Chi-squared distribution

Take $k$ independent standard normals, square each, and add them up. The result is chi-squared with $k$ degrees of freedom.

X = Z_{1}^{2} + Z_{2}^{2} + \dots + Z_{k}^{2}

The density itself is unlovely, but every interactive feature follows from that definition:

f (x; k) = \frac{x ^{k /2 - 1} e ^{- x /2}}{2 ^{k /2} Γ ( k /2 )}

Chi-squared (mean = k = 4.0) Normal(k, √(2k))

k (degrees of freedom) 4.0 n (samples) 1500

PDF

f (x; k) = \frac{x ^{k /2 - 1} e ^{- x /2}}{2 ^{k /2} Γ ( k /2 )} (x \geq 0)

. The sum of k squared standard normals:

X = Z_{1}^{2} + Z_{2}^{2} + \dots + Z_{k}^{2}, Z_{i} \sim N (0, 1)

. The dashed Normal(k, √2k) approximation tightens as k grows.

What to notice

Small k is strongly right-skewed. At $k = 1$ the mode sits at zero and the density blows up there — a single Normal squared spends most of its time near zero.
Large k goes Normal. The dashed curve is Normal(k, √(2k)). By $k \approx 30$ the two are nearly indistinguishable, which is why variance-based tests quietly revert to z-tests when degrees of freedom are high.
Mean and variance are both linear in k. Adding one more squared Normal adds 1 to the mean and 2 to the variance.

Every statistical test whose null distribution is “something squared divided by something” turns out to involve chi-squared:

Sample variance. If your data are iid Normal, the scaled sample variance is exactly chi-squared:

$\frac{( n - 1 ) S ^{2}}{σ ^{2}} \sim χ_{n - 1}^{2}$
This is the denominator of Student’s t.
Pearson’s χ² test. Summing squared residuals of observed vs. expected counts produces an approximate chi-squared.
Likelihood ratios. Under regularity conditions, −2 log(LR) is chi-squared in the sample-size limit (Wilks’ theorem).

Together with the F distribution — the ratio of two chi-squareds — it covers most of the classical frequentist toolkit.