Confidence Interval for Difference of Means Calculator

It is a range of plausible values for the true difference between two population means, based on sample data. If you repeated the study many times, the stated percentage of intervals (e.g. 95%) would contain the true difference. A CI that does not include zero suggests a statistically significant difference between the groups.

Use the equal variances (pooled) method when you have good reason to believe both populations have the same standard deviation — for example, when a Levene's or F-test for equality of variances is non-significant. When in doubt, the unequal variances (Welch's) method is safer because it is robust even when variances are similar, and it is the default recommendation in most modern statistics software.

The CI is: (x̄₁ − x̄₂) ± t* × SE. For equal variances, the pooled standard deviation is used to compute SE = s_p × √(1/n₁ + 1/n₂), with df = n₁ + n₂ − 2. For unequal variances (Welch's), SE = √(s₁²/n₁ + s₂²/n₂), and df is estimated using the Welch–Satterthwaite equation. t* is the critical value from the t-distribution for the chosen confidence level.

A 95% CI means that if you took 100 random samples and computed the interval each time, approximately 95 of those intervals would contain the true population parameter. It does not mean there is a 95% probability that the true value lies within this specific interval — the true value is fixed; it is the interval that varies across samples.

The confidence level is the long-run proportion of confidence intervals (constructed from repeated sampling) that will contain the true population parameter. Common choices are 90%, 95%, and 99%. A higher confidence level produces a wider interval because more certainty requires a broader range.

The two-tailed p-value tests the null hypothesis that the two population means are equal (difference = 0). If p < 0.05 (for a 95% CI), the CI will not include zero, and you reject the null hypothesis. Both tell the same story: a p-value below your significance threshold corresponds to a CI that excludes zero.

The larger the sample size, the narrower and more precise the confidence interval. As a rule of thumb, n ≥ 30 per group is often considered sufficient for the Central Limit Theorem to apply, but the required size depends on the expected effect size, variance, and desired precision. Power analysis tools can help you determine an appropriate sample size before data collection.

No — this calculator is designed for two independent samples. For paired data (e.g. before-and-after measurements on the same subjects), you should compute the difference for each pair and then use a one-sample confidence interval on those differences. Using the two-sample formula on paired data ignores the correlation between pairs and produces incorrect results.