Means: Correlated, Paired, and Cross-Over Designs
Calculate a two-sided confidence interval for the mean difference in paired observations using the Student’s t distribution. Enter the observed mean paired difference, standard deviation of the paired differences, number of pairs, and confidence level. Runs entirely in your browser.
For paired observations, the analysis is performed on the difference within each pair. Let di denote the difference for pair i. The sample mean of these differences estimates the population mean paired difference, μD.
When the standard deviation of the paired differences is estimated from the sample and the differences are approximately normally distributed, the two-sided confidence interval is based on the Student’s t distribution:
Here, d̄ is the observed mean paired difference, sd is the sample standard deviation of the paired differences, n is the number of complete pairs, and the degrees of freedom are n − 1. The standard error is sd/√n, and the margin of error is the t critical value multiplied by that standard error.
The resulting interval gives a range of values compatible with the observed paired data under the stated confidence procedure. The confidence level describes the long-run coverage of intervals constructed using this same procedure, rather than the probability that a particular fixed parameter is inside the already-calculated interval.
The procedure assumes that the distribution of the paired differences is approximately normal, particularly when the number of pairs is small. Pairing should also be appropriate for the study design, with the two measurements in each pair meaningfully related.
The PASS documentation gives a validation example using 42 pairs, a standard deviation of paired differences of 9.6, and a two-sided 95% confidence interval. PASS reports an actual distance from the mean difference to either confidence limit of 2.99157. Using a constructed mean paired difference of 10 for the interval calculation gives:
The calculator retains full numerical precision internally and rounds only the displayed results.
NCSS, LLC. PASS Sample Size Software — Confidence Intervals for Paired Means,
Chapter 496. PASS documentation. The documentation specifies the
Student’s t-based confidence interval for a paired mean difference
when the variance is unknown and provides the validation example used above.
PASS Confidence Intervals for Paired Means documentation
Hahn, G. J., & Meeker, W. Q. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley & Sons, New York. This reference is cited by the PASS documentation for the statistical-interval methodology.