Calculator Input
Example Data Table
| Case | Sample One | Sample Two | Use Case |
|---|---|---|---|
| 1 | 18 | 16 | Paired before and after check |
| 2 | 21 | 18 | Mean difference review |
| 3 | 19 | 20 | Two sample comparison |
| 4 | 24 | 19 | Evidence strength test |
| 5 | 20 | 17 | Confidence interval estimate |
Formula Used
One sample: t = (x̄ - μ0) / (s / √n), with df = n - 1.
Paired: t = (d̄ - d0) / (sd / √n), using the paired differences.
Equal variance two sample: t = ((x̄1 - x̄2) - Δ0) / (sp √(1/n1 + 1/n2)).
Welch two sample: t = ((x̄1 - x̄2) - Δ0) / √(s1²/n1 + s2²/n2).
Confidence interval: estimate ± t critical × standard error.
How to Use This Calculator
- Select the t test type that matches your study design.
- Choose raw data or summary statistics mode.
- Enter the null mean or null difference.
- Set alpha and confidence level.
- Press calculate to view the t statistic, p value, confidence interval, and decision.
- Use the CSV or PDF buttons to save the output.
T Test Statistics Guide
Purpose of a T Test
A t test checks whether a sample mean, paired mean difference, or two group mean difference is large enough to question a null claim. It is useful when population standard deviation is unknown. It also works well with small samples when assumptions are reasonable. This calculator supports common research, quality, education, business, and clinical comparison tasks.
Choosing the Correct Test
Use a one sample test when one group is compared with a target value. Use a paired test when every observation has a matched partner. Before and after scores are a common example. Use a two sample test when two independent groups are compared. Welch testing is often safer when group variances or sample sizes are different.
Reading the Output
The t statistic measures how many standard errors the estimate sits from the null value. A larger absolute value gives stronger evidence. The p value estimates how unusual the result is under the null claim. If p is below alpha, the calculator rejects the null hypothesis. The confidence interval gives a practical range for the estimated mean or difference.
Why Degrees of Freedom Matter
Degrees of freedom control the shape of the t curve. Smaller samples have heavier tails. That makes critical values larger. As sample size grows, the t curve becomes closer to the normal curve. The calculator computes degrees of freedom automatically. Welch tests may produce decimal degrees of freedom. That is normal and expected.
Practical Interpretation
Statistical significance is not the same as practical importance. Always review the effect size and confidence interval. A tiny difference can become significant with a large sample. A useful difference may fail significance with weak data. Check the study design, outliers, independence, and measurement quality before making final conclusions. Use the downloads for reports, audits, and repeatable documentation.
FAQs
1. What does a t test measure?
It measures whether an observed mean or mean difference is far enough from a null value after accounting for sample variation.
2. When should I use a one sample t test?
Use it when one sample mean must be compared with a known target, standard, benchmark, or expected value.
3. When is a paired t test best?
Use it when observations are matched, such as before and after scores, twins, repeated measures, or paired product tests.
4. What is the Welch t test?
Welch testing compares two independent means without assuming equal variances. It is often preferred for unequal group spreads.
5. What does the p value mean?
The p value shows how unusual the result would be if the null hypothesis were true under the selected alternative.
6. What does alpha control?
Alpha sets the rejection cutoff. A common value is 0.05, but stricter work may use 0.01.
7. Why is the confidence interval useful?
It gives a likely range for the true mean or difference and helps judge practical importance.
8. Can I download my results?
Yes. Use the CSV button for spreadsheets and the PDF button for a simple report copy.