Paired Samples T-Test

What Is the Paired Samples T-Test?

The paired samples t-test (sometimes called the dependent samples t-test) is used when you measure the same group of people — or matched pairs — on two occasions or under two different conditions. Rather than comparing two separate groups, you are comparing two sets of scores that are linked because they come from the same individuals.

Why Do We Need It?

Imagine a clinical psychologist who wants to know whether a six-week mindfulness programme reduces anxiety. She measures the anxiety levels of 25 patients before the programme begins and again after it ends. Because the same patients provide both sets of scores, any differences between the "before" and "after" measurements cannot be explained by differences between people — each person acts as their own control. This is a classic scenario for the paired samples t-test.

If we ignored the pairing and used an independent samples t-test instead, we would treat the before and after scores as though they came from different people. That approach throws away valuable information. The paired design removes the variability that comes from individual differences (some people are naturally more anxious than others), which makes it easier to detect a genuine treatment effect. In statistical terms, pairing increases thepower of the test — its ability to find a real difference when one exists.

How Does It Work?

The first step is to calculate a difference score for every participant by subtracting their score in one condition from their score in the other. If the mindfulness programme has no effect, these difference scores should hover around zero. The paired samples t-test then asks whether the average of those difference scores is significantly different from zero.

The test produces a t-statistic, which is the mean difference divided by the standard error of those differences. A larger absolute t-value indicates that the average change is large relative to how much individual changes vary. Alongside the t-statistic you will get a p-value. If the p-value is less than your chosen significance level (commonly 0.05), you conclude that the difference is statistically significant — meaning it is unlikely to have occurred by chance.

What Does the Result Mean?

A significant result tells you that, on average, scores changed from one measurement to the next by more than you would expect from random variation alone. Going back to our example, if the psychologist finds a significant drop in anxiety scores, she has evidence that the programme is associated with reduced anxiety. A non-significant result means she cannot rule out that any observed change is simply due to chance.

As with all significance tests, the p-value does not tell you how large or meaningful the change is in practical terms. Reporting an effect size — such as Cohen's d calculated from the difference scores — helps convey the magnitude of the effect.

Key Assumptions

• Paired observations: Each score in one condition must be meaningfully linked to exactly one score in the other condition. This usually means the same person is measured twice, but it can also involve matched pairs (for example, twins assigned to different conditions).
• Normality of difference scores: The distribution of the difference scores should be approximately normal. The raw scores themselves do not need to be normal; what matters is that the differences are. With moderate to large sample sizes (around 30 or more pairs), the test is fairly robust to violations of this assumption.
• No extreme outliers: A single very unusual difference score can pull the mean difference strongly in one direction and distort the result. It is good practice to check for outliers before running the test.

When to Use It

Use the paired samples t-test whenever you are comparing two measurements that are linked. Common designs include before-and-after studies, cross-over trials where every participant experiences both treatments, and studies that use matched pairs. If the two sets of scores come from entirely different people with no pairing, you need an independent samples t-test instead. If you have three or more related measurements from the same subjects, you should look at a repeated measures ANOVA.

A Quick Example

A sports scientist measures the resting heart rate of 20 athletes before and after a 12-week training programme. Before training, the average heart rate is 72 beats per minute; after training it is 66. She calculates each athlete's difference score (before minus after), finds that the mean difference is 6 beats per minute, and runs a paired samples t-test. The result is t = 4.38 with a p-value of 0.0003. Because this p-value is well below 0.05, she concludes that the training programme significantly reduced resting heart rate.

The paired samples t-test is a simple yet powerful tool for before-and-after comparisons. By taking advantage of the natural link between paired observations, it gives you a sharper, more sensitive test than comparing two independent groups would.