graphic data?
How did each result contribute to answering the question or testing the hypothesis raised in the introduction?
How were the results interpreted? How were they related to the original problem (authors’ view of evidence rather than objective ﬁndings)?
Were the authors able to answer the question (test the hypothesis) raised?
Did the research provide new factual information, a new understanding of a phenomenon in the ﬁeld, or a new research technique?
How was the signiﬁcance of the work described?
Do the authors relate the findings of the study to literature in the field?
Did the reported observations or interpretations support or refute observations or interpretations made by other researchers?

These questions were adapted from the following sources: Kuyper, B.J. (1991). Bringing up scientists in the art of critiquing research. Bioscience 41(4), 248-250. Wood, J.M. (2003). Research Lab Guide. MICR*3260 Microbial Adaptation and Development Web Site . Retrieved July 31, 2006.

Once you are familiar with the article, you can establish the research context by asking the following questions:

Who conducted the research? What were/are their interests?
When and where was the research conducted?
Why did the authors do this research?
Was this research pertinent only within the authors’ geographic locale, or did it have broader (even global) relevance?
Were many other laboratories pursuing related research when the reported work was done? If so, why?
For experimental research, what funding sources met the costs of the research?
On what prior observations was the research based? What was and was not known at the time?
How important was the research question posed by the researchers?

Remember that simply disagreeing with the material is not considered to be a critical assessment of the material. For example, stating that the sample size is insufficient is not a critical assessment. Describing why the sample size is insufficient for the claims being made in the study would be a critical assessment.

Use the questions below to help you evaluate the quality of the authors’ research:

Does the title precisely state the subject of the paper?
Read the statement of purpose in the abstract. Does it match the one in the introduction?

Acknowledgments

Could the source of the research funding have influenced the research topic or conclusions?
Check the sequence of statements in the introduction. Does all the information lead coherently to the purpose of the study?
Review all methods in relation to the objective(s) of the study. Are the methods valid for studying the problem?
Check the methods for essential information. Could the study be duplicated from the methods and information given?
Check the methods for ﬂaws. Is the sample selection adequate? Is the experimental design sound?
Check the sequence of statements in the methods. Does all the information belong there? Is the sequence of methods clear and pertinent?
Was there mention of ethics? Which research ethics board approved the study?
Carefully examine the data presented in the tables and diagrams. Does the title or legend accurately describe the content?
Are column headings and labels accurate?
Are the data organized for ready comparison and interpretation? (A table should be self-explanatory, with a title that accurately and concisely describes content and column headings that accurately describe information in the cells.)
Review the results as presented in the text while referring to the data in the tables and diagrams. Does the text complement, and not simply repeat data? Are there discrepancies between the results in the text and those in the tables?
Check all calculations and presentation of data.
Review the results in light of the stated objectives. Does the study reveal what the researchers intended?
Does the discussion clearly address the objectives and hypotheses?
Check the interpretation against the results. Does the discussion merely repeat the results?
Does the interpretation arise logically from the data or is it too far-fetched?
Have the faults, ﬂaws, or shortcomings of the research been addressed?
Is the interpretation supported by other research cited in the study?
Does the study consider key studies in the ﬁeld?
What is the significance of the research? Do the authors mention wider implications of the findings?
Is there a section on recommendations for future research? Are there other research possibilities or directions suggested?

Consider the article as a whole

Reread the abstract. Does it accurately summarize the article?
Check the structure of the article (ﬁrst headings and then paragraphing). Is all the material organized under the appropriate headings? Are sections divided logically into subsections or paragraphs?
Are stylistic concerns, logic, clarity, and economy of expression addressed?

After you have evaluated the research, consider whether the research has been successful. Has it led to new questions being asked, or new ways of using existing knowledge? Are other researchers citing this paper?

You should consider the following questions:

How did other researchers view the signiﬁcance of the research reported by your authors?
Did the research reported in your article result in the formulation of new questions or hypotheses (by the authors or by other researchers)?
Have other researchers subsequently supported or refuted the observations or interpretations of these authors?
Did the research make a signiﬁcant contribution to human knowledge?
Did the research produce any practical applications?
What are the social, political, technological, medical implications of this research?
How do you evaluate the signiﬁcance of the research?

To answer these questions, look at review articles to ﬁnd out how reviewers view this piece of research. Look at research articles and databases like Web of Science to see how other people have used this work. What range of journals have cited this article?

These questions were adapted from the following sources:

Kuyper, B.J. (1991). Bringing up scientists in the art of critiquing research. Bioscience 41(4), 248-250. Wood, J.M. (2003). Research Lab Guide. MICR*3260 Microbial Adaptation and Development Web Site . Retrieved July 31, 2006.

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1 Important points to consider when critically evaluating published research papers

Simple review articles (also referred to as ‘narrative’ or ‘selective’ reviews), systematic reviews and meta-analyses provide rapid overviews and ‘snapshots’ of progress made within a field, summarising a given topic or research area. They can serve as useful guides, or as current and comprehensive ‘sources’ of information, and can act as a point of reference to relevant primary research studies within a given scientific area. Narrative or systematic reviews are often used as a first step towards a more detailed investigation of a topic or a specific enquiry (a hypothesis or research question), or to establish critical awareness of a rapidly-moving field (you will be required to demonstrate this as part of an assignment, an essay or a dissertation at postgraduate level).

The majority of primary ‘empirical’ research papers essentially follow the same structure (abbreviated here as IMRAD). There is a section on Introduction, followed by the Methods, then the Results, which includes figures and tables showing data described in the paper, and a Discussion. The paper typically ends with a Conclusion, and References and Acknowledgements sections.

The Title of the paper provides a concise first impression. The Abstract follows the basic structure of the extended article. It provides an ‘accessible’ and concise summary of the aims, methods, results and conclusions. The Introduction provides useful background information and context, and typically outlines the aims and objectives of the study. The Abstract can serve as a useful summary of the paper, presenting the purpose, scope and major findings. However, simply reading the abstract alone is not a substitute for critically reading the whole article. To really get a good understanding and to be able to critically evaluate a research study, it is necessary to read on.

While most research papers follow the above format, variations do exist. For example, the results and discussion sections may be combined. In some journals the materials and methods may follow the discussion, and in two of the most widely read journals, Science and Nature, the format does vary from the above due to restrictions on the length of articles. In addition, there may be supporting documents that accompany a paper, including supplementary materials such as supporting data, tables, figures, videos and so on. There may also be commentaries or editorials associated with a topical research paper, which provide an overview or critique of the study being presented.

Box 1 Key questions to ask when appraising a research paper

Is the study’s research question relevant?
Does the study add anything new to current knowledge and understanding?
Does the study test a stated hypothesis?
Is the design of the study appropriate to the research question?
Do the study methods address key potential sources of bias?
Were suitable ‘controls’ included in the study?
Were the statistical analyses appropriate and applied correctly?
Is there a clear statement of findings?
Does the data support the authors’ conclusions?
Are there any conflicts of interest or ethical concerns?

There are various strategies used in reading a scientific research paper, and one of these is to start with the title and the abstract, then look at the figures and tables, and move on to the introduction, before turning to the results and discussion, and finally, interrogating the methods.

Another strategy (outlined below) is to begin with the abstract and then the discussion, take a look at the methods, and then the results section (including any relevant tables and figures), before moving on to look more closely at the discussion and, finally, the conclusion. You should choose a strategy that works best for you. However, asking the ‘right’ questions is a central feature of critical appraisal, as with any enquiry, so where should you begin? Here are some critical questions to consider when evaluating a research paper.

Look at the Abstract and then the Discussion : Are these accessible and of general relevance or are they detailed, with far-reaching conclusions? Is it clear why the study was undertaken? Why are the conclusions important? Does the study add anything new to current knowledge and understanding? The reasons why a particular study design or statistical method were chosen should also be clear from reading a research paper. What is the research question being asked? Does the study test a stated hypothesis? Is the design of the study appropriate to the research question? Have the authors considered the limitations of their study and have they discussed these in context?

Take a look at the Methods : Were there any practical difficulties that could have compromised the study or its implementation? Were these considered in the protocol? Were there any missing values and, if so, was the number of missing values too large to permit meaningful analysis? Was the number of samples (cases or participants) too small to establish meaningful significance? Do the study methods address key potential sources of bias? Were suitable ‘controls’ included in the study? If controls are missing or not appropriate to the study design, we cannot be confident that the results really show what is happening in an experiment. Were the statistical analyses appropriate and applied correctly? Do the authors point out the limitations of methods or tests used? Were the methods referenced and described in sufficient detail for others to repeat or extend the study?

Take a look at the Results section and relevant tables and figures : Is there a clear statement of findings? Were the results expected? Do they make sense? What data supports them? Do the tables and figures clearly describe the data (highlighting trends etc.)? Try to distinguish between what the data show and what the authors say they show (i.e. their interpretation).

Moving on to look in greater depth at the Discussion and Conclusion : Are the results discussed in relation to similar (previous) studies? Do the authors indulge in excessive speculation? Are limitations of the study adequately addressed? Were the objectives of the study met and the hypothesis supported or refuted (and is a clear explanation provided)? Does the data support the authors’ conclusions? Maybe there is only one experiment to support a point. More often, several different experiments or approaches combine to support a particular conclusion. A rule of thumb here is that if multiple approaches and multiple lines of evidence from different directions are presented, and all point to the same conclusion, then the conclusions are more credible. But do question all assumptions. Identify any implicit or hidden assumptions that the authors may have used when interpreting their data. Be wary of data that is mixed up with interpretation and speculation! Remember, just because it is published, does not mean that it is right.

O ther points you should consider when evaluating a research paper : Are there any financial, ethical or other conflicts of interest associated with the study, its authors and sponsors? Are there ethical concerns with the study itself? Looking at the references, consider if the authors have preferentially cited their own previous publications (i.e. needlessly), and whether the list of references are recent (ensuring that the analysis is up-to-date). Finally, from a practical perspective, you should move beyond the text of a research paper, talk to your peers about it, consult available commentaries, online links to references and other external sources to help clarify any aspects you don’t understand.

The above can be taken as a general guide to help you begin to critically evaluate a scientific research paper, but only in the broadest sense. Do bear in mind that the way that research evidence is critiqued will also differ slightly according to the type of study being appraised, whether observational or experimental, and each study will have additional aspects that would need to be evaluated separately. For criteria recommended for the evaluation of qualitative research papers, see the article by Mildred Blaxter (1996), available online. Details are in the References.

Activity 1 Critical appraisal of a scientific research paper

A critical appraisal checklist, which you can download via the link below, can act as a useful tool to help you to interrogate research papers. The checklist is divided into four sections, broadly covering:

some general aspects
research design and methodology
the results
discussion, conclusion and references.

Science perspective – critical appraisal checklist [ Tip: hold Ctrl and click a link to open it in a new tab. ( Hide tip ) ]

Identify and obtain a research article based on a topic of your own choosing, using a search engine such as Google Scholar or PubMed (for example).
The selection criteria for your target paper are as follows: the article must be an open access primary research paper (not a review) containing empirical data, published in the last 2–3 years, and preferably no more than 5–6 pages in length.
Critically evaluate the research paper using the checklist provided, making notes on the key points and your overall impression.

Critical appraisal checklists are useful tools to help assess the quality of a study. Assessment of various factors, including the importance of the research question, the design and methodology of a study, the validity of the results and their usefulness (application or relevance), the legitimacy of the conclusions, and any potential conflicts of interest, are an important part of the critical appraisal process. Limitations and further improvements can then be considered.

