Width distribution of a Student confidence interval ($\frac{n(n-1)W^2}{4t_{n-1,1-\alpha /2}^2{\sigma }^2}\sim {\chi }_{n-1}^2$)

For a normal sample of size $n\ge 2$, 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 $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ gives the displayed law. It quantifies the random variability in interval precision even when the sample size and coverage level are held fixed.