T-Test Calculator – One-Sample, Two-Sample & Paired

How the t-test works

A Student’s t-test asks whether a sample mean (or a difference of means) is far enough from a hypothesized value that the difference is unlikely under H₀. Unlike a z-test, the t-test uses the sample standard deviation and a t distribution with degrees of freedom that depend on sample size. This calculator covers three classic designs: one-sample (compare x̄ to μ₀), independent two-sample (Welch by default, optional pooled variance), and paired (matched before/after differences). For every run you get the t statistic, degrees of freedom, p-value, critical t at your α, a reject / fail-to-reject decision, and a one-sentence conclusion in plain English.

Required Inputs & Typical Use Cases

Use one-sample mode for quality-control checks against a target mean, lab calibration problems, or textbook “is μ equal to 5?” exercises. Use two-sample mode to compare two independent groups (treatment vs control, men vs women, batch A vs batch B). Prefer Welch unless you have a strong reason to assume equal variances; turn on “Pooled variance” only for the classic Student’s t with df = n₁ + n₂ − 2. Use paired mode for before/after measurements on the same subjects, twin studies, or any matched design where independence between the two lists would be wrong.

The Formula

One-sample: t = (x̄ − μ₀) / (s/√n),  df = n − 1
Welch two-sample: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂),  df = Welch–Satterthwaite
Pooled two-sample: t = (x̄₁ − x̄₂) / (sₚ√(1/n₁ + 1/n₂)),  df = n₁ + n₂ − 2
Paired: t = (d̄ − μ₀) / (s_d/√n) on differences dᵢ,  df = n − 1
Critical t: two-sided → tInv(1 − α/2, df); left → tInv(α, df); right → tInv(1 − α, df)

• x̄, s, n

  Sample mean, sample SD, and sample size

• μ₀

  Hypothesized mean (or mean difference for paired)

• t

  Test statistic under the Student-t null distribution

• df

  Degrees of freedom (integer or Welch–Satterthwaite)

• α

  Significance level (default 0.05)

• p-value

  Probability of a result at least as extreme as observed under H₀

Worked Example

1. One-sample, two-tailed, α = 0.05

   Data 5.1, 4.9, 5.3, 5.5, 5.0 with μ₀ = 5. Then x̄ = 5.16, s ≈ 0.2408, n = 5, df = 4, SE ≈ 0.1077, t ≈ 1.4856, and two-sided p ≈ 0.2116. Because p ≥ 0.05, fail to reject H₀ — there is not significant evidence that the mean differs from 5. For reference, tInv(0.975, 10) ≈ 2.228139 (two-sided 5% critical value at df = 10).

2. Choosing Welch vs pooled

   If the two samples have clearly different spreads, keep Welch (default). If a syllabus or paper assumes equal variances, enable pooled variance so df = n₁ + n₂ − 2 and SE uses the pooled SD sₚ.

Tips, Assumptions & Limitations

• Paste raw lists with commas, spaces, tabs, or semicolons — or enter summary stats x̄, s, n.
• Two-tailed is the default; switch to left or right when your alternative is directional.
• Welch is safer for independent samples; pooled only when equal variances are justified.
• Paired lists must be the same length; or paste the differences directly with μ₀ = 0 by default.
• Common mistake: using a two-sample test on matched pairs (use paired instead).
• Common mistake: treating a large |t| as significant without checking the p-value against α.
• Never report NaN: if inputs are incomplete the calculator shows a clear error instead.