Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 223 / 3 / b / Solution

The analogous statement is false under only a finite-variance assumption. Let $X=0$ with probability $3/4$ and $X=4$ with probability $1/4$. Then $\mu =1$, while the unique population median is zero. With fixed $k$ and group size $n/k\to \infty$, every group median converges in probability to zero, so their mean also converges to zero. Its error from $\mu$ tends to one rather than having order $n^{-1/2}$. Medians inside groups estimate the population median, while means inside groups preserve the population mean.

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