= Jackknife variance estimator
{title2=$\widehat V_J$}
The <jackknife variance estimator> estimates the <sampling variance> of the full-data <estimator>, using the deletion estimates and their <mean> from <jackknife resampling>:
$$
\widehat V_J=\frac{n-1}{n}\sum_{i=1}^n
(\widehat\theta_{(-i)}-\overline\theta_J)^2,\qquad
\widehat{\operatorname{SE}}_J=\sqrt{\widehat V_J}.
$$
The square root is the estimated <standard error>. The factor $n-1$ multiplies the divisor-$n$ <variance> of the deletion estimates: those estimates vary less than estimates from new <independent> experiments because almost all their observations are shared. Ryan Tibshirani's https://www.stat.cmu.edu/~ryantibs/statcomp-F15/lectures/design.html[Carnegie Mellon notes on computing jackknife standard errors]{ref} give the same scaling in terms of the divisor-$(n-1)$ <sample variance> of the deletion estimates.
For the <sample mean>, this formula has an exact algebraic simplification. With $s_n^2$ denoting the usual <sample variance>,
$$
\widehat\theta_{(-i)}=\frac{n\overline x-x_i}{n-1},\qquad
\overline\theta_J=\overline x,\qquad
\widehat\theta_{(-i)}-\overline\theta_J
=-\frac{x_i-\overline x}{n-1},
$$
and therefore
$$
\widehat V_J=\frac{1}{n(n-1)}\sum_{i=1}^n(x_i-\overline x)^2
=\frac{s_n^2}{n}.
$$
Under the <independent and identically distributed> assumption and a finite second <moment>, $s_n^2$ is an <unbiased estimator> of the population <variance> $\sigma^2$. Consequently $\widehat V_J$ is an <unbiased estimator> of the <sample mean>'s <sampling variance> $\sigma^2/n$. The equality $\widehat V_J=s_n^2/n$ itself holds for any observed data with $n>1$; its sampling interpretation uses the stated assumptions.
For $(1,2,3,4,10)$, the sum of squared deviations from the <sample mean> is $50$, giving <sample variance> $s_5^2=50/4=12.5$. The deletion <sample means> have squared deviations totaling $3.125$, so
$$
\widehat V_J=\frac45(3.125)=2.5=\frac{12.5}{5},\qquad
\widehat{\operatorname{SE}}_J=\sqrt{2.5}\approx1.58114.
$$
Here $12.5$ estimates the <variance> of individual observations, while $2.5$ estimates the <sampling variance> of their <sample mean>. These answer different questions.
For more general <statistics>, validity requires regularity beyond the deletion formula, such as a sufficiently <smooth function> of <sample means> with suitable <moments>. Ordinary delete-one <jackknife resampling> can fail for nonsmooth <quantiles>, including the <median>; see Larry Wasserman's https://www.stat.cmu.edu/~brian/valerie/617-2022/0%20-%20books/2006%20-%20Wasserman%20All%20Of%20Nonparametric%20Statistics.pdf[All of Nonparametric Statistics, Section 3.1]{ref}. With <statistical dependence>, deleting individual observations need not reproduce the relevant <sampling variance>; the deletion scheme must reflect that <statistical dependence>. Shalizi's https://stat.cmu.edu/~cshalizi/490/23/2023-11-29-jackknife.pdf[jackknife notes, Sections 4 and 7]{ref} distinguish the basic <independent and identically distributed> case from extensions.
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