A t–Test in statistics is a hypothesis testing method that facilitates the comparison of the results (more specifically, the means) of two scenarios. 

These tests are termed “t-Tests” because they boil the sample data down to one number called the t-value.

Paired t-Tests are a variation of regular t-Tests, and these tests only work on dependent samples. Dependent samples are results of tests on the same subject – could be a person or a thing – but in different circumstances.

In contrast, if independent samples are involved, you would use the Independent Samples t-Test.

While Paired t-Tests can have very useful applications, the tests can only be utilized when specific conditions are met. 

How can you identify the right circumstances to use the Paired t-Test? More importantly, how do you perform a Paired t-Test?

We answer those questions and more in this post.

You can also check our online t-test calculator

When to Use Paired t-Test?

Before you can understand when it is appropriate to use the Paired t-Tests, you must understand what exactly the test is and how it works.

What is Paired t-Test?

A Paired t-Test is a parametric test, which means it is used to estimate unknown parameters. The test is run on dependent samples, which are paired measurements for the same set of items.

Some examples for dependent samples include:

• A blood sugar test taken at two different times.
• A temperature test taken in different weather conditions.
• A hearing test taken for a person’s left and right ears.

As you can tell, a pair of measurements is involved independent samples, which is why they’re sometimes referred to as paired samples. The nature of dependent samples also provides us with insight into the naming of the Paired t-Test.

The goal of using Paired t-Tests is to find statistical evidence that the mean difference between the two tests in the dependent samples is zero.

The Paired t-Test has some other names, including the Dependent t-Test, the Repeated Measures t-Test, the Correlated Pairs t-Test, and the Matched Pairs t-Test.

Before you can perform the test on a dependent sample, you must ensure that it meets the requirements mentioned below.

When Can You Use Paired t-Test?

A Paired T-Test is most commonly to:

• Test the statistical difference in data at two different points in time;
• Test the statistical difference in data in two different conditions;
• Test the statistical difference in two measurements in the data; and
• Test the statistical difference between a matched pair.

It is vital for you to note that Paired t-Tests can only be used to compare the means of strictly two paired units on a continuous outcome. 

It is also critical that the continuous outcome be normally distributed.

How to Determine When to Use It

Besides the two measurements being of the same subject (be it an item or a person), the data that you want to run the test on must meet all of the following requirements:

• The dependent variable must be continuous.
• The paired values must be recorded in different variables.
• The samples/groups in the data must be related. In other words, the subjects in the first group should also be in the second group.
• The sample of data used must be randomly selected from the population.
• Normal distributions of the data should approximately be of the difference between the paired values.
• There mustn’t be any outliers in the difference between the groups.

You could also do the Paired t-Test if the two items being measured have a unique condition.

If you intend to program a Paired t-Test, you must remember to use a variable that represents the difference between the paired values when testing assumptions. Using the original variables to test hypotheses is not recommended since the variables may get altered.

In a scenario where the assumptions are not met, instead of running a Paired t-Test, you could try running the non-parametric Wilcoxon Signed-Ranks Test.

Also, if the sample sizes are too small, you should try applying a non-parametric test that doesn’t assume any normality.

Scenarios Where You Shouldn’t Use the Paired t-Test

There are some scenarios where using a Paired t-Test is not appropriate. These scenarios include:

• If the sample has unpaired data;
• If the comparisons in the data involve more than two groups;
• If the continuous outcome is not normally distributed; or
• If the outcome is ranked or ordinal.

How To Use Paired T-Test

To understand how the Paired t-Test works, you must first understand the hypotheses involved.

Hypotheses

The hypotheses involved can be expressed in two ways – but the two express the same intention and are equivalent mathematically.

One way of representing the null hypothesis and the alternative hypothesis is:

Ho: 1=2

(Which means “the paired means are equal”)

H1: 12

(Which means “the paired means are not equal”)

The hypotheses can also be represented as:

H0: 1–2=0

(Which means “the difference between the means is zero”)

H1: 1–20

(Which means “the difference between the means is not zero”)

In the expressions above, “µ1” represents the population mean of the first variable, and “µ2” represents the population mean of the second variable.

Procedure

Let’s say “x” represents the data before a certain event, and “y” represents the data after the event has occurred. 

Now, to test the null hypothesis and find out if the mean difference is zero, you must:

1. First, calculate the difference between the two observations for every pair of data (di=yi–xi). You must do this while ensuring that you distinguish between positive and negative differences.
2. Next, calculate the mean difference (d).
3. You must then calculate the standard deviation of the mean differences (sd).
4. Use the standard deviation to find the standard error of sd, which is SE (d)=sdn.
5. The next step is to determine the t-statistic, given by the formula T=dSE (d). Note that under this hypothesis, the statistic has a t-distribution with n-1 degrees of freedom.
6. Finally, use the tables of the t-distribution to get the value of the tn-1 distribution. Comparing the value of T to the derived distribution will give the p-value for the paired t-test.

You can learn more on how to interpret t-test results  or see how to perform a step by step T-test on a calculator

Example of Paired t-Test Usage

Paired t-Tests are most commonly used for case-control studies. They are also used for repeated-measures designs.

You could, for example, use a Paired Sample t-Test to evaluate the effectiveness of a company’s training program. To do this, you would first measure the performance of employees before and after completing training, then analyze the means difference with a Paired t-Test.

But the application of the test can be explained with a much simpler example. Let’s say a teacher is trying to find whether tests from the year before and the current year are equally difficult by measuring the difference between scores.

The data is as follows:

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Step #1: Subtract the Y scores from X scores.

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Step #2: Add the values calculated in step #1. 

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Step #3: Square the values derived in step #1.

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Step #4: Add the values calculated in step #3.

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Step #5: Use the formula below to find the t-score

t= (D)ND2–(D)2N (N-1)(N)

 

In the formula above,

D represents the sum of the differences (from step #2)

D2 represents the sum of the squared differences (from step #4)

(D)2 represents the sum of the differences (from step #2) squared

Substituting all the values:

t= -73111131 –(-73)211 (11-1)(11)

 

t= -73111131 –532911 110

 

t= -2.74

Step #6: Subtract 1 from the sample to get the required degrees of freedom. Since there are 11 items in the sample, the degrees of freedom are 10.

Step #7: Determine the p-value from the t-table using the determined degrees of freedom. Since there is no alpha-level specified, we will use 0.05 (or 5%). 

Since the value of df is 10, the t-value will be 2.228.

Step #8: Compare the value found from the t-table to the calculated value. The calculated value is greater than the table value at the alpha level (we can ignore the polarity of the figure).

Furthermore, the p-value is less than the alpha level (p<0.05). Hence, we can reject the null hypothesis in the test.

Conclusion

Doing Paired t-Tests may seem difficult at first, but once you get familiar with the order of the steps, applying the test will become more intuitive to you.

Follow along with the steps above a few times, and you should start feeling comfortable with it soon.

We have various comprehensive calculators that you can use online for free. You can choose from t-test calculator, graphing, matrix, the standard deviation to statistics, and scientific calculators. Check it here.