What is Pooled Variance and T‑Test (statistics tutoring)

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Pooled Variance t‑Test Explained (statistics tutoring)
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The pooled variance t‑test is a classical method in statistical inference and statistics tutoring. It compares the means of two independent samples under the assumption that the two populations have equal variances. When this assumption holds, pooling the variances increases efficiency and yields a more powerful test.

If you need help deciding between pooled variance and Welch’s test, or want support with hypothesis testing and inference, visit the tutoring services page.

Pooled Variance t‑Test: Use when σ₁² = σ₂². t = (x̄₁ − x̄₂) / √(sₚ²(1/n₁ + 1/n₂)) where sₚ² is the pooled variance.

The pooled variance estimator combines information from both samples to produce a more stable estimate of the common variance when the equal‑variance assumption is justified.

Why the Pooled Variance t‑Test Matters

The pooled test is important because it:

  • is more powerful than Welch’s test when variances are equal
  • provides a clean, classical derivation of the two‑sample t‑statistic
  • is widely used in introductory statistics and biostatistics
  • forms the basis for ANOVA and linear regression inference

How to Compute the Pooled Variance t‑Test Step by Step

  1. Compute sample means:
    x̄₁ and x̄₂.
  2. Compute sample variances:
    s₁² and s₂².
  3. Compute the pooled variance:
    sₚ² = [ (n₁−1)s₁² + (n₂−1)s₂² ] / (n₁ + n₂ − 2 ).
  4. Compute the standard error:
    SE = √(sₚ²(1/n₁ + 1/n₂)).
  5. Compute the test statistic:
    t = (x̄₁ − x̄₂) / SE.
  6. Degrees of freedom:
    df = n₁ + n₂ − 2.
  7. Compare t to the t‑distribution with df degrees of freedom.

Numerical Example

Sample 1: n₁ = 10, x̄₁ = 22, s₁² = 25 Sample 2: n₂ = 14, x̄₂ = 18, s₂² = 24

Pooled variance: sₚ² = [9·25 + 13·24] / (10 + 14 − 2) = (225 + 312) / 22 = 537 / 22 ≈ 24.41.

Standard error: SE = √(24.41(1/10 + 1/14)) = √(24.41(0.1 + 0.0714)) = √(24.41 × 0.1714) = √4.185 ≈ 2.046.

Test statistic: t = (22 − 18) / 2.046 ≈ 1.95.

Degrees of freedom: df = 10 + 14 − 2 = 22.

Compare t = 1.95 to t₀.₀₂₅,₂₂ ≈ 2.07 → fail to reject H₀ at the 5% level.

Common Mistakes

  • Using the pooled test when variances are unequal
  • Assuming equal variances without checking diagnostics
  • Confusing pooled variance with pooled standard deviation
  • Ignoring the df = n₁ + n₂ − 2 formula

Why This Matters in Statistical Inference

The pooled variance test is central to:

  • classical two‑sample inference
  • ANOVA derivations
  • regression residual analysis
  • applied research with equal‑variance assumptions

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