Report a Paired Mean Difference

Make the pair explicit

A paired analysis starts with a defensible match between two measurements on the same observational unit. Specify whether the difference is follow-up minus baseline, method A minus method B, or another direction. Keep identifiers and measurement times long enough to verify the match before generating the summary statistics.

The R t-test documentation supports paired observations and removes missing values in pairs. That software behavior is a reminder to report how many complete pairs remain, rather than using the total number of measurements or the enrollment count as n.

Summarize differences rather than separate occasions

For each eligible pair, compute its signed difference. The inputs needed here are the number of complete pairs, the mean difference, and the sample standard deviation of those differences. The baseline SD and follow-up SD cannot be substituted individually. Their relationship to the difference SD also depends on within-pair correlation.

The paired t interval is the ordinary one-sample mean interval applied to the difference variable. Its standard error is SD of differences divided by sqrt(n), with n−1 degrees of freedom. The NIST mean confidence-limit reference gives the Student t construction.

Check a fictional summary

Suppose 25 complete pairs have mean difference 4 units and sample SD of differences 10 units. The SE is 10/5 = 2. With 24 degrees of freedom, the two-sided 95% critical value is approximately 2.06390. The interval is 4 ± 4.12780, or approximately −0.1278 to 8.1278 units.

The paired mean tool uses a Student t critical value rather than replacing it with 1.96. This distinction matters most for small numbers of pairs. Reversing the subtraction gives mean −4 and interval approximately −8.1278 to 0.1278; the scientific interpretation must reverse with it.

Examine what the summary cannot establish

The exact small-sample t model assumes independent, normally distributed differences. It does not require the two original marginal distributions to have equal variances. Plot the differences and investigate extreme values, recording errors, and dependence across participants or clusters. A summary-only tool cannot perform these checks.

Use the paired mean worksheet to record matching rules, difference direction, completeness, and the reason for selecting the interval. Do not interpret a confidence interval for the mean difference as the range expected for individual changes or as limits of agreement between measurement methods.

Finally, describe the complete-pair population. If missing follow-up is related to outcomes, the available pairs may not represent all enrolled participants. A mathematically correct t interval does not repair that selection problem or establish that an observed before–after difference was caused by an intervention.

Sources checked on 5 November 2026; all numerical examples are fictional and independently recomputed.