Chi-squared

The story

Take $k$ independent draws from a standard Normal, square each one and add them up. The result is Chi-squared with $k$ degrees of freedom. Squaring makes everything positive, and because each squared term has mean 1 and variance 2, the total has mean $k$ and variance $2k$.

This is why it appears throughout statistical testing: whenever you add up squared standardised discrepancies, such as $(\text{observed} - \text{expected})^2/\text{expected}$ in a contingency table, or the squared deviations in a sample variance, a Chi-squared is lurking. For Normal data, $(n - 1)S^2/\sigma^2$ follows Chi-squared with $n - 1$ degrees of freedom.

Formulas

Parameters

Where it shows up

• The Pearson goodness-of-fit test comparing observed and expected counts.
• Testing for independence in a contingency table.
• Confidence intervals for a population variance.
• The squared distance from the origin of a random point in $k$-dimensional space with independent standard Normal coordinates.

Try this

1. Set $k = 1$ and shade from 0 to 3.84. The probability is 0.95, which matches the Normal: $P(Z^2 \le 3.84) = P(\lvert Z \rvert \le 1.96)$.
2. Step $k$ from 1 to 2 to 3. The density at zero goes from infinite, to exactly $1/2$ (the Exponential with rate $1/2$), to zero with a hump at $k - 2 = 1$. For $k \ge 2$ the mode is always $k - 2$.
3. Set $k = 20$, press Draw 1000 and compare the sample variance with $2k = 40$. The shape is already fairly close to a Normal with mean 20 and $\sigma \approx 6.32$, though still slightly right-skewed.

Related distributions

• It is the sum of $k$ squared standard Normals; in particular $Z^2$ is Chi-squared with $k = 1$.
• Chi-squared with $k$ degrees of freedom is $\text{Gamma}(k/2, 2)$.
• With $k = 2$ it is the Exponential with rate $1/2$.
• If $Z$ is standard Normal and $V$ is an independent $\text{Chi-squared}(\nu)$, then $Z/\sqrt{V/\nu}$ is Student t with $\nu$ degrees of freedom.