Difference of Means Confidence Interval Calculator

Mean A - Mean B
Variance ratio (A/B)
Welch CI
Welch df
Pooled (equal-variance) CI

What this answers

This calculator answers "how large is the true gap between these two group means, and how precisely can I pin that gap down?" rather than only testing whether a difference exists. An interval is more informative than a bare significance test because it shows a plausible range of effect sizes, not just a yes-or-no verdict at one threshold.

How it is calculated

The interval is the observed difference in sample means plus and minus a margin of error. By default, the margin of error uses Welch's standard error, which combines each group's own variance divided by its own sample size rather than assuming a shared pooled variance, together with the Welch-Satterthwaite degrees of freedom. This calculator also reports the classic equal-variance pooled interval as an explicit comparison, since it is only appropriate when the two groups genuinely share similar spread, a condition this page does not assume by default.

Worked example

For Group A of 1, 2, 3 and Group B of 3, 4, 5, both groups share identical variance, so the Welch and pooled intervals coincide exactly for this input: a mean difference of negative 2, on 4 degrees of freedom either way. When the two groups have unequal variance, the Welch and pooled intervals will diverge, and the Welch interval is the one you should trust.

Assumption audit

Calculated from your data: each group's sample size, its own variance, the observed variance ratio between groups, and whether either group falls below the minimum of 2 values the interval requires.

Evidence to review: whether each group's shape looks roughly symmetric rather than heavily skewed, using the Histogram Maker. The Welch interval is more robust to unequal variance than the pooled alternative but both assume the underlying data is not wildly non-normal in small samples.

You must verify: that observations within each group are independent of one another and that the two groups were assigned or sampled in a way that makes them comparable except for whatever difference you are trying to estimate.

Common mistakes

Treating the pooled interval as the default when the two groups have visibly different spread is a common error; the variance ratio shown above flags exactly this situation. A second mistake is reporting only whether the interval crosses zero without looking at the interval's width, since a wide interval that happens to exclude zero is still a weak, imprecise estimate of the true gap.

Limitations

A wider confidence level always widens the interval, trading precision for confidence, all else equal. This calculator cannot detect a biased sampling design or a confound between the groups; a technically correct interval built on an unrepresentative comparison still estimates the wrong quantity.