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Critical Analysis – Types, Examples and Writing Guide

Table of Contents

Critical Analysis

Definition:

Critical analysis is a process of examining a piece of work or an idea in a systematic, objective, and analytical way. It involves breaking down complex ideas, concepts, or arguments into smaller, more manageable parts to understand them better.

Types of Critical Analysis

Types of Critical Analysis are as follows:

Literary Analysis

This type of analysis focuses on analyzing and interpreting works of literature , such as novels, poetry, plays, etc. The analysis involves examining the literary devices used in the work, such as symbolism, imagery, and metaphor, and how they contribute to the overall meaning of the work.

Film Analysis

This type of analysis involves examining and interpreting films, including their themes, cinematography, editing, and sound. Film analysis can also include evaluating the director’s style and how it contributes to the overall message of the film.

Art Analysis

This type of analysis involves examining and interpreting works of art , such as paintings, sculptures, and installations. The analysis involves examining the elements of the artwork, such as color, composition, and technique, and how they contribute to the overall meaning of the work.

Cultural Analysis

This type of analysis involves examining and interpreting cultural artifacts , such as advertisements, popular music, and social media posts. The analysis involves examining the cultural context of the artifact and how it reflects and shapes cultural values, beliefs, and norms.

Historical Analysis

This type of analysis involves examining and interpreting historical documents , such as diaries, letters, and government records. The analysis involves examining the historical context of the document and how it reflects the social, political, and cultural attitudes of the time.

Philosophical Analysis

This type of analysis involves examining and interpreting philosophical texts and ideas, such as the works of philosophers and their arguments. The analysis involves evaluating the logical consistency of the arguments and assessing the validity and soundness of the conclusions.

Scientific Analysis

This type of analysis involves examining and interpreting scientific research studies and their findings. The analysis involves evaluating the methods used in the study, the data collected, and the conclusions drawn, and assessing their reliability and validity.

Critical Discourse Analysis

This type of analysis involves examining and interpreting language use in social and political contexts. The analysis involves evaluating the power dynamics and social relationships conveyed through language use and how they shape discourse and social reality.

Comparative Analysis

This type of analysis involves examining and interpreting multiple texts or works of art and comparing them to each other. The analysis involves evaluating the similarities and differences between the texts and how they contribute to understanding the themes and meanings conveyed.

Critical Analysis Format

Critical Analysis Format is as follows:

I. Introduction

Provide a brief overview of the text, object, or event being analyzed
Explain the purpose of the analysis and its significance
Provide background information on the context and relevant historical or cultural factors

II. Description

Provide a detailed description of the text, object, or event being analyzed
Identify key themes, ideas, and arguments presented
Describe the author or creator’s style, tone, and use of language or visual elements

III. Analysis

Analyze the text, object, or event using critical thinking skills
Identify the main strengths and weaknesses of the argument or presentation
Evaluate the reliability and validity of the evidence presented
Assess any assumptions or biases that may be present in the text, object, or event
Consider the implications of the argument or presentation for different audiences and contexts

IV. Evaluation

Provide an overall evaluation of the text, object, or event based on the analysis
Assess the effectiveness of the argument or presentation in achieving its intended purpose
Identify any limitations or gaps in the argument or presentation
Consider any alternative viewpoints or interpretations that could be presented
Summarize the main points of the analysis and evaluation
Reiterate the significance of the text, object, or event and its relevance to broader issues or debates
Provide any recommendations for further research or future developments in the field.

VI. Example

Provide an example or two to support your analysis and evaluation
Use quotes or specific details from the text, object, or event to support your claims
Analyze the example(s) using critical thinking skills and explain how they relate to your overall argument

VII. Conclusion

Reiterate your thesis statement and summarize your main points
Provide a final evaluation of the text, object, or event based on your analysis
Offer recommendations for future research or further developments in the field
End with a thought-provoking statement or question that encourages the reader to think more deeply about the topic

How to Write Critical Analysis

Writing a critical analysis involves evaluating and interpreting a text, such as a book, article, or film, and expressing your opinion about its quality and significance. Here are some steps you can follow to write a critical analysis:

Read and re-read the text: Before you begin writing, make sure you have a good understanding of the text. Read it several times and take notes on the key points, themes, and arguments.
Identify the author’s purpose and audience: Consider why the author wrote the text and who the intended audience is. This can help you evaluate whether the author achieved their goals and whether the text is effective in reaching its audience.
Analyze the structure and style: Look at the organization of the text and the author’s writing style. Consider how these elements contribute to the overall meaning of the text.
Evaluate the content : Analyze the author’s arguments, evidence, and conclusions. Consider whether they are logical, convincing, and supported by the evidence presented in the text.
Consider the context: Think about the historical, cultural, and social context in which the text was written. This can help you understand the author’s perspective and the significance of the text.
Develop your thesis statement : Based on your analysis, develop a clear and concise thesis statement that summarizes your overall evaluation of the text.
Support your thesis: Use evidence from the text to support your thesis statement. This can include direct quotes, paraphrases, and examples from the text.
Write the introduction, body, and conclusion : Organize your analysis into an introduction that provides context and presents your thesis, a body that presents your evidence and analysis, and a conclusion that summarizes your main points and restates your thesis.
Revise and edit: After you have written your analysis, revise and edit it to ensure that your writing is clear, concise, and well-organized. Check for spelling and grammar errors, and make sure that your analysis is logically sound and supported by evidence.

When to Write Critical Analysis

You may want to write a critical analysis in the following situations:

Academic Assignments: If you are a student, you may be assigned to write a critical analysis as a part of your coursework. This could include analyzing a piece of literature, a historical event, or a scientific paper.
Journalism and Media: As a journalist or media person, you may need to write a critical analysis of current events, political speeches, or media coverage.
Personal Interest: If you are interested in a particular topic, you may want to write a critical analysis to gain a deeper understanding of it. For example, you may want to analyze the themes and motifs in a novel or film that you enjoyed.
Professional Development : Professionals such as writers, scholars, and researchers often write critical analyses to gain insights into their field of study or work.

