T-Statistic Calculator
Compute the t-statistic, degrees of freedom, p-values, and critical t for one-sample and two-sample tests (Welch or pooled), and from correlation r.
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Results
t-statistic
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Degrees of freedom (df)
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Critical t
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p (two-tailed)
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p (left)
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p (right)
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Decision at α
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Two-tailed comparison unless you chose left/right tail.
Understanding the T-Statistic (Simple Guide)
The t-statistic measures how far your sample result is from what you’d expect under a null hypothesis, in units of standard error. It helps you decide whether a difference is likely due to random chance.
Common t-tests
- One-sample: Compare a sample mean (x̄) to a known value (μ₀).
- Two-sample (Welch): Compare means of two groups with possibly unequal variances and sizes.
- Two-sample (pooled): Compare means assuming equal variances.
- From r: Test if a correlation is different from 0.
Formulas
- One-sample: t = (x̄ − μ₀) / (s / √n), df = n − 1
- Welch two-sample: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), df ≈ [(s₁²/n₁ + s₂²/n₂)²] / [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]
- Pooled two-sample: sp² = ((n₁−1)s₁² + (n₂−1)s₂²)/(n₁+n₂−2), t = (x̄₁ − x̄₂) / √(sp²(1/n₁ + 1/n₂)), df = n₁ + n₂ − 2
- From r: t = r √((n−2)/(1−r²)), df = n − 2
Tails and decisions
Two-tailed tests check for any difference; one-tailed tests check direction (less or greater). If |t| ≥ tcrit (or p ≤ α), you reject H₀.
Assumptions (quick check)
- Independence of observations
- Approximately normal data (small n); CLT helps for larger n
- For pooled test: equal variances across groups
Example (one-sample)
x̄ = 12.5, μ₀ = 10, s = 3.2, n = 25 ⇒ t = (12.5−10)/(3.2/√25) = 3.91, df = 24. With α = .05 (two-tailed), tcrit ≈ 2.064. Since 3.91 > 2.064, reject H₀.