Calculator
Example Data Table
| Case | Sample 1 | Sample 2 | Method | Use |
|---|---|---|---|---|
| Training scores | mean 82.4, sd 7.1, n 35 | mean 78.9, sd 8.3, n 32 | Welch t | Compare average score change |
| Before and after | Raw before list | Raw after list | Paired t | Use matched rows |
| Conversion test | 48 successes from 120 | 37 successes from 115 | Two proportion z | Compare rates |
Formula Used
Welch two sample means: t = ((x̄1 - x̄2) - d0) / sqrt(s1²/n1 + s2²/n2). Degrees of freedom use the Welch-Satterthwaite formula.
Pooled two sample means: sp² = ((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2). Then t = ((x̄1 - x̄2) - d0) / (sp sqrt(1/n1 + 1/n2)).
Known deviation means: z = ((x̄1 - x̄2) - d0) / sqrt(σ1²/n1 + σ2²/n2).
Paired means: t = (d̄ - d0) / (sd / sqrt(n)). The degrees of freedom equal n - 1.
Two proportions: z = ((p1 - p2) - d0) / SE. A pooled SE is used when d0 equals zero.
How To Use This Calculator
Select the test family first. Choose summary input or raw values for mean tests. Use raw paired lists only when each row is matched. Enter alpha, confidence level, and the null difference. Choose the alternative direction. Press Calculate. Review the p value, interval, effect size, assumption note, and decision. Use CSV or PDF downloads for records.
Two Sample Hypothesis Testing Guide
Purpose
A two sample hypothesis test compares two independent groups, paired measurements, or two proportions. It asks whether an observed difference is large enough to reject a stated null difference. This calculator keeps the workflow clear. You enter sample summaries or raw values. The page then computes the statistic, degrees of freedom, p value, confidence interval, and final decision.
What The Result Means
The null hypothesis usually says that both populations have the same mean or proportion. The alternative hypothesis describes the direction you want to test. A two tailed test checks any difference. A left tailed test checks whether sample one is smaller. A right tailed test checks whether sample one is larger. The p value measures how unusual the result is when the null claim is true. If it is below alpha, the test rejects the null claim.
Choosing The Correct Method
Use Welch testing when two independent means may have unequal variances. It is often the safest default. Use pooled testing only when equal variance is reasonable. Use known variance testing when population standard deviations are supplied from a trusted source. Use paired testing when each value in sample one is naturally matched with a value in sample two. Use the proportion test when your data are counts of successes and trials.
Interpreting Practical Size
Statistical significance does not always mean practical importance. Review the difference estimate and confidence interval. For means, the calculator also reports Cohen's d and Hedges' g when possible. These effect sizes express the difference in standard deviation units. A narrow interval gives more precise information. A wide interval suggests more data may be needed.
Good Data Practice
Check sample independence before using independent methods. Inspect raw data for impossible values, missing entries, and extreme outliers. Use paired mode only when both lists are aligned row by row. For proportion testing, successes must be between zero and the trial count. Report alpha, test type, assumption choice, and confidence level with the result. Export files help keep this record consistent. The calculator supports classroom examples, research screening, business experiments, and quality checks. It should support decisions, not replace subject knowledge. Use the conclusion beside context, design limits, and uncertainty notes.
FAQs
What is a two sample hypothesis test?
It is a statistical test that compares two groups. It checks whether their means or proportions differ beyond random sampling variation.
When should I use Welch testing?
Use Welch testing when independent samples may have different variances. It is a good default for many real data sets.
When is pooled testing suitable?
Pooled testing is suitable when equal population variance is reasonable. Use subject knowledge, design details, or variance checks before selecting it.
What does the p value show?
The p value shows how unusual the observed statistic is under the null hypothesis. Smaller values give stronger evidence against that claim.
What is alpha?
Alpha is the chosen significance level. Common values are 0.05 and 0.01. The calculator compares the p value with alpha.
Can I use raw data?
Yes. Select raw entry for mean tests. Enter comma separated values. For paired testing, both lists must have matching order and length.
Does significance prove importance?
No. A result can be statistically significant but practically small. Always review the estimated difference, confidence interval, and effect size.
What should I export?
Export the result table when reporting work. It records the method, statistic, p value, interval, assumptions, alpha, and decision.