= Jackknife and bootstrap variance of a sample second moment
{title2=$\widehat v_J=\frac{1}{n(n-1)}\sum_i(x_i^2-\overline{x^2})^2$}
For independent identically distributed observations with finite fourth <moment>, the <sample mean> of their squares has <sampling variance> $\operatorname{Var}(X^2)/n$. Its <jackknife variance estimator> is exactly the displayed unbiased estimator, since its leave-one-out values differ from their average by $-(x_i^2-\overline{x^2})/(n-1)$. The <conditional bootstrap variance of a sample mean> is $\widehat v_B=n^{-2}\sum_i(x_i^2-\overline{x^2})^2=(n-1)\widehat v_J/n$. Thus the two methods have a known finite-sample scaling difference, despite sharing the same large-sample target.
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