The Chi-Squared, t, and F Distributions

Every test statistic built by comparing an estimate to an estimated standard error — rather than a known one — stops being normal. The previous lesson's $Z=(\bar X-\mu)/(\sigma/\sqrt n)$ was exactly normal only because $\sigma$ was assumed known; the moment $S$ replaces $\sigma$, the extra randomness in the denominator fattens the tails, and the resulting statistic follows a new family. This note builds the three distributions that arise, in order of construction: the chi-squared distribution ($\chi^2$), built from sums of squared normals and the natural distribution of a sample variance; the Student's $t$ distribution, built from a normal divided by the square root of a scaled chi-squared, and the exact distribution of the one-sample test statistic; and Snedecor's $F$ distribution, built from a ratio of two scaled chi-squareds, used whenever two estimated variances are compared. All three are reference distributions — a practitioner reads their quantiles off a table or a scipy/R call the same way they read $z$-quantiles off the standard normal, and this note is what makes those quantiles mean something rather than being magic numbers.

Learning objectives. By the end of this note you should be able to: