Bootstrap for variance estimation I think I might have misunderstood the purpose of the bootstrap, because I think the following argument proves that bootstrapping is useless.
Let $N \in \mathbb{N}^*$, $(X_n)_{n < N}$ a $N$-tuple of iid random variables, and $(x_n)_{n < N}$ be a $N$-tuple of real numbers, assumed to be realizations of the $X_n$'s.
Let $m := \frac{1}{N}\sum_n x_n$ be the estimator of the mean, for which we would like to estimate the variance; the exact variance is $\frac{1}{N}Var(X_n)$, which is unknown if the law of the $X_n$'s is unknown.
Let us consider $\sigma := \frac{1}{N-1}\sum_n (x_n - m)^2$ the sample variance. Then $\frac{1}{N}\sigma$ is already an unbiased estimator of the variance of $m$.
However, let us now denote $\sigma_{BS}$ the exact bootstrapped variance, given by the following formula (if I am mistaken, please tell me!): we choose $\tau : \{0,\cdots, N-1\} \rightarrow \{0,\cdots, N-1\}$ uniformly; then we define $\sigma_{BS} := Var \left[\frac{1}{N}\sum_n x_{\tau(n)}\right]$.
Then $\sigma_{BS}$ is an estimator of the variance of $m$, and $\sigma_{BS}$ is itself estimated by choosing a lot of $\tau$'s at random and taking the mean.
However, if my computations are correct, then $\sigma_{BS} = \frac{N-1}{N^2}\sigma$. Therefore, why bother resampling instead of just computing the sample variance?
So, where am I mistaken? Is bootstrapping something else than I described?

In case of the mean, there is nice theory that can be used to estimate the variance of the sampling distribution directly. So indeed one doesn't need the bootstrap for this. The bootstrap can be used to estimate the variance of a statistic for which a straight general method as for the mean does not exist. Furthermore, the bootstrap can be used to assess the shape of the sampling distribution, which even for the mean is unknown unless the underlying distribution is known. This can be used for constructing confidence intervals (the formula sample mean plus/minus constant times estimated standard deviation relies on normality).
