How to tell if scores changed between two measurements

01What the paired t-test does

Many studies measure the same people twice: a knowledge test before and after a course, blood pressure before and after a treatment, a wellbeing score at the start and end of a term. Because both measurements come from the same person, they are linked. Someone who scores high the first time will tend to score high the second time as well.

The paired t-test uses that link. Rather than comparing two group averages, it calculates the change for each person and then tests whether the average change is different from zero. This removes the differences between people from the comparison, which usually makes the test more sensitive than an independent t-test would be on the same data.

02When to use it

Use it when the outcome is numeric and you have two measurements per person, or two measurements on matched pairs such as twins, spouses, or matched schools. Each row of your dataset should hold both measurements for the same case.

A worked example: 60 teachers complete a teaching-skills assessment before and after a training course. The average score rises from 52 to 58. The paired t-test checks whether an average gain of 6 points is larger than you would expect from measurement noise and sampling variation.

For three or more time points, use repeated-measures ANOVA. For two groups that are unrelated, use the independent t-test.

03Assumptions

The differences between the pairs, not the original scores, should be roughly normally distributed. Pairs should be independent of one another. With fewer than about 20 pairs or clearly skewed differences, the Wilcoxon signed-rank test is a more dependable option.

Only complete pairs are used. Anyone missing either measurement drops out of the test, so check how many pairs remain, particularly in studies with attrition between the first and second measurement.

04Reading the result

The mean difference

The average change across all pairs is the headline number. The module subtracts Variable 2 from Variable 1, so with a before score first and an after score second, a rise appears as a negative number. Reading the direction wrongly is the most common error with this test.

The confidence interval

The 95% interval around the mean difference shows the plausible range for the true average change. An interval that excludes zero corresponds to a significant result at the 5% level, and its width shows how precisely the change has been estimated.

Effect size

The output does not include an effect size, but you can work out Cohen's d from two numbers it does report: divide the mean difference by the standard deviation of the differences. Below 0.2 is negligible, 0.2 to 0.5 small, 0.5 to 0.8 medium, and above 0.8 large. It helps judge whether a statistically significant gain is large enough to matter.

05How to run it in AnalyZ Solutions

Open Inferential Analysis, choose Compare Means, then Paired t-test.

1. Select Variable 1
   Choose the first measurement, usually the earlier or pre-test column.
2. Select Variable 2
   Choose the second measurement, usually the later or post-test column.
3. Choose the tails
   Use two-tailed unless you decided in advance that Variable 1 would be larger. The one-tailed option tests only whether Variable 1 is greater than Variable 2.
4. Filter if required
   For example, to test the change for women only, or for one site.
5. Click Run
   The output includes descriptives for both variables, the mean and standard deviation of the differences, the t statistic, degrees of freedom, significance, and the 95% confidence interval, plus a plain-language interpretation.

06Interpreting the result

The figures below are invented to show the steps, and they follow the results tables in AnalyZ Solutions, with a few columns left out to save space. Suppose 60 teachers took a skills assessment before and after a training course. Variable 1 is the before score and Variable 2 is the after score.

Reading it step by step

1. Read the sign carefully
   The module subtracts Variable 2 from Variable 1, which here means before minus after. A mean difference of -6.0 therefore means scores were 6 points higher after the course. Confirm the order of your variables before describing a rise or a fall.
2. Check significance
   The p-value is below 0.001 and carries three stars, so the change is significant at the 0.1% level.
3. Read the confidence interval
   The interval for the mean difference runs from -8.583 to -3.417. Zero is well outside it, so the change is clearly not zero, and the true gain is probably between about 3.4 and 8.6 points.
4. Judge the size
   The output does not include an effect size, but you can work one out. Divide the mean difference by the SD of the differences: 6.0 divided by 10.0 gives a Cohen's d of 0.60, a medium effect.

In plain words

Assessment scores rose by 6 points on average after the course, from 52 to 58. The gain is medium in size and significant at the 0.1% level, and the true average gain is likely to lie between about 3.4 and 8.6 points. Because there is no comparison group, the test shows that scores changed and cannot show that the course caused the change.

07Common mistakes

Using an independent test on paired data. This ignores the link between the two measurements and wastes statistical power.

Not checking the sign. Confirm that the difference is Variable 1 minus Variable 2 before describing an increase or a decrease.

Ignoring attrition. If the people who dropped out differ from those who stayed, the pairs that remain may not represent your full sample.

Testing many pairs of time points separately. With three or more waves, one repeated-measures test is better than several paired tests.