Calculator Inputs
Choose the study design first. Then enter raw observations or summaries.
Automatic Test Recommendation
| Study question | Recommended method | Reason |
|---|
| One group versus a known target | One-sample t-test | Tests one sample mean against μ₀. |
| Two unrelated groups | Welch t-test | Works without assuming equal variances. |
| Two unrelated groups with supported equal variances | Pooled t-test | Uses a shared variance estimate. |
| Before and after measurements | Paired t-test | Tests the mean within-pair difference. |
| Severe nonnormality or influential outliers | Nonparametric or permutation method | Reduces reliance on t-test assumptions. |
Consider Wilcoxon signed-rank testing for paired data. Consider Mann–Whitney testing for unrelated groups. A permutation test can directly estimate evidence under exchangeability.
Example Data Table
These values compare two independent teaching methods.
| Student | Method A | Method B | Difference |
|---|
| 1 | 72 | 68 | 4 |
| 2 | 75 | 71 | 4 |
| 3 | 78 | 73 | 5 |
| 4 | 69 | 66 | 3 |
| 5 | 81 | 75 | 6 |
| 6 | 77 | 70 | 7 |
| 7 | 74 | 72 | 2 |
| 8 | 80 | 74 | 6 |
| 9 | 76 | 69 | 7 |
| 10 | 79 | 71 | 8 |
Formula Used
One-sample t-test
Independent pooled t-test
Welch unequal-variance t-test
Paired-samples t-test
Effect sizes
Cohen's d standardizes the estimated difference. Hedges' g applies a small-sample correction. The effect correlation converts t into an association measure.
Understanding Student's T-Tests
A Student's t-test compares means using estimated sampling uncertainty. It helps evaluate whether an observed difference exceeds random variation. The method uses the t distribution. That distribution has heavier tails than the normal curve. Its shape depends on calculated degrees of freedom.
Choosing the Correct Test
Use a one-sample test for one group. Compare its mean with a meaningful target. Use an independent test for unrelated groups. Welch's method is usually the safer default. It does not require equal population variances. Pooled testing assumes both groups share one variance. Use paired testing for linked measurements. Common examples include before and after scores. Matched participants also require paired analysis.
Reading the Main Results
The t statistic measures standardized distance from the null value. Larger absolute values indicate stronger disagreement. The p-value describes evidence under the null hypothesis. Compare that value with the selected alpha. Reject the null when the p-value is smaller. Otherwise, fail to reject the null. This outcome never proves the null hypothesis.
Confidence Intervals and Effects
A confidence interval shows plausible difference values. Intervals crossing zero often match nonsignificant two-tailed results. Cohen's d describes standardized practical magnitude. Hedges' g adjusts that estimate for small samples. Effect labels provide rough guidance only. Context should determine meaningful practical importance.
Checking Assumptions
Observations should match the selected study design. Independent tests require unrelated participants. Paired tests require correctly aligned pairs. Strong outliers can distort means and standard deviations. Small samples need reasonably normal values. Paired tests require normal differences instead. Welch testing handles unequal variances better. It cannot repair dependence or poor sampling.
Using Diagnostics Carefully
This page estimates normality using expected normal scores. It also checks IQR outliers. Independent raw data receive a Brown-Forsythe Levene test. Diagnostics should guide inspection, not replace judgment. Large samples can flag harmless departures. Small samples may miss serious departures. Review plots alongside subject knowledge.
Reporting a T-Test
Report the design, statistic, degrees, and p-value. Add the confidence interval and effect size. State the comparison direction clearly. Describe sampling methods and important limitations. Type I errors reject true nulls. Type II errors sometimes miss real differences during testing. Avoid claiming causation from significance alone. Clear reporting makes statistical conclusions easier to evaluate.
Frequently Asked Questions
What does a Student's t-test calculate?
It compares an estimated mean difference with its standard error. The resulting t statistic determines a p-value using the t distribution.
Should I choose Welch or pooled testing?
Welch testing is generally safer for unrelated groups. Choose pooled testing only when equal variances are defensible and the design supports that assumption.
Can I enter summary statistics?
Yes. Enter sample sizes, means, and standard deviations. Paired summary mode requires the mean and standard deviation of paired differences.
What does failing to reject mean?
It means the evidence did not meet your alpha threshold. It does not prove that the null hypothesis is true.
Why are Welch degrees of freedom decimal?
Welch's formula approximates sampling uncertainty using both variances and sample sizes. That calculation naturally produces fractional degrees of freedom.
What happens when paired lengths differ?
The calculator stops and requests equal lengths. Every first measurement must match one corresponding second measurement.
Does significance prove practical importance?
No. Statistical significance concerns evidence against a null model. Effect sizes and confidence intervals better describe possible practical importance.
Can this calculator establish causation?
No. Causal conclusions require suitable design, assignment, controls, and assumptions. A t-test alone only evaluates a mean difference.
Statistical Disclaimer
This calculator supports education and preliminary analysis. Results depend on correct data, sampling, independence, and assumptions. Verify important work with qualified statistical review and established software.