Calculator
Example Data Table
| Case | Mean 1 | Mean 2 | SD 1 | SD 2 | n1 | n2 | Method | Alpha |
|---|---|---|---|---|---|---|---|---|
| Exam score comparison | 84.2 | 79.5 | 10.8 | 11.4 | 45 | 40 | Welch | 0.05 |
| Known sigma study | 52.1 | 49.8 | 8.2 | 7.9 | 60 | 55 | z | 0.01 |
| Equal variance trial | 128.6 | 121.3 | 18.4 | 17.9 | 32 | 34 | Pooled | 0.05 |
Formula Used
General statistic
Standardized statistic = [(x̄1 − x̄2) − D0] / SE. Here D0 is the null mean difference.
Known standard deviations
SE = √[(σ1² / n1) + (σ2² / n2)]. The calculator reports a z statistic for this method.
Pooled sample standard deviations
sp² = [((n1 − 1)s1²) + ((n2 − 1)s2²)] / (n1 + n2 − 2). SE = sp × √[(1 / n1) + (1 / n2)]. df = n1 + n2 − 2.
Welch unequal variance method
SE = √[(s1² / n1) + (s2² / n2)]. df = [(s1² / n1 + s2² / n2)²] / [((s1² / n1)² / (n1 − 1)) + ((s2² / n2)² / (n2 − 1))].
How to Use This Calculator
Enter the two sample means, standard deviations, and sample sizes.
Enter the null difference. Use zero when testing equal population means.
Select Welch, pooled, or known standard deviation method.
Choose the tail direction and significance level.
Press calculate to view the statistic above the form.
Use CSV or PDF buttons to save the current result.
Understanding the test statistic
A standardized test statistic for μ1 − μ2 compares two independent population means. It turns a raw mean difference into a scale based on sampling error. That scale helps you judge whether the observed difference is unusual under the null claim. A value near zero suggests the sample difference fits the claim. A large positive or negative value suggests stronger evidence against it.
Choosing the right method
This calculator supports common two sample settings. Use the known sigma option when both population standard deviations are known. Use the pooled option when sample standard deviations are used and equal variances are reasonable. Use the Welch option when variances may differ. Welch is often safer for real data because it adjusts the degrees of freedom.
What the result means
The statistic is only one part of inference. The p value shows how extreme the result is for your selected tail. The critical value gives a decision boundary. The confidence interval gives a likely range for the true mean difference. If the interval excludes the null difference, the test often rejects for a matching two tailed level.
Practical notes
Enter summary data from independent samples. Keep units consistent. Use positive standard deviations. Use sample sizes greater than one. The calculator also reports standard error, degrees of freedom, margin of error, and a decision. These items help you explain the test in homework, audit notes, dashboards, and reports.
Using results responsibly
A small p value does not prove a cause. It only measures evidence against the null model. Sampling design, data quality, outliers, and assumptions still matter. Check whether the samples are independent. Review whether the distributions are highly skewed. For small samples, outliers can change the conclusion.
Why standardization helps
Standardization removes the original measurement scale. Dollars, seconds, scores, or weights become comparable through standard errors. This makes reporting easier. It also lets the calculator pair the observed statistic with probability rules for z or t distributions.
Export and review
Use the CSV export for spreadsheets. Use the PDF export for a compact record. The example table shows typical entries. Replace those values with your own study data. Then compare the statistic, p value, and interval before writing the conclusion.
FAQs
What is μ1 − μ2?
It is the difference between two population means. The calculator tests whether the observed sample difference supports a selected null difference.
When should I use Welch method?
Use Welch method when the two samples may have different variances. It is often a safe default for independent samples.
When should I use the pooled method?
Use the pooled method when equal variances are reasonable. It combines both sample variances into one shared variance estimate.
What does the p value mean?
The p value shows how unusual the statistic is under the null hypothesis. Smaller values give stronger evidence against the null claim.
What is the null difference?
The null difference is the claimed value of μ1 − μ2. Use zero when testing whether two population means are equal.
Can I use this for paired data?
No. This calculator is for independent samples. Paired data needs a paired mean difference test instead.
Why are degrees of freedom shown?
Degrees of freedom help define the t distribution. They affect p values, critical values, and confidence intervals.
Why do z and t results differ?
The z method uses known population standard deviations. The t methods use sample standard deviations, so they include extra uncertainty.