Paired T Test Calculator — Free & Interactive

The Paired-Sample t-Test. Why should before-and-after data be analyzed as differences?

A paired sample t-test evaluates the mean of within-pair differences when two measurements are deliberately linked, such as before and after values on the same participant or measurements on matched units. The pairing is part of the design, not a formatting choice. The test reduces each pair to one difference and applies a one sample t-test to those differences, so the relevant normality and independence assumptions concern the differences rather than the two raw columns separately.

With differences d one through d n, the statistic t divides d bar minus mu d by s d divided by the square root of n, with n minus 1 degrees of freedom. Under independent normally distributed differences and H naught, this statistic follows a Student t distribution. Pairing can improve precision when within-pair measurements are positively correlated because between-unit variation cancels in the subtraction. Poor or arbitrary matching can instead add noise and reduce interpretability. The p-value is a tail probability under the null mean-difference model. It does not give the probability that treatment caused the change, that every subject improved, or that the null is true. Use the test when every value in Sample 1 has exactly one meaningful partner in Sample 2 and the research question concerns the average signed change or matched difference. Pair identities must be preserved in the same row order. Report the mean difference, its standard deviation, n pairs, t, df, and p. The sign follows the input order. A positive average difference can coexist with some negative individual differences, so do not generalize the mean result to every pair. Rejecting H naught supports evidence that the population mean difference follows the stated alternative, conditional on the design. Failing to reject leaves that mean difference uncertain. The current tool does not provide a confidence interval or a distribution-free alternative.

Six participants have systolic measurements before a program of 142, 135, 151, 146, 139, 144 and after measurements of 137, 132, 145, 143, 136, 139. The plan defines d equals before minus after and tests H naught, mu d equals 0 against a two-sided alternative at alpha equals 0.05. Enter the before values as Sample 1 and after values in the matching order as Sample 2. The paired differences, in millimeters of mercury, are five, three, six, three, three, and five. Their mean is about four point one seven millimeters of mercury. The difference standard deviation is about one point three three, so its standard error is about zero point five four three. A t statistic of seven point six eight with five degrees of freedom gives a two tailed probability near zero point zero zero zero six. Inspect whether any pair was misaligned and whether the measurement protocol was comparable across occasions. A numerically strong t result cannot correct swapped rows or systematic time effects.

The test addresses whether the population’s average before-minus-after change differs from zero under this paired design. Evidence of a positive difference would be consistent with lower after values, but causality still depends on controls for regression to the mean, concurrent treatment, measurement drift, and the absence of a comparison group.

Try it free at Distri Scope dot com.