= Width distribution of a Student confidence interval
{title2=$\frac{n(n-1)W^2}{4t_{n-1,1-\alpha/2}^2\sigma^2}\sim\chi^2_{n-1}$}
For a <normal sample> of size $n\geq2$, the central <Student t confidence interval> has width $W=2t_{n-1,1-\alpha/2}S/\sqrt n$. Its squared width is therefore a fixed multiple of the <sample variance>. The exact sampling identity $(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$ gives the displayed law. It quantifies the random variability in interval precision even when the <sample size> and coverage level are held fixed.
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