Paired Samples t-Test

Use this test when you have two measurements from the same individuals, such as before and after a treatment, or measurements from matched pairs. For example, comparing patient blood pressure before and after medication, or comparing left-eye and right-eye measurements from the same subjects.

• The differences between paired observations are continuous.
• The differences are approximately normally distributed (or the sample size is large enough for the Central Limit Theorem to apply).
• Observations are paired: each value in one group corresponds to exactly one value in the other group.
• The pairs are independent of each other.

• If Pr > |t| is less than your alpha level, the mean difference between conditions is statistically significant.
• A positive mean difference indicates that values in the first condition are, on average, higher than in the second condition (and vice versa).
• The confidence interval for the mean difference shows the range of plausible true differences. If it excludes zero, the result is significant at that confidence level.
• The paired design is more powerful than the independent design because it controls for between-subject variability.

• Treating paired data as independent (using an independent t-test instead) throws away the pairing information and usually reduces statistical power.
• Outliers in the differences can strongly affect the result. Consider the Wilcoxon signed-rank test if differences are non-normal or contain extreme values.
• Carry-over effects in repeated measures designs (e.g., learning or fatigue) can bias the result. Counterbalancing helps address this.

1. Compute the difference for each pair (d_i = x1_i - x2_i).
2. Calculate the mean and standard deviation of these differences.
3. Compute the t-statistic as the mean difference divided by the standard error of the differences (SD / sqrt(N)).
4. Compare the t-statistic to the t-distribution with N-1 degrees of freedom to obtain the p-value.