Critical Analysis Example

An Example of Critical Analysis Could be as follow:

Research Topic:

The Impact of Online Learning on Student Performance

Introduction:

The introduction of the research topic is clear and provides an overview of the issue. However, it could benefit from providing more background information on the prevalence of online learning and its potential impact on student performance.

Literature Review:

The literature review is comprehensive and well-structured. It covers a broad range of studies that have examined the relationship between online learning and student performance. However, it could benefit from including more recent studies and providing a more critical analysis of the existing literature.

Research Methods:

The research methods are clearly described and appropriate for the research question. The study uses a quasi-experimental design to compare the performance of students who took an online course with those who took the same course in a traditional classroom setting. However, the study may benefit from using a randomized controlled trial design to reduce potential confounding factors.

The results are presented in a clear and concise manner. The study finds that students who took the online course performed similarly to those who took the traditional course. However, the study only measures performance on one course and may not be generalizable to other courses or contexts.

Discussion :

The discussion section provides a thorough analysis of the study’s findings. The authors acknowledge the limitations of the study and provide suggestions for future research. However, they could benefit from discussing potential mechanisms underlying the relationship between online learning and student performance.

Conclusion :

The conclusion summarizes the main findings of the study and provides some implications for future research and practice. However, it could benefit from providing more specific recommendations for implementing online learning programs in educational settings.

Purpose of Critical Analysis

There are several purposes of critical analysis, including:

To identify and evaluate arguments : Critical analysis helps to identify the main arguments in a piece of writing or speech and evaluate their strengths and weaknesses. This enables the reader to form their own opinion and make informed decisions.
To assess evidence : Critical analysis involves examining the evidence presented in a text or speech and evaluating its quality and relevance to the argument. This helps to determine the credibility of the claims being made.
To recognize biases and assumptions : Critical analysis helps to identify any biases or assumptions that may be present in the argument, and evaluate how these affect the credibility of the argument.
To develop critical thinking skills: Critical analysis helps to develop the ability to think critically, evaluate information objectively, and make reasoned judgments based on evidence.
To improve communication skills: Critical analysis involves carefully reading and listening to information, evaluating it, and expressing one’s own opinion in a clear and concise manner. This helps to improve communication skills and the ability to express ideas effectively.

Importance of Critical Analysis

Here are some specific reasons why critical analysis is important:

Helps to identify biases: Critical analysis helps individuals to recognize their own biases and assumptions, as well as the biases of others. By being aware of biases, individuals can better evaluate the credibility and reliability of information.
Enhances problem-solving skills : Critical analysis encourages individuals to question assumptions and consider multiple perspectives, which can lead to creative problem-solving and innovation.
Promotes better decision-making: By carefully evaluating evidence and arguments, critical analysis can help individuals make more informed and effective decisions.
Facilitates understanding: Critical analysis helps individuals to understand complex issues and ideas by breaking them down into smaller parts and evaluating them separately.
Fosters intellectual growth : Engaging in critical analysis challenges individuals to think deeply and critically, which can lead to intellectual growth and development.

Advantages of Critical Analysis

Some advantages of critical analysis include:

Improved decision-making: Critical analysis helps individuals make informed decisions by evaluating all available information and considering various perspectives.
Enhanced problem-solving skills : Critical analysis requires individuals to identify and analyze the root cause of a problem, which can help develop effective solutions.
Increased creativity : Critical analysis encourages individuals to think outside the box and consider alternative solutions to problems, which can lead to more creative and innovative ideas.
Improved communication : Critical analysis helps individuals communicate their ideas and opinions more effectively by providing logical and coherent arguments.
Reduced bias: Critical analysis requires individuals to evaluate information objectively, which can help reduce personal biases and subjective opinions.
Better understanding of complex issues : Critical analysis helps individuals to understand complex issues by breaking them down into smaller parts, examining each part and understanding how they fit together.
Greater self-awareness: Critical analysis helps individuals to recognize their own biases, assumptions, and limitations, which can lead to personal growth and development.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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33 Critical Analysis Examples

critical analysis examples and definition, explained below

Critical analysis refers to the ability to examine something in detail in preparation to make an evaluation or judgment.

It will involve exploring underlying assumptions, theories, arguments, evidence, logic, biases, contextual factors, and so forth, that could help shed more light on the topic.

In essay writing, a critical analysis essay will involve using a range of analytical skills to explore a topic, such as:

Evaluating sources
Exploring strengths and weaknesses
Exploring pros and cons
Questioning and challenging ideas
Comparing and contrasting ideas

If you’re writing an essay, you could also watch my guide on how to write a critical analysis essay below, and don’t forget to grab your worksheets and critical analysis essay plan to save yourself a ton of time:

Grab your Critical Analysis Worksheets and Essay Plan Here

Critical Analysis Examples

1. exploring strengths and weaknesses.

Perhaps the first and most straightforward method of critical analysis is to create a simple strengths-vs-weaknesses comparison.

Most things have both strengths and weaknesses – you could even do this for yourself! What are your strengths? Maybe you’re kind or good at sports or good with children. What are your weaknesses? Maybe you struggle with essay writing or concentration.

If you can analyze your own strengths and weaknesses, then you understand the concept. What might be the strengths and weaknesses of the idea you’re hoping to critically analyze?

Strengths and weaknesses could include:

Does it seem highly ethical (strength) or could it be more ethical (weakness)?
Is it clearly explained (strength) or complex and lacking logical structure (weakness)?
Does it seem balanced (strength) or biased (weakness)?

You may consider using a SWOT analysis for this step. I’ve provided a SWOT analysis guide here .

2. Evaluating Sources

Evaluation of sources refers to looking at whether a source is reliable or unreliable.

