The Chi-Squared Distribution

You estimate a stock's daily volatility from 30 trading days of returns. That estimate is itself a random number — a different 30 days would give a slightly different volatility estimate. How much slack should you put around it? Answering that requires knowing the distribution of a sum of squared normal deviations — which is exactly what the chi-squared distribution is built to describe.

The analogy: total distance from the bullseye, across many darts

Imagine throwing $k$ darts at a target, where each dart's horizontal miss and vertical miss are independent, standard-normal-sized errors. If you square each individual miss distance and add them all up across $k$ throws, the total is chi-squared distributed with $k$ "degrees of freedom." It measures accumulated squared error from a bunch of independent, zero-centered noise sources — which is precisely the situation you're in every time you compute a sample variance: you're summing squared deviations from a mean.

Writing it down

If $Z_1, \ldots, Z_k$ are independent standard normal random variables, then

$$X = Z_1^2 + Z_2^2 + \cdots + Z_k^2 \;\sim\; \chi^2_k.$$

In words: the chi-squared distribution with $k$ degrees of freedom is exactly the distribution of a sum of $k$ squared independent standard normals. It has mean $k$ and variance $2k$ — each squared standard normal contributes 1 to the mean and 2 to the variance on average, so both scale linearly with how many you add. The connection to sample variance: if $S^2$ is the sample variance of $n$ i.i.d. normal observations,

$$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}.$$

In words: take your sample variance, rescale it by the true variance and by $(n-1)$, and the result follows a chi-squared distribution with $n-1$ degrees of freedom — which is the entire basis for building a confidence interval on volatility itself, not just on the mean.

Picture squaring every draw from this curve and summing $k$ of them: with $k=1$ the result piles up near zero with a long right tail (a single squared normal is usually small, occasionally large); as $k$ grows the sum's distribution shifts right and becomes more symmetric — the chi-squared distribution's shape moving from heavily skewed toward roughly normal as degrees of freedom increase.

Worked example 1: confidence interval on volatility

You compute a daily return standard deviation of $S = 1.2\%$ from $n=30$ days. Using $\chi^2_{29}$, the 2.5th and 97.5th percentiles are roughly 16.05 and 45.72 (standard table values). A 95% confidence interval for $\sigma^2$ is

$$\left(\frac{(n-1)S^2}{45.72}, \frac{(n-1)S^2}{16.05}\right) = \left(\frac{29(1.44)}{45.72}, \frac{29(1.44)}{16.05}\right) = (0.914, 2.603).$$

Taking square roots, $\sigma$ lies in roughly $(0.96\%, 1.61\%)$ with 95% confidence — a genuinely wide band from only 30 days of data, illustrating how noisy short-window volatility estimates really are.

Worked example 2: a goodness-of-fit check

A model predicts trades should split 40/35/25 across three venues. Over 200 trades, you observe 95, 65, 40. The chi-squared goodness-of-fit statistic sums $(observed - expected)^2/expected$ across categories: expected counts are $80, 70, 50$.

$$\frac{(95-80)^2}{80} + \frac{(65-70)^2}{70} + \frac{(40-50)^2}{50} = \frac{225}{80} + \frac{25}{70} + \frac{100}{50} = 2.81 + 0.36 + 2.00 = 5.17.$$

With 2 degrees of freedom (3 categories minus 1), the 95th percentile of $\chi^2_2$ is 5.99. Since $5.17 < 5.99$, you fail to reject the model — the deviation, while visible, is within what random noise alone would produce.

Try a low positive exponent here — that rising-then-flattening shape mirrors a chi-squared density with few degrees of freedom: a fast climb from zero followed by a long decaying tail, which flattens toward something more symmetric as degrees of freedom (like exponent here) increase.

What this means in practice

• Volatility confidence intervals. Any time you report a realized volatility number, the chi-squared distribution is what tells you how much to trust it, especially with short lookback windows.
• Goodness-of-fit and independence tests. Checking whether observed trade routing, defaults, or category counts match a model's prediction routinely uses a chi-squared test statistic.
• Building block for other distributions. The $t$-distribution and $F$-distribution, both central to regression inference, are constructed directly from ratios involving chi-squared variables.