Two-Sample t-Test (Welch's)

Test whether two means differ. t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂).

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How to use this calculator

  • Enter mean, SD, n for each group.

About this calculator

Welch's t-test compares means of two independent samples without assuming equal variances. Formula: t = mean diff / SE. Welch-Satterthwaite df is more conservative than Student's. p < 0.05 conventionally significant. For paired observations: paired t-test (different formula). For >2 groups: ANOVA. Source: Welch (1947); Student's "Probable Error of a Mean" (1908).

Frequently asked

Welch vs. Student?+
Welch doesn't assume equal variances. More conservative. Default in modern stats.
When to use t vs. z?+
t: small sample (n<30), unknown population SD. z: large sample, known SD. Modern practice: just use t.
Paired vs. independent?+
Paired: same subjects measured twice. Independent: two different groups. Different formulas.
Effect size?+
Cohen's d = (m₁−m₂)/pooled SD. d = 0.2 small, 0.5 medium, 0.8 large.
When p > 0.05?+
Fail to reject null (no significant difference). Not "no difference" — just not enough evidence.

Related calculators

One-Sample z-Test
z = (x̄ − μ) / (σ/√n). Compare sample mean to known population.
P-Value Calculator (z to p)
Convert a z-score to one-tail or two-tail p-value via the standard normal CDF.
Confidence Interval Calculator
CI = mean ± z × (σ/√n). 90 / 95 / 99 % standard intervals.
Chi-Square Goodness-of-Fit
χ² = Σ(O−E)²/E. Test if observed frequencies match expected.
One-Way ANOVA F-Statistic
F = MS_between / MS_within. Test if 3+ group means differ.
Linear Regression (Slope + Intercept)
y = mx + b via least squares. Returns m, b, r², SE.

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