T-Test Calculator

One & Two Sample Paired & Welch Cohen’s d Free · No Signup

Free online t-test calculator for one-sample, two-sample, paired, and Welch t-tests. Compute t-statistic, p-value, degrees of freedom, critical value, confidence interval, Cohen’s d effect size, interactive t-distribution chart, and Python scipy export.

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Perform one-sample, two-sample, paired, or Welch t-tests with full results.

T-Distribution Visualization

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What Is a T-Test?

A t-test is a statistical hypothesis test used to determine whether there is a significant difference between the means of one or two groups. It uses the t-distribution, which accounts for small sample sizes and unknown population standard deviations.

0 t α/2 α/2
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Hypothesis Testing

Compare a sample mean to a known value or compare two group means to determine if the difference is statistically significant.

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Statistical Significance

The p-value tells you the probability of observing your data under the null hypothesis. Small p-values suggest significant results.

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Effect Size

Cohen’s d measures the practical significance of the difference. A result can be statistically significant but have a small effect.

Types of T-Tests

One-Sample T-Test:  t = (x̄ − μ₀) / (s / √n)
Compares a sample mean to a known or hypothesized population mean μ₀. Uses n−1 degrees of freedom.
Two-Sample T-Test (Pooled):  t = (x̄1 − x̄2) / (sp × √(1/n1 + 1/n2))
Compares means of two independent groups assuming equal variances. sp is the pooled standard deviation with n1+n2−2 degrees of freedom.
Paired T-Test:  t = d̄ / (sd / √n)
Tests whether the mean difference d̄ of paired observations differs from zero. Uses n−1 degrees of freedom where n is the number of pairs.
Welch’s T-Test:  t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2)
Compares means of two independent groups without assuming equal variances. Uses Welch-Satterthwaite approximation for degrees of freedom.

Choosing the Right T-Test

ScenarioTest TypeWhen to Use
Compare sample mean to a known valueOne-SampleTesting if a batch mean equals the specification value
Compare two independent groups with similar variancesTwo-SampleTreatment vs control, males vs females
Before and after measurements on same subjectsPairedPre-test vs post-test, left eye vs right eye
Two independent groups with unequal or unknown variancesWelchDefault choice when unsure about equal variances

Tip: When in doubt, use the Welch t-test. It is robust to unequal variances and performs well even when variances are actually equal.

Interpreting Results

✅ Significant Result

When p < α, reject the null hypothesis. The observed difference is unlikely due to random chance alone. Report the t-statistic, df, and p-value.

❌ Not Significant

When p ≥ α, fail to reject the null hypothesis. There is insufficient evidence of a significant difference. This does not prove the groups are equal.

📏 Effect Size (Cohen’s d)

Small: d ≈ 0.2, Medium: d ≈ 0.5, Large: d ≈ 0.8. A significant p-value with tiny d may lack practical importance.

↔️ Confidence Interval

The CI for the mean difference shows the range of plausible values. If it excludes zero, the result is significant at that confidence level.

Assumptions & When to Use Alternatives

  • 1. Normality: Data should be approximately normally distributed, or sample size should be large enough (n ≥ 30) for the Central Limit Theorem to apply.
  • 2. Independence: Observations must be independent of each other (except in paired tests where pairs are dependent but differences are independent).
  • 3. Equal Variances: The two-sample t-test assumes equal variances. Use Welch’s t-test if this assumption is violated.
  • 4. Continuous Data: T-tests are designed for continuous (interval or ratio) data, not ordinal or categorical data.

Non-parametric alternatives: If your data violates normality with small samples, consider the Mann-Whitney U test (for independent samples) or Wilcoxon signed-rank test (for paired samples). For comparing more than two groups, use ANOVA instead.

Frequently Asked Questions

Use one-sample to compare a mean to a known value. Use independent two-sample when groups are unrelated with similar variances. Use paired for before/after or matched data. Use Welch when group variances differ or you are unsure about equal variances.
The p-value is the probability of observing data as extreme as yours if the null hypothesis is true. A small p-value below alpha suggests the data is unlikely under the null, so you reject it.
Cohen’s d measures effect size, or practical significance. A statistically significant result with tiny d may not be meaningful. Small d is about 0.2, medium about 0.5, and large about 0.8.
Data should be approximately normal or sample size large enough for CLT. Observations must be independent. Two-sample tests assume equal variances unless using Welch. Paired tests assume the differences are normally distributed.
Use two-tailed when you want to detect a difference in either direction. Use one-tailed only when you have a strong prior hypothesis about the direction of the effect before collecting data.
Avoid t-tests with very small non-normal samples or heavily skewed data. Use non-parametric alternatives like Mann-Whitney U or Wilcoxon signed-rank test. For more than two groups, use ANOVA instead.

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