How this calculation works
Preserve the entered pair order and form one difference per matched observation. The sample of differences, rather than the two separate column variances, determines standard error and degrees of freedom.
dᵢ = Aᵢ − Bᵢ; t = (mean d − null difference) / (sample SD of d / sqrt(number of pairs))A worked example
The example differences are 1, 2, 2 and 3. Their mean is 2, sample variance 2/3 and standard error about 0.408248, giving t about 4.898979 with three degrees of freedom.
Good to know
Independent pairs and an appropriate model for the differences are required. No sorting to create matches, missing-pair repair, multiplicity correction or causal inference.
Reference: NIST: paired observations and t inference (opens in a new tab) ↗
Pairing is part of the data
Row positions identify which A and B measurement belong together. Sorting each column independently would replace the actual pairing with an invented one and can change the variance of differences.
Equal sample lengths are necessary but do not prove the study is paired. Use the matched design that generated the observations.
The welch two sample t test calculator adds run Welch’s unequal-variance t test from two independent samples, with noninteger degrees of freedom, chosen p-value tail and a separate two-sided interval. The mcnemar paired binary test calculator adds compare discordant matched binary outcomes using exact or explicitly corrected asymptotic McNemar conventions, with concordant counts kept separate.
Model the differences rather than separate column spreads
The paired standard error uses the sample standard deviation of within-pair differences. Correlation between the two measurements is reflected through those differences.
The relevant normality approximation concerns the difference distribution. Independence is required across pairs, not between the two measurements inside each pair.
Read the sign and interval explicitly
A positive difference means A exceeds B under the entered ordering. Reversing columns reverses the signed estimate. The null difference is a separate hypothesis input.
The confidence interval remains two-sided even if a directional p value is requested. A zero variance of differences is rejected because the ordinary finite t statistic is not defined.
Use the result with the next measurement
The Welch two sample t test calculator assumes independent samples. It does not use the matched covariance captured by a paired-difference analysis.
Independent pairs and an appropriate model for the differences are required. No sorting to create matches, missing-pair repair, multiplicity correction or causal inference.
A few questions, answered
Should I sort the columns first?
No, preserve the actual pair correspondence.
Which variance enters the test?
Sample variance of within-pair differences.
What is n?
The number of complete pairs.