Calculator
Example Data Table
| Case | Mean 1 | SD 1 | N 1 | Mean 2 | SD 2 | N 2 | Null difference | Typical note |
|---|---|---|---|---|---|---|---|---|
| Exam score study | 82 | 10 | 36 | 76 | 12 | 34 | 0 | Moderate evidence with Welch mode |
| Process timing | 5.4 | 1.2 | 25 | 5.1 | 1.1 | 25 | 0 | Small difference in standard error units |
| Product test | 120 | 18 | 42 | 113 | 16 | 40 | 0 | Check alpha and interval before reporting |
Formula Used
Mean difference: d = x̄1 - x̄2
Centered difference: d0 = d - Δ0
Welch standard error: SE = sqrt(s12 / n1 + s22 / n2)
Pooled standard error: SE = sp sqrt(1 / n1 + 1 / n2)
Standardized test statistic: t = d0 / SE
Welch degrees of freedom: df = (a + b)2 / [a2 / (n1 - 1) + b2 / (n2 - 1)], where a = s12 / n1 and b = s22 / n2.
Confidence interval: d ± tcritical × SE
Cohen's d: d / sp. Hedges' g: Cohen's d × small sample correction.
How to Use This Calculator
- Enter the mean, standard deviation, and size for both samples.
- Enter the hypothesized difference. Use zero for a common equality test.
- Select Welch mode unless equal population variance is a safe assumption.
- Choose the tail direction that matches your research question.
- Set alpha, confidence level, and decimal places.
- Press Calculate. The result appears above the form.
- Use CSV or PDF to save the computed table.
Article
About the Statistic
A two sample standardized test statistic compares two group means. It shows how far the observed difference sits from a null difference. The distance is measured in standard error units. That makes results easier to compare across studies.
This calculator is designed for summary data. You enter both means, both standard deviations, and both sample sizes. You also choose the hypothesized difference. Most studies use zero, but a planned margin can also be tested.
Choosing a Method
Welch mode is useful when spreads or sample sizes differ. It estimates the standard error from both samples and uses adjusted degrees of freedom. Pooled mode assumes equal population variance. It combines the two variances before computing the standard error.
The result includes the observed mean difference, standard error, degrees of freedom, t value, p value, and confidence interval. It also reports Cohen’s d and Hedges’ g. These effect sizes help describe practical size, not only statistical evidence.
Reading the Result
Use the tail option to match your research question. A two tailed test checks for any difference. A right tailed test checks whether the first mean is higher after the null adjustment. A left tailed test checks whether it is lower.
The decision line compares the p value with alpha. A small p value means the sample result would be unusual if the null difference were true. It does not prove causation. It also does not show importance by itself.
Confidence limits show a plausible range for the true mean difference. When a two sided interval misses the null difference, it often agrees with a two tailed decision at the matching alpha level.
Practical Notes
Good inputs matter. Use independent samples unless your design is paired. Use sample standard deviations, not standard errors. Avoid rounded data when possible. Larger samples usually give steadier estimates.
This tool is helpful for exam work, lab reports, A/B tests, product studies, health comparisons, and quality checks. It saves the result as a simple table, then lets you export CSV or PDF for records. Before you rely on a conclusion, review assumptions and context. Outliers can change both means and deviations. Unequal sample quality can also mislead. Treat the calculation as evidence, then combine it with design knowledge and practical judgment. Before reporting results.
FAQs
What does this calculator test?
It tests whether the difference between two independent sample means is unusual compared with a chosen null difference.
Should I use Welch or pooled mode?
Use Welch mode for most practical cases. Use pooled mode only when equal population variance is reasonable and supported by study design.
What is the null difference?
It is the difference expected under the null claim. Zero means the two population means are assumed equal for testing.
What does the p value show?
It shows how unusual the observed statistic would be if the null difference were true under the selected tail rule.
What does a two tailed test mean?
A two tailed test checks for any difference. It counts evidence in both positive and negative directions.
Can I use standard errors instead of deviations?
No. Enter sample standard deviations. The calculator uses them with sample sizes to compute the standard error.
What is Hedges' g?
Hedges' g is a corrected effect size. It adjusts Cohen's d to reduce small sample bias.
Why are degrees of freedom decimal values?
Welch mode uses an adjusted formula. That formula often gives decimal degrees of freedom, which is expected.