IID Monte Carlo sample mean, variance, and standard error

Let $(Y_i{)}_{i\ge 1}$ be real IID random variables on one probability space, where IID means that the variables have a common law and are mutually independent (Identical distribution and IID families). When $\mathbb{E}\mathrm{\mid }{Y}_1\mathrm{\mid }<\mathrm{\infty }$, write $\mu :=\mathbb{E}[Y_1]$ and define the Monte Carlo estimate of this expectation by the sample mean ${\bar{Y}}_n:=n^{-1}{\sum }_{i=1}^n{Y}_i$; this is the partial-sum convention of Partial sums, row sums and sample means. For each $n\ge 2$ and any realized finite real sample, define its sample variance and nonnegative square root by $s_n^2:=\frac{1}{n-1}{\sum }_{i=1}^n(Y_i-{\bar{Y}}_n{)}^2,s_n:=\sqrt{s_n^2}\mathrm{.}$ When $\mathbb{E}[Y_1^2]<\mathrm{\infty }$, write ${\sigma }^2:=\mathrm{Var}(Y_1)$, using Moments, variance, and covariance on a probability space. The population standard error of ${\bar{Y}}_n$ is $\sigma \mathrm{/}\sqrt{n}$, and its plug-in estimate is ${\widehat{se}}_n:=s_n\mathrm{/}\sqrt{n}$. Although the finite-sample quantity $s_n$ is defined for any finite real sample, its interpretation as an estimate of a finite population standard error is used here only under the finite-second-moment condition.

The formula above defines the mathematical sample variance. Computing it as the difference between the sample second moment and the squared sample mean can suffer catastrophic cancellation in floating-point arithmetic; no stable-update or floating-point error bound is asserted on this page.