This is a fundamental media literacy skill .

Steps involved in evaluating sources include asking questions like:

Who is the author and are they trustworthy?
Is this written by an expert?
Is this sufficiently reviewed by an expert?
Is this published in a trustworthy publication?
Are the arguments sound or common sense?

For more on this topic, I’d recommend my detailed guide on digital literacy .

3. Identifying Similarities

Identifying similarities encompasses the act of drawing parallels between elements, concepts, or issues.

In critical analysis, it’s common to compare a given article, idea, or theory to another one. In this way, you can identify areas in which they are alike.

Determining similarities can be a challenge, but it’s an intellectual exercise that fosters a greater understanding of the aspects you’re studying. This step often calls for a careful reading and note-taking to highlight matching information, points of view, arguments or even suggested solutions.

Similarities might be found in:

The key themes or topics discussed
The theories or principles used
The demographic the work is written for or about
The solutions or recommendations proposed

Remember, the intention of identifying similarities is not to prove one right or wrong. Rather, it sets the foundation for understanding the larger context of your analysis, anchoring your arguments in a broader spectrum of ideas.

Your critical analysis strengthens when you can see the patterns and connections across different works or topics. It fosters a more comprehensive, insightful perspective. And importantly, it is a stepping stone in your analysis journey towards evaluating differences, which is equally imperative and insightful in any analysis.

4. Identifying Differences

Identifying differences involves pinpointing the unique aspects, viewpoints or solutions introduced by the text you’re analyzing. How does it stand out as different from other texts?

To do this, you’ll need to compare this text to another text.

Differences can be revealed in:

The potential applications of each idea
The time, context, or place in which the elements were conceived or implemented
The available evidence each element uses to support its ideas
The perspectives of authors
The conclusions reached

Identifying differences helps to reveal the multiplicity of perspectives and approaches on a given topic. Doing so provides a more in-depth, nuanced understanding of the field or issue you’re exploring.

This deeper understanding can greatly enhance your overall critique of the text you’re looking at. As such, learning to identify both similarities and differences is an essential skill for effective critical analysis.

My favorite tool for identifying similarities and differences is a Venn Diagram:

To use a venn diagram, title each circle for two different texts. Then, place similarities in the overlapping area of the circles, while unique characteristics (differences) of each text in the non-overlapping parts.

6. Identifying Oversights

Identifying oversights entails pointing out what the author missed, overlooked, or neglected in their work.

Almost every written work, no matter the expertise or meticulousness of the author, contains oversights. These omissions can be absent-minded mistakes or gaps in the argument, stemming from a lack of knowledge, foresight, or attentiveness.

Such gaps can be found in:

Missed opportunities to counter or address opposing views
Failure to consider certain relevant aspects or perspectives
Incomplete or insufficient data that leaves the argument weak
Failing to address potential criticism or counter-arguments

By shining a light on these weaknesses, you increase the depth and breadth of your critical analysis. It helps you to estimate the full worth of the text, understand its limitations, and contextualize it within the broader landscape of related work. Ultimately, noticing these oversights helps to make your analysis more balanced and considerate of the full complexity of the topic at hand.

You may notice here that identifying oversights requires you to already have a broad understanding and knowledge of the topic in the first place – so, study up!

7. Fact Checking

Fact-checking refers to the process of meticulously verifying the truth and accuracy of the data, statements, or claims put forward in a text.

Fact-checking serves as the bulwark against misinformation, bias, and unsubstantiated claims. It demands thorough research, resourcefulness, and a keen eye for detail.

Fact-checking goes beyond surface-level assertions:

Examining the validity of the data given
Cross-referencing information with other reliable sources
Scrutinizing references, citations, and sources utilized in the article
Distinguishing between opinion and objectively verifiable truths
Checking for outdated, biased, or unbalanced information

If you identify factual errors, it’s vital to highlight them when critically analyzing the text. But remember, you could also (after careful scrutiny) also highlight that the text appears to be factually correct – that, too, is critical analysis.

8. Exploring Counterexamples

Exploring counterexamples involves searching and presenting instances or cases which contradict the arguments or conclusions presented in a text.

Counterexamples are an effective way to challenge the generalizations, assumptions or conclusions made in an article or theory. They can reveal weaknesses or oversights in the logic or validity of the author’s perspective.

Considerations in counterexample analysis are:

Identifying generalizations made in the text
Seeking examples in academic literature or real-world instances that contradict these generalizations
Assessing the impact of these counterexamples on the validity of the text’s argument or conclusion

Exploring counterexamples enriches your critical analysis by injecting an extra layer of scrutiny, and even doubt, in the text.

By presenting counterexamples, you not only test the resilience and validity of the text but also open up new avenues of discussion and investigation that can further your understanding of the topic.

9. Assessing Methodologies

Assessing methodologies entails examining the techniques, tools, or procedures employed by the author to collect, analyze and present their information.

The accuracy and validity of a text’s conclusions often depend on the credibility and appropriateness of the methodologies used.

Aspects to inspect include:

The appropriateness of the research method for the research question
The adequacy of the sample size
The validity and reliability of data collection instruments
The application of statistical tests and evaluations
The implementation of controls to prevent bias or mitigate its impact

One strategy you could implement here is to consider a range of other methodologies the author could have used. If the author conducted interviews, consider questioning why they didn’t use broad surveys that could have presented more quantitative findings. If they only interviewed people with one perspective, consider questioning why they didn’t interview a wider variety of people, etc.

See Also: A List of Research Methodologies

10. Exploring Alternative Explanations

Exploring alternative explanations refers to the practice of proposing differing or opposing ideas to those put forward in the text.

An underlying assumption in any analysis is that there may be multiple valid perspectives on a single topic. The text you’re analyzing might provide one perspective, but your job is to bring into the light other reasonable explanations or interpretations.

