StatReason

Paired Difference Confidence Interval Calculator

Estimate the average change between two measurements taken on the same subjects or matched pairs, with a confidence interval built directly from the paired differences.

What this answers

This calculator answers "by how much did things actually change, and how precisely can I pin that change down?" for matched data, such as the same people measured before and after an intervention, or naturally paired items such as left and right. It never pairs rows by position without you confirming that pairing is meaningful.

How it is calculated

Each pair's difference (after minus before) is computed first, and every subsequent step works only with that list of differences, exactly as a paired t-test does internally. The interval is the mean difference plus and minus a t critical value (based on n minus 1 degrees of freedom and your confidence level) multiplied by the standard error of the mean difference. Using the differences directly, rather than treating before and after as two independent groups, is what makes this method appropriate for matched data.

Worked example

For before values of 2, 4, and 6 paired with after values of 1, 3, and 5, every pair changes by exactly negative 1, so the standard deviation of the differences is 0 and the interval collapses to a single point: negative 1 to negative 1. That degenerate result reflects perfectly consistent change across pairs, not an error in the calculation; real data with any variation in the size of the change will produce a proper interval with positive width.

Assumption audit

Calculated from your data: the number of pairs, whether both lists have matching lengths, the mean and standard deviation of the differences, and whether the differences are constant (giving a zero-width interval).
Evidence to review: whether the distribution of differences looks roughly symmetric rather than heavily skewed, since the t interval on differences is more robust with a roughly symmetric spread and less reliable with a small, lopsided sample.
You must verify: that each row genuinely represents a matched pair, not an arbitrary row-position alignment of two otherwise unrelated lists, and that the pairing itself was not influenced by the outcome being measured.

Common mistakes

The most common error is pasting two independent groups into this calculator instead of genuinely matched pairs; if there is no real subject-to-subject correspondence between row 1 of "before" and row 1 of "after," use the Difference of Means Confidence Interval Calculator for independent groups instead. A second mistake is reversing which list is "before" and which is "after," which only flips the sign of the reported change but is easy to misread if not checked.

Limitations

This interval only describes the observed pairs; it does not correct for a poorly chosen pairing scheme or for changes that would have happened anyway without any intervention. A wider confidence level always produces a wider interval, trading precision for confidence.