T-Distribution Formula

The T-Statistic Formula

Variables

Two-Sample T-Test

Compares the means of two independent groups (Welch's t-test).

Degrees of Freedom

• One-sample: df = n - 1
• Two-sample (Welch's): df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

When to Use T vs Z

• Use t-test when population σ is unknown (most real-world situations)
• Use z-test when population σ is known or n is very large (>30)
• The t-distribution approaches the z-distribution as sample size increases

Example

Key Notes

• For small df (≤ 5), the t-distribution has heavy tails — extreme values occur more often than in a normal distribution; interpret p-values cautiously with very small samples
• Statistical significance (p < 0.05) ≠ practical significance — a sample of 10,000 can show "significant" differences that are too small to matter; always check effect size (Cohen's d)
• The t-test assumes the underlying population is approximately normal — for strongly skewed data with small samples, use non-parametric alternatives (Mann-Whitney U, Wilcoxon signed-rank)
• Paired t-test (same subjects measured twice, e.g., before/after treatment) uses the differences directly: t = d̄/(s_d/√n); it is more powerful than an independent t-test when subjects are matched

Key Notes

• t-statistic: t = (x̄ − μ) / (s/√n): Used when the population standard deviation σ is unknown and estimated by the sample standard deviation s. Degrees of freedom ν = n − 1. The t-distribution has heavier tails than the standard normal — reflecting extra uncertainty from estimating σ.
• Heavier tails than normal: At ν = 1 (Student's t with 1 df), the tails are extremely heavy — the distribution has no finite variance. As ν increases, the t-distribution converges to the standard normal. By ν ≥ 30, the difference is practically negligible for most purposes.
• t-test varieties: One-sample t-test: compare a sample mean to a known μ. Independent two-sample: compare means of two separate groups (assuming equal or unequal variances). Paired t-test: compare pre/post measurements on the same subjects (most powerful when measurements are correlated).
• t-distribution for confidence intervals: x̄ ± t_(α/2, ν) × (s/√n): For small n, the t critical value is larger than z, giving a wider CI — correctly reflecting greater uncertainty. At n = 10, the 95% CI uses t ≈ 2.26 instead of z = 1.96.
• Applications: The t-distribution is used in clinical trials (comparing treatment vs control with small samples), quality control (sample means against specifications), A/B testing with limited data, educational research, and any analysis where sample size is too small to rely on the normal approximation.