Cultivating alternative explanations often involves:

Formulating hypotheses or theories that differ from those presented in the text
Referring to other established ideas or models that offer a differing viewpoint
Suggesting a new or unique angle to interpret the data or phenomenon discussed in the text

Searching for alternative explanations challenges the authority of a singular narrative or perspective, fostering an environment ripe for intellectual discourse and critical thinking . It nudges you to examine the topic from multiple angles, enhancing your understanding and appreciation of the complexity inherent in the field.

A Full List of Critical Analysis Skills

Exploring Strengths and Weaknesses
Evaluating Sources
Identifying Similarities
Identifying Differences
Identifying Biases
Hypothesis Testing
Fact-Checking
Exploring Counterexamples
Assessing Methodologies
Exploring Alternative Explanations
Pointing Out Contradictions
Challenging the Significance
Cause-And-Effect Analysis
Assessing Generalizability
Highlighting Inconsistencies
Reductio ad Absurdum
Comparing to Expert Testimony
Comparing to Precedent
Reframing the Argument
Pointing Out Fallacies
Questioning the Ethics
Clarifying Definitions
Challenging Assumptions
Exposing Oversimplifications
Highlighting Missing Information
Demonstrating Irrelevance
Assessing Effectiveness
Assessing Trustworthiness
Recognizing Patterns
Differentiating Facts from Opinions
Analyzing Perspectives
Prioritization
Making Predictions
Conducting a SWOT Analysis
PESTLE Analysis
Asking the Five Whys
Correlating Data Points
Finding Anomalies Or Outliers
Comparing to Expert Literature
Drawing Inferences
Assessing Validity & Reliability

Analysis and Bloom’s Taxonomy

Benjamin Bloom placed analysis as the third-highest form of thinking on his ladder of cognitive skills called Bloom’s Taxonomy .

This taxonomy starts with the lowest levels of thinking – remembering and understanding. The further we go up the ladder, the more we reach higher-order thinking skills that demonstrate depth of understanding and knowledge, as outlined below:

Here’s a full outline of the taxonomy in a table format:

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 5 Top Tips for Succeeding at University
Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 50 Durable Goods Examples
Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 100 Consumer Goods Examples
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2 thoughts on “33 Critical Analysis Examples”

THANK YOU, THANK YOU, THANK YOU! – I cannot even being to explain how hard it has been to find a simple but in-depth understanding of what ‘Critical Analysis’ is. I have looked at over 10 different pages and went down so many rabbit holes but this is brilliant! I only skimmed through the article but it was already promising, I then went back and read it more in-depth, it just all clicked into place. So thank you again!

You’re welcome – so glad it was helpful.

6 Common Leadership Styles — and How to Decide Which to Use When

Rebecca Knight

Being a great leader means recognizing that different circumstances call for different approaches.

Research suggests that the most effective leaders adapt their style to different circumstances — be it a change in setting, a shift in organizational dynamics, or a turn in the business cycle. But what if you feel like you’re not equipped to take on a new and different leadership style — let alone more than one? In this article, the author outlines the six leadership styles Daniel Goleman first introduced in his 2000 HBR article, “Leadership That Gets Results,” and explains when to use each one. The good news is that personality is not destiny. Even if you’re naturally introverted or you tend to be driven by data and analysis rather than emotion, you can still learn how to adapt different leadership styles to organize, motivate, and direct your team.

Much has been written about common leadership styles and how to identify the right style for you, whether it’s transactional or transformational, bureaucratic or laissez-faire. But according to Daniel Goleman, a psychologist best known for his work on emotional intelligence, “Being a great leader means recognizing that different circumstances may call for different approaches.”

RK Rebecca Knight is a journalist who writes about all things related to the changing nature of careers and the workplace. Her essays and reported stories have been featured in The Boston Globe, Business Insider, The New York Times, BBC, and The Christian Science Monitor. She was shortlisted as a Reuters Institute Fellow at Oxford University in 2023. Earlier in her career, she spent a decade as an editor and reporter at the Financial Times in New York, London, and Boston.

Mel Kiper, Jr Fills Two Critical Needs for Giants in Final 2024 Mock Draft

Mel Kiper Jr addresses two key needs for the Giants in his final mock draft of 2024.

Author: Ezekiel Trezevant IV
Publish date: Apr 17, 2024

In this story:

Despite a busy off-season, the New York Giants still have a few glaring remaining needs that they're expected to address in next week's draft.

Among the most glaring are WR1 and quarterback, two positions with fans and media split as far as which position the Giants should address in Round 1. Given the depth of both position groups, the Giants will have options. Still, if it's a quarterback that they want from among the projected top prospects (Caleb Williams, Drake Maye, Jayden Daniels, and J.J. McCarthy), they will more than likely have to trade up from the sixth spot to get one of them, a risky yet bold type of move.

Taking that into consideration, ESPN draft analyst Mel Kuper Jr., in his final mock draft of 2024, has the Giants going receiver in the first round in a "stick and pick" scenario and then engaging in a trade-up in the second round to get their quarterback.

In Round 1, Kiper has the Giants taking LSU receiver Malik Nabers at six. He is a yards-after-catch specialist who has WR1 written all over him.

Notes Kiper of the pick: "The Giants' pass offense fell apart last season as they finished 30th in the league in yards per dropback (4.9). Sure, most of that was without quarterback Daniel Jones, but he struggled in his six games before his ACL injury, too, throwing three times as many interceptions (six) as touchdown passes. New York also ranked 30th in yards after the catch (1,601), which showed its need for outside playmakers.

"Nabers could solve that problem, as he's ferocious after the catch. He can run any route, creating separation on even the best cornerbacks. This is how the Giants could instantly turn around their offense."

The Giants have a solid receiver room with Darius Slayton, Wan'Dale Robinson, and Jalin Hyatt all under contract. However, none of them are legitimate WR1s, which, by the way, have been missing from the Giants' offense since the team traded away Odell Beckham Jr. Being able to land a receiver of Nabers' skillset would indeed significantly help the Giants' offense, regardless of who the quarterback is.

