Student-t Distribution

The Student-t distribution accounts for the extra uncertainty of estimating a standard deviation from a small sample.

Why not just use the normal

When you standardize a sample mean but must estimate σ from the same small sample, the result is not quite normal: the estimated σ adds noise. The Student-t distribution captures this, with heavier tails that reflect the extra uncertainty.

Definition

A t-distributed variable is T = Z / √(V/k), where Z is standard normal and V is an independent χ²(k). The parameter k is the degrees of freedom. As k grows the t-distribution approaches the standard normal, because the estimate of σ becomes reliable.

Heavier tails

With few degrees of freedom the t-distribution puts noticeably more mass in the tails than the normal, so its critical values are larger. For a 95% two-sided interval with 5 degrees of freedom the multiplier is about 2.57, versus 1.96 for the normal. This wider interval is the honest reflection of a small sample.

The t-test

The t-statistic (x̄ − μ₀)/(s/√n) compares a sample mean to a hypothesized value using the sample standard deviation s. Under the null it follows a t-distribution with n − 1 degrees of freedom. The one-sample, two-sample, and paired t-tests all rest on this.

Practical guidance

Use t rather than normal critical values whenever σ is estimated and the sample is small, roughly n below 30. The t-test assumes approximately normal data; for heavy-tailed or highly skewed samples, a nonparametric test or a resampling approach is safer.

With large samples the distinction vanishes, so the choice only matters when data are scarce — precisely when getting it right matters most.