T-Test Calculator

Frequently Asked Questions

When do I use a t-test?

To compare means when the population standard deviation is unknown and the sample is small/normal-ish: one-sample (vs a value), two-sample (two groups), or paired (before/after).

What is the difference between pooled and Welch?

Pooled assumes equal group variances; Welch's does not and is generally safer. The tool reports Welch by default for two-sample tests.

What are degrees of freedom?

Roughly the number of independent pieces of information: n−1 for one-sample, more complex for two-sample (Welch–Satterthwaite). They set the t-distribution's shape.

What if my data are not normal?

For large n the t-test is robust. For small, skewed samples consider a non-parametric test (Mann–Whitney, Wilcoxon).

Provided by AllCalculators.io
Free online calculators for everyday. No registration required.

Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

JavaScript is required to use the interactive calculator above. The questions and answers below remain readable without JavaScript.

How This Calculator Works

A t-test compares means when the population standard deviation is unknown and is estimated from the sample. The one-sample form tests H₀: μ = μ₀ with t = (x̄ − μ₀) ÷ (s ÷ √n) and df = n − 1. The independent two-sample form compares two group means; the pooled version assumes equal variances and uses df = n₁ + n₂ − 2, while Welch's version does not, computing df from the Welch–Satterthwaite equation. The paired form reduces matched pairs to their differences and runs a one-sample test on those, with df = n − 1. The two-tailed p-value is obtained from the Student's t CDF, which this tool evaluates through the regularized incomplete beta function I_x(df÷2, ½) using a Lentz continued fraction - a numerical approximation good to about four decimals.

Worked Example: One-Sample, μ₀ = 50

A sample of n = 30 has mean x̄ = 52 and s = 6, and we test against μ₀ = 50. The standard error is 6 ÷ √30 ≈ 1.0954, so t = (52 − 50) ÷ 1.0954 ≈ 1.826 with df = 29. The two-tailed p-value is about 0.078. Against α = 0.05 this is not significant, so we fail to reject H₀: the 2-point difference could plausibly be sampling noise given this much spread and sample size.

When to Use Each Mode

One-sample: compare a single group mean to a known target or historical value.

Two-sample (Welch): the safe default for comparing two independent groups - robust to unequal variances and unequal n.

Two-sample (pooled): only when variances are genuinely similar (e.g. a Levene test does not reject equality).

Paired: before/after measurements on the same subjects, or naturally matched pairs - it removes between-subject variation.

Common Mistakes & Assumptions

The t-test assumes the sampling distribution of the mean is approximately normal - reasonable for n ≥ 30 by the Central Limit Theorem, but fragile for small, skewed samples. Using a pooled test when variances differ inflates the error rate; prefer Welch. Treating paired data as independent (or vice versa) is a frequent error: pairing, when valid, dramatically increases power by cancelling subject-level variation. Always feed sample standard deviations (n − 1 denominator), not population ones.

Interpretation & Limitations

A significant t says the observed mean difference is unlikely under H₀; it does not measure how large or important the difference is - report a confidence interval for the difference and an effect size such as Cohen's d. With very large samples a negligible difference becomes significant; with tiny samples a real one is missed. The Welch df is generally non-integer, which is expected. Far-tail p-values from the continued-fraction beta routine are approximate; treat values beyond p ≈ 10⁻⁶ as indicative only.

Frequently Asked Questions

Related Statistics Calculators