In the second round, Kiper projects the Giants trading up to the top of the round, where they would be in a position to grab Oregon quarterback Bo Nix, of whom Kiper wrote, "Nix started a whopping 61 college games in stints at Auburn and Oregon, throwing for 15,352 total yards. He leveled up in 2023, with 45 touchdown passes and three picks while completing 77.4% of his throws."

Nix, like any other draft prospect, has some questions, one of which Kiper notes is arm strength. "He was asked to get the ball out quickly and around the line of scrimmage for the Ducks, putting his receivers in a position to make plays after the catch. That’s why I have a Round 2 grade on him."

Kiper is correct in that the Giants' plan for the coming year, regardless of who they might draft at quarterback (assuming they do draft one), is to allow that prospect to sit for a bit as he acclimates to life in the NFL and, in the interim, roll with incumbent Daniel Jones once he's cleared medically from his torn ACL.

If Jones isn't green-lighted to return, the offense would then be led by Drew Lock, acquired in the off-season to replace Tyrod Taylor.

As for Kiper's suggestion to trade up in Round 2 for Nix, the cost would likely be less than if the Giants tried to trade up from six to four or five in the draft order. The Giants, remember, traded away the higher of their two second-round picks in the Brian Burns acquisition, so they might not want to risk waiting for Nix or, if they like him, Michael Penix Jr to drop to them.

General manager Joe Schoen has shown that he's not afraid to trade up to get the guy he wants, but at the same time, he hasn't been wreckless in going all out while ignoring the other needs he has to fill.

We don't expect this year to be any different, not with the Giants needing a CB2, additional interior defensive line help, a tackle prospect, and a safety, among other things.

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As Iranian rockets and drones lit up the sky over Israel, the Middle East entered a dangerous new phase

Analysis As Iranian rockets and drones lit up the sky over Israel, the Middle East entered a dangerous new phase

Up in the sky, dozens of bright orange balls flew in an arc through the darkness.

It almost looked like a cluster of shooting stars, except they were much more dangerous and serious.

They were rockets and drones fired at Israel by Iran.

For days I have been preparing myself, our ABC team here in Jerusalem, and my family, for an attack by Iran on Israeli soil.

This week, the United States warned Tehran was planning a "significant attack" in response to several Iranian military leaders being killed at the start of April in an air strike – believed to have been launched by Israel – on the Iranian consulate in Damascus, Syria.

At the time, Iran vowed to seek revenge. The tension has been building across Israel ever since.

Regular citizens were warned to stock up their emergency kits, including with a battery-operated radio and bottled water — in a similar way some Australians are warned to get emergency supplies ready during extreme weather.

Stores were stripped of supplies and everyone was on edge.

People were discussing what to do if an attack happened, or whether they should try to flee the country altogether.

Days of tension and then bright lights in the sky

The signs that Israeli authorities knew something was imminent started on Saturday local time.

Israel's Home Front Command announced schools would be closed and educational activities banned for 48 hours.

And then, just before 11pm on Saturday night, reports started filtering through that Iran had launched hundreds of unmanned drones and missiles towards Israel.

It marked a serious escalation between the two longtime arch-enemies.

We were told by Israel's military that we had a few hours until the first wave of drones and missiles made it into Israeli airspace, and that aerial capabilities, including the Iron Dome, and the Arrow, had been readied for defence.

It was about three hours later when the first missiles started appearing in the sky above me.

Dressed in our personal protective equipment, ABC camera operator Haidarr Jones and I rushed onto my balcony to film.

Flashes of bright, orange light flew through the sky.

Some of them exploded before my eyes as they were intercepted by Israel's defence array.

On my phone, I'm registered to receive security alert notifications — here we call them rocket alerts — that are put out by the government as a warning that something is entering Israeli airspace.

My phone was pinging non-stop in what felt like dozens of alerts within seconds.

And a moment later, those pings notified me of a threat right above me.

Rocket alarm sirens around my neighbourhood in Jerusalem started blaring, and I knew I had a few seconds to get to safety.

We ran from the balcony, collected my husband and daughter, and dashed into the bomb shelter inside our house.

I could hear the sonic booms as Israel's defence systems shot down drones or missiles as they arced across Jerusalem.

Earlier in the day, I prepared our bomb shelter with mattresses and water.

As the room shook with each explosion and sirens continued to ring out, I tried to get my child back to sleep.

Why the next 24 hours are critical

The sirens stopped after about five minutes, so I moved back upstairs to the balcony and looked up.

Bright balls of light were still flying overhead, signifying the attack was ongoing.

From my balcony, I could hear the cheers of some Palestinians in East Jerusalem who were celebrating a response on Israel, which they felt was long overdue following six months of war in Gaza.

I watched from my balcony as they let off fireworks.

In October last year, Hamas-led militants stormed across the border into southern Israel, killing hundreds of civilians and taking 253 hostages back into Gaza.

Israel's government swiftly launched a retaliatory military campaign, which it said was aimed at eliminating Hamas from the coastal enclave.

Since then, more than 33,000 people in Gaza have been killed, according to the local health ministry.

It has been a tremendously long six months in this region, and as I looked around and continued my work, I wondered: What will the following days hold?

This is the first time that Iran has directly attacked Israel despite decades of a shadow war between the pair.

Israel has said it could use "unprecedented action" to respond, which is expected within the next 24 hours.

Iran has already warned it will be ready to retaliate again.

No-one knows what will happen here, but there is a lot of fear and anxiety.

The situation is threatening to spiral into an all-out war between Iran and its allies, and Israel and the United States.

It could change the Middle East as we know it.

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Biden says he expects Iran to attack Israel 'sooner rather than later'

Joe Biden is pictured from the side as he leaves his Marine One aircraft.

