Calculator Inputs
Enter raw values or summary statistics. Use independent samples for separate groups and paired samples for matched observations.
Example Data Table
This example shows two small independent samples, such as average scores from two training groups.
| Observation | Sample 1 | Sample 2 |
|---|---|---|
| 1 | 14.2 | 13.5 |
| 2 | 15.1 | 13.9 |
| 3 | 16.0 | 14.1 |
| 4 | 15.8 | 14.8 |
| 5 | 14.9 | 13.7 |
| 6 | 15.4 | 14.0 |
Formula Used
For two independent samples, the observed difference is the first mean minus the second mean.
Difference = x̄₁ - x̄₂
Welch standard error allows unequal variances across groups.
SE = √[(s₁² / n₁) + (s₂² / n₂)]
Welch degrees of freedom use the Satterthwaite approximation.
df = [(s₁² / n₁) + (s₂² / n₂)]² / {[(s₁² / n₁)² / (n₁ - 1)] + [(s₂² / n₂)² / (n₂ - 1)]}
Pooled analysis assumes equal variances in both groups.
sₚ² = [((n₁ - 1)s₁²) + ((n₂ - 1)s₂²)] / (n₁ + n₂ - 2)
SE = sₚ √[(1 / n₁) + (1 / n₂)]
Paired analysis studies within pair differences.
d̄ = average(xᵢ - yᵢ)
SE = s_d / √n
t = (d̄ - μ₀) / SE
The test statistic and confidence interval are:
t = (Observed Difference - Hypothesized Difference) / SE
CI = Observed Difference ± t* × SE
Effect sizes summarize practical importance. Cohen d and Hedges g are shown for independent groups, while dz is shown for paired samples.
How to Use This Calculator
- Choose raw data when you know each observation. Choose summary statistics when you only know means, standard deviations, and sample sizes.
- Select independent samples for separate groups. Select paired samples for before-after studies or matched observations.
- Pick Welch when group variances may differ. Pick pooled only when an equal variance assumption is reasonable.
- Set the confidence level and alternative hypothesis that match your study question.
- Enter a hypothesized difference, usually zero, then submit the form.
- Review the difference, p value, interval, effect sizes, sample summary, and graph before exporting the report.
Frequently Asked Questions
1. When should I use Welch instead of pooled?
Use Welch when group variances or sample sizes differ noticeably. It is more robust and is usually the safer default when you are unsure about equal variance assumptions.
2. What does the p value tell me?
The p value measures how compatible your data are with the null hypothesis. Smaller values suggest stronger evidence that the true mean difference is not the hypothesized value.
3. Why is the confidence interval important?
A confidence interval gives a plausible range for the population mean difference. It shows estimate precision and helps you judge practical importance, not just statistical significance.
4. What is the difference between independent and paired samples?
Independent samples come from separate groups. Paired samples come from linked observations, such as before-and-after measurements or matched participants, where each value in one sample corresponds to one value in the other.
5. What does Cohen d mean here?
Cohen d expresses the mean difference in standard deviation units. It helps compare practical impact across studies, even when measurements use different original scales.
6. Can I use summary statistics instead of raw data?
Yes. Summary mode works when you know each group mean, standard deviation, and sample size, or when you know the paired difference summary statistics.
7. Why might a result be non significant?
A non significant result may reflect a small difference, high variability, or limited sample size. It does not always mean the groups are identical.
8. What should I export in a report?
Export the method, sample summaries, observed difference, t statistic, degrees of freedom, p value, confidence interval, and effect size so readers can verify your interpretation.