How this calculation works
Compute sample means and unbiased sample variances separately. Use the Welch standard error and Satterthwaite degrees of freedom rather than pooling variances. Tail selection changes the hypothesis p value; the displayed confidence interval remains explicitly two-sided.
t = ((mean A − mean B) − null difference) / sqrt(sA²/nA + sB²/nB)A worked example
Samples 1–5 and 2–6 have means 3 and 4 and equal sample variance 2.5. Their mean difference is −1, standard error 1 and Welch t statistic −1 with eight degrees of freedom.
Good to know
Independent-sample inference only. No causal, equivalence or practical-significance conclusion, imputation or multiple-testing correction. Zero standard error is rejected.
Reference: NIST: unequal-variance two-sample t test (opens in a new tab) ↗
Keep independence separate from unequal variance
Welch’s method allows the two population variances to differ; it does not remove the need for independent observations and an appropriate sampling design. A before-and-after study on the same subjects is a different design.
Sample variances use n minus one. Each sample can have a different length, and the degrees of freedom usually need not be a whole number.
The paired t test differences calculator adds analyze matched measurements through their within-pair differences, showing t, degrees of freedom, p value, an interval and a complete paired ledger. The correlated measurement uncertainty budget calculator adds propagate standard or symmetric half-width uncertainty sources with signed sensitivities and correlations, checking the matrix before reporting expanded uncertainty.
A one-sided p value does not relabel the interval
The greater alternative concerns A minus B being greater than the entered null difference. Reversing the samples reverses that direction. Two-sided evidence is unchanged only when the null difference is zero or its sign is reversed too.
The interval shown here is two-sided at the selected confidence level. It is not silently converted into a one-sided bound when a one-sided hypothesis is selected.
Separate effect estimate from null hypothesis
Changing the null difference changes the t statistic and p value. It does not change the observed mean difference or its standard-error-based confidence interval.
A small p value is not the probability the null is true, proof of causation or a guarantee that the difference matters in practice. Interpret the magnitude and interval with the study context.
Use the result with the next measurement
The paired t test differences calculator uses within-pair differences. Choosing Welch because the two columns have similar lengths does not make paired observations independent.
Independent-sample inference only. No causal, equivalence or practical-significance conclusion, imputation or multiple-testing correction. Zero standard error is rejected.
A few questions, answered
Are variances pooled?
No, Welch uses separate variance contributions.
Can degrees of freedom be fractional?
Yes, Satterthwaite degrees of freedom commonly are.
Does a one-sided test change the displayed interval?
No, it stays labeled two-sided.