A shock vote at the UN and a fit of pique by Bibi reveals the new low in American-Israeli relations

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Hypothesis Testing
Step 2: Collect data. For a statistical test to be valid, it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in. Hypothesis testing example.
7.1: Basics of Hypothesis Testing
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
1.2
Step 7: Based on Steps 5 and 6, draw a conclusion about H 0. If F calculated is larger than F α, then you are in the rejection region and you can reject the null hypothesis with ( 1 − α) level of confidence. Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting ...
Mastering Hypothesis Testing: A Comprehensive Guide for ...
1. Introduction to Hypothesis Testing - Definition and significance in research and data analysis. - Brief historical background. 2. Fundamentals of Hypothesis Testing - Null and Alternative…
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$. An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
S.3.1 Hypothesis Testing (Critical Value Approach)
The critical value for conducting the left-tailed test H0 : μ = 3 versus HA : μ < 3 is the t -value, denoted -t( α, n - 1), such that the probability to the left of it is α. It can be shown using either statistical software or a t -table that the critical value -t0.05,14 is -1.7613. That is, we would reject the null hypothesis H0 : μ = 3 ...
Hypothesis Testing
Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions. In this Blog post we will learn: What is Hypothesis Testing? Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3.
Developing and Testing Hypotheses
Statistical hypothesis testing is sometimes known as confirmatory data analysis. It is a way of drawing inferences from data. In the process, you develop a hypothesis or theory about what you might see in your research. You then test that hypothesis against the data that you collect. Hypothesis testing is generally used when you want to compare ...
How Do You Formulate (Important) Hypotheses?
In our experience, presenting research questions is a more common form of stating the goal of a research study than presenting well-formulated hypotheses. Authors sometimes present a hypothesis, often as a simple prediction of what they might find. The hypothesis is then forgotten and not used to guide the analysis or interpretations of the ...
Critical Value: Definition, Finding & Calculator
Critical values (CV) are the boundary between nonsignificant and significant results in a hypothesis test. Test statistics that exceed a critical value have a low probability of occurring if the null hypothesis is true. Therefore, when test statistics exceed these cutoffs, you can reject the null and conclude that the effect exists in the population. . In other words, they define the rejection ...
Introduction to Hypothesis Testing
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
PDF Step'by-step guide to critiquing research. Part 1: quantitative research
critiquing the literature, critical analysis, reviewing the literature, evaluation and appraisal of the literature which are in essence the same thing (Bassett and Bassett, 2003). Terminology in research can be confusing for the novice research reader where a term like 'random' refers to an organized manner of selecting items or participants ...
How to Write a Hypothesis
Aim for clarity and simplicity in your wording. State direction, if applicable: If your hypothesis involves a directional outcome (e.g., "increase" or "decrease"), make sure to specify this. You also need to think about how you will measure whether or not the outcome moved in the direction you predicted.
Critical Analysis: The Often-Missing Step in Conducting Literature
In using a critical analysis process, the authors will have contributed a level of analysis beyond summary that includes pattern recognition within the body of work studied; thereby, defining the state of the science about the targeted topic. State of the science articles (literature reviews) are very useful to other researchers, educators ...
5. Critically Analyze and Evaluate
Sample Template for Critical Analysis of the Literature. Tip: Read and Annotate PDFs! Opening an article in PDF format in Acrobat Reader will allow you to use "sticky notes" and "highlighting" to make notes on the article without printing it out. Make sure to save the edited file so you don't lose your notes!
A Practical Guide to Writing Quantitative and Qualitative Research
In quantitative research, hypotheses predict the expected relationships among variables.15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable (simple hypothesis) or 2) between two or more independent and dependent variables (complex hypothesis).4,11 Hypotheses may ...
Statistical Hypothesis Testing: How to Calculate Critical Values
Step 1: Identify the test statistic. Before you can figure out the key values, you need to choose the right test statistic for your hypothesis test. The "test statistic" is a number that shows that the data are different from the "null value.". This is a list of test statistics. Which one to use depends on the data or hypothesis being ...
Developing a Hypothesis
The first is to raise a research question, answer that question by conducting a new study, and then offer one or more theories (usually more) to explain or interpret the results. This format works well for applied research questions and for research questions that existing theories do not address. The second way is to describe one or more ...
Analyzing the Text
Remember that simply disagreeing with the material is not considered to be a critical assessment of the material. For example, stating that the sample size is insufficient is not a critical assessment. Describing why the sample size is insufficient for the claims being made in the study would be a critical assessment.
Succeeding in postgraduate study: 4.4 Applying critical and reflective
1 Important points to consider when critically evaluating published research papers. Simple review articles (also referred to as 'narrative' or 'selective' reviews), systematic reviews and meta-analyses provide rapid overviews and 'snapshots' of progress made within a field, summarising a given topic or research area.
Critical Analysis
Critical Analysis Format is as follows: I. Introduction. Provide a brief overview of the text, object, or event being analyzed. Explain the purpose of the analysis and its significance. Provide background information on the context and relevant historical or cultural factors. II.
33 Critical Analysis Examples (2024)
33 Critical Analysis Examples. Critical analysis refers to the ability to examine something in detail in preparation to make an evaluation or judgment. It will involve exploring underlying assumptions, theories, arguments, evidence, logic, biases, contextual factors, and so forth, that could help shed more light on the topic.
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Most of the more than 300 Iranian munitions, the majority of which are believed to have been launched from inside of Iran's territory during a five-hour attack, were intercepted before they got ...
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Adding to an already miserable start to 2024, Boeing has now been accused of routinely ignoring a whistleblower's complaints about the allegedly critically flawed manufacturing process for its ...
As Iranian rockets and drones lit up the sky over Israel, the Middle
Why the next 24 hours are critical The sirens stopped after about five minutes, so I moved back upstairs to the balcony and looked up. Bright balls of light were still flying overhead, signifying ...