An interval for a difference
Two group means invite one question: how big is the gap between the populations? The confidence interval for μ₁ − μ₂ answers it directly: the plausible range for the true difference, built from the observed gap plus or minus a margin. This calculator uses Welch's version, which does not pretend the two groups share a variance, and computes its degrees of freedom from the Welch–Satterthwaite formula.
Two classes' quiz marks: group 1 (78…91, mean 84.625) and group 2 (72…80, mean 75.75). The gap is 8.875 marks; the 95% Welch interval runs 4.53 to 13.22. Zero is nowhere near it: at 95% confidence the first class's true mean is higher, by somewhere between roughly 4.5 and 13 marks.
The zero check
The quickest read of any difference interval: does it contain 0? If it does, a zero true difference is plausible at that confidence level (and the matching two-tailed test fails to reject at the corresponding α). If it does not, the difference is statistically significant, and the interval tells you far more than the test alone, because it also bounds how large the difference plausibly is. "Significant" differences that could be as small as 0.2 units often should not drive decisions; the interval shows you that at a glance.
Judging two groups by whether their separate intervals overlap. Overlap of individual 95% intervals is a stricter, wrong test. Two means can differ significantly while their intervals still overlap. Always build the interval for the difference itself.
Frequently asked questions
Why Welch instead of pooled?
Pooled assumes equal population variances; when that assumption is wrong, its interval is wrong too. Welch costs almost nothing when variances are equal and protects you when they are not, so it is the safer default.
The interval is entirely negative. What does that mean?
Group 2's mean is higher: the interval estimates μ₁ − μ₂, so negative values say the second group leads, by the amounts shown.