Enter your sample summary values
Sample summary example
| Group | Mean | Standard Deviation | Sample Size |
|---|---|---|---|
| Training Method A | 82.0 | 9.5 | 36 |
| Training Method B | 76.0 | 8.8 | 34 |
Use this sample summary to test whether two independent group means differ meaningfully. The calculator estimates the difference, interval, test statistic, and p value.
Core formulas for comparing two means
Welch's t-test
Difference: Δ = x̄1 - x̄2
Standard error: SE = √(s12/n1 + s22/n2)
Test statistic: t = (x̄1 - x̄2 - Δ0) / SE
Degrees of freedom: df = (A + B)2 / [(A2/(n1-1)) + (B2/(n2-1))], where A = s12/n1 and B = s22/n2.
Pooled two-sample t-test
Pooled variance: sp2 = [((n1-1)s12) + ((n2-1)s22)] / (n1 + n2 - 2)
Standard error: SE = sp √(1/n1 + 1/n2)
Degrees of freedom: df = n1 + n2 - 2
Confidence interval
(x̄1 - x̄2) ± t* × SE
Steps for accurate comparisons
- Enter a label, mean, standard deviation, and sample size for each group.
- Choose Welch's method when variances may differ between groups.
- Choose the pooled method when equal variances are defensible.
- Set the null difference, significance level, and confidence level.
- Press Compare Means to view the result above the form.
- Review the difference, p value, confidence interval, effect size, and graph.
- Use the CSV or PDF button to export your results.
Frequently asked questions
1. What does this calculator compare?
It compares the means of two independent groups using summary statistics. You enter means, standard deviations, and sample sizes instead of raw observations.
2. When should I use Welch's test?
Use Welch's test when group variances may be different or sample sizes are unequal. It is usually the safer default for independent samples.
3. When is the pooled method appropriate?
Use the pooled method when both groups have similar variances and independent observations. It assumes one shared population variance for both groups.
4. What does the p value tell me?
The p value measures how compatible your observed difference is with the null hypothesis. Smaller values indicate stronger evidence against that null assumption.
5. What does the confidence interval show?
It gives a plausible range for the true difference between population means. If the interval excludes the null difference, the result is statistically significant at the matching level.
6. Can I run one-sided tests?
Yes. Choose whether the first group is greater, less, or simply different. The calculator adjusts the p value to match your selected hypothesis direction.
7. What if one standard deviation is zero?
A zero standard deviation means every observed value in that group is identical. The test can still work unless both groups create a zero standard error.
8. Is significance the same as importance?
No. Statistical significance reflects evidence strength, while practical importance reflects real-world impact. Always review effect size, interval width, and context together.