Estimate mean differences from two samples precisely. Review pooled, Welch, margins, confidence levels, and assumptions. Plot outputs, export reports, and explain interval decisions clearly.
Enter summary statistics for two independent samples. The page stays in a single-column flow. The calculator fields adapt across screen sizes.
This example uses independent samples with summary statistics only.
| Group | Mean | Standard Deviation | Sample Size | Method | Confidence Level | Estimated Interval |
|---|---|---|---|---|---|---|
| Training Team | 84.2 | 10.6 | 36 | Welch | 95% | [0.74, 10.66] |
| Comparison Team | 78.5 | 11.1 | 40 | Welch | 95% |
Difference in means: x̄₁ - x̄₂
Welch interval: (x̄₁ - x̄₂) ± t* × √[(s₁² / n₁) + (s₂² / n₂)]
Welch degrees of freedom: [(s₁² / n₁ + s₂² / n₂)²] / [((s₁² / n₁)² / (n₁ - 1)) + ((s₂² / n₂)² / (n₂ - 1))]
Pooled standard deviation: sp = √{[((n₁ - 1)s₁²) + ((n₂ - 1)s₂²)] / (n₁ + n₂ - 2)}
Pooled interval: (x̄₁ - x̄₂) ± t* × sp × √[(1 / n₁) + (1 / n₂)]
Margin of error: t* × standard error
Use Welch when variances may differ. Use pooled when equal-variance assumptions are reasonable.
1. Enter labels for both independent samples.
2. Provide each sample mean, standard deviation, and size.
3. Choose a confidence level, such as 90%, 95%, or 99%.
4. Select Welch for unequal variances or pooled for similar variances.
5. Choose how many decimal places you want in the output.
6. Click Calculate Interval to see the result above the form.
7. Review the interval, standard error, margin, graph, and interpretation.
8. Download a CSV summary or PDF report when needed.
It estimates a plausible range for the true difference between two population means. The calculator reports sample one minus sample two, so direction matters.
Use Welch when group variances or sample sizes differ noticeably. It is more flexible and usually the safer default for independent samples.
The pooled method fits best when both populations have similar variances and the independent-samples assumption is reasonable. It uses a shared variance estimate.
If zero is inside the interval, the observed difference may be small enough to be consistent with no real mean difference at that confidence level.
This version uses summary statistics only. Convert raw data into group means, standard deviations, and sample sizes before entering values.
Higher confidence uses a larger critical value. That creates a wider margin of error and therefore a wider confidence interval.
Standard deviation and t-based uncertainty need at least two observations in each sample. Smaller values cannot support a valid interval estimate.
The samples should be independent, measurements should be quantitative, and the data should be reasonably normal or supported by adequate sample sizes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.