T-Test Calculator

About the T-Test Calculator

The t-test is a standard way to compare means when the population standard deviation is unknown. It lets you test a sample mean against a reference value, compare two independent groups, or compare paired measurements from the same subjects.

This calculator supports one-sample, pooled two-sample, Welch's, and paired t-tests. Enter summary statistics, choose the test type and tail direction, and it returns the t statistic, p-value, degrees of freedom, effect size, and a confidence interval for the mean difference.

That makes it useful whenever you want a compact hypothesis test for averages rather than a model with many predictors.

When This Page Helps

T-tests are often the quickest way to answer a mean-comparison question, but the details change depending on whether the groups are paired, whether variances are equal, and whether you need a one- or two-tailed result. Keeping those choices in one place reduces the chance of mixing formulas or reading the wrong degrees of freedom.

How to Use the Inputs

1. Select the t-test variant: one-sample, two-sample, Welch's, or paired.
2. Choose tail direction: two-tailed, right-tailed, or left-tailed.
3. Enter the summary statistics for your data (means, standard deviations, sample sizes).
4. Set your significance level alpha (default 0.05).
5. Review the t statistic, p-value, and reject/fail-to-reject decision.
6. Check Cohen's d effect size to assess practical significance.
7. Examine the confidence interval for the mean difference.

Example Calculation

Tips & Best Practices

• Use Welch's t-test by default for two-sample comparisons — it's robust to unequal variances and doesn't lose much power when variances are equal.
• The paired t-test is much more powerful than the two-sample t-test when study designs allow pairing, because it eliminates between-subject variability.
• Cohen's d benchmarks: 0.2 = small, 0.5 = medium, 0.8 = large. Always report it alongside the p-value.
• If the confidence interval for the mean difference includes zero, the result is non-significant (for two-tailed tests).
• T-tests assume approximate normality. For small samples from skewed distributions, consider the Wilcoxon test.
• With very large samples (n > 100), the t-distribution approaches the normal distribution and t-test results match z-test results.

Choosing the Right T-Test

One-sample t-test: comparing a single group to a known value (e.g., testing if average reaction time differs from 250ms). Two-sample t-test: comparing independent groups (e.g., treatment vs control). Paired t-test: comparing the same subjects under two conditions (e.g., before and after treatment, left vs right hand). When in doubt between pooled and Welch's two-sample tests, use Welch's — it's valid under both equal and unequal variances.

Interpreting Results Beyond the P-Value

A complete t-test report includes the t statistic, degrees of freedom, p-value, mean difference, confidence interval, and effect size. The confidence interval is arguably more informative than the p-value: it shows the range of plausible values for the true difference, giving you both significance and magnitude information in one quantity.

Common Pitfalls

Multiple comparisons: running many t-tests inflates the false positive rate — use ANOVA or correction methods instead. Violated assumptions: the pooled t-test with very unequal variances and unequal sample sizes gives unreliable results — switch to Welch's. Confusing statistical and practical significance: a p-value of 0.001 with d = 0.05 means a real but trivially small effect.

Frequently Asked Questions

• When should I use a t-test vs a z-test?

  Use a t-test when the population standard deviation is unknown, which is the usual case. Use a z-test only when the population standard deviation is known or you have a special design that justifies it.

• What is the difference between pooled and Welch's t-test?

  The pooled (Student's) t-test assumes equal variances in both groups. Welch's t-test does not make this assumption and uses a modified degrees of freedom formula. Welch's is recommended as the default because it's valid regardless of variance equality.

• What does degrees of freedom (df) mean?

  Degrees of freedom determines the shape of the t-distribution used for p-value calculation. Higher df means the distribution is closer to normal. For one-sample: df = n−1. For two-sample: df = n₁+n₂−2. Welch's df is typically fractional.

• Can I use a t-test for non-normal data?

  The t-test is robust to moderate non-normality, especially with larger samples (n > 30) due to the Central Limit Theorem. For small samples from heavily skewed or outlier-prone distributions, use non-parametric alternatives like the Wilcoxon test.

• What assumptions does the t-test require?

  Independence of observations, approximate normality of the sampling distribution, and (for the pooled test) equal variances in both groups. Welch's test relaxes the equal variance assumption.

• What if I have raw data instead of summary statistics?

  Compute the mean, standard deviation, and sample size from your raw data first, then enter those values. Alternatively, use our standard deviation calculator to get these summary statistics.