A normal distribution, but not exactly a normal distribution

Overview

• Topic: A distribution like the normal but not exactly: the Student's t-distribution.

Key Concepts

• Symmetric about 0 and centered at 0.
• Shape depends on degrees of freedom (
  \nu).
• Heavier tails than the normal for finite \nu.
• Converges to the normal as \nu \to \infty.

Formulas

• Probability density function:
  f(t|\nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)}\left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}
• Mean:
  E[T] = 0\quad (\nu > 1)
• Variance:
  \operatorname{Var}(T) = \frac{\nu}{\nu-2}\quad (\nu > 2)
• Relation to normal:
  T_{\nu} \xrightarrow{\nu \to \infty} N(0,1)

Use Cases

• Unknown population standard deviation with small sample sizes.
• Used in t-tests and small-sample confidence intervals.

Quick Take

• Heavier tails for small \nu; approaches normal as \nu grows.