A paired t-test is used when two measurements belong together, such as before and after scores for the same students or measurements from matched subjects. Instead of comparing two separate group means, it focuses on the difference inside each pair. This matters because pairing can remove person-to-person variation and make a real effect easier to detect.
It is one of the most useful tests for experiments with repeated measurements or carefully matched cases.
The method turns each pair into a single difference, then tests whether the mean difference is far enough from a hypothesized value, usually 0. The test statistic compares the observed mean difference to the standard error of the differences. A large absolute t value gives stronger evidence that the average change is not due to random variation alone.
The test works best when the paired differences are independent from pair to pair and roughly normal, especially for small samples.
Key Facts
- Use a paired t-test when each observation in one condition is naturally linked to one observation in the other condition.
- Compute differences as d = after - before, or choose one direction and use it consistently.
- Mean difference: dbar = sum(d_i) / n.
- Standard error of the mean difference: SE = s_d / sqrt(n).
- Test statistic: t = (dbar - mu_0) / (s_d / sqrt(n)), where mu_0 is usually 0.
- Degrees of freedom: df = n - 1, where n is the number of pairs, not the total number of measurements.
Vocabulary
- Paired t-test
- A hypothesis test that checks whether the mean difference between paired measurements is equal to a specified value.
- Matched pairs
- Observations that are linked because they come from the same subject, related subjects, or deliberately matched units.
- Difference score
- The value found by subtracting one measurement in a pair from the other measurement in the same pair.
- Standard error
- The estimated standard deviation of the sample mean difference, equal to s_d / sqrt(n).
- Degrees of freedom
- The number of independent pieces of information used to estimate variation, equal to n - 1 for a paired t-test.
Common Mistakes to Avoid
- Using an independent two-sample t-test for paired data is wrong because it ignores the link between measurements and can lose statistical power.
- Using the total number of measurements as n is wrong because the paired t-test sample size is the number of pairs, not twice the number of pairs.
- Mixing difference directions is wrong because some differences become after - before while others become before - after, which can cancel real effects.
- Checking normality of the two raw variables instead of the differences is wrong because the paired t-test assumption concerns the distribution of the difference scores.
Practice Questions
- 1 Five students take a quiz before and after tutoring. Their score changes, computed as after - before, are 4, 6, 3, 7, and 5 points. Find dbar, s_d, SE, and the paired t statistic for testing mu_0 = 0.
- 2 A lab measures reaction time before and after caffeine for 8 people. The differences after - before have dbar = -18 ms and s_d = 24 ms. Compute the t statistic and degrees of freedom for a paired t-test of no mean change.
- 3 A researcher compares blood pressure in 20 patients before treatment and after treatment. Explain why a paired t-test is more appropriate than an independent two-sample t-test, and state one condition that should be checked.