# Standard error for expected value [duplicate]

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I have a computer simulation from which I get a sample of values of some microscopic quantity $X$, i.e. $\{ x_1,\ldots,x_N \}$. I'm interested in estimating the expected value of $X$, i.e. $E[ X]\approx\frac{1}{N}\sum x_i$, and the variance associated with this estimate.

Now, do I understand correctly that:

1. taking the estimate of the variance of $X$, $Var[X]\approx\frac{1}{N}\sum (x_i-\bar{x})^2$, is NOT what I'm looking for (which in this notation would be $Var[E[X]]$)?

2. I need to make use of some statistical technique, such as bootstrapping?

## marked as duplicate by whuber♦Aug 20 '14 at 21:29

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## migrated from physics.stackexchange.comAug 20 '14 at 21:21

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• I am voting to migrate to math.SE! – Danu Aug 20 '14 at 20:38
• @Danu probably better for stats SE... can't auto-migrate there, but if ponadto requests it (which I would suggest, since there doesn't appear to be any physics in this question), one of the mods can move it over. – Kyle Aug 20 '14 at 20:41
• Oh, stats is even better! I forgot it existed... – Danu Aug 20 '14 at 20:43
• Sorry, but how can I "request a migration" (gracefully)? – ponadto Aug 20 '14 at 20:45
• The duplicate showed up in a site search for "standard error" + mean. Many more related posts can be found in the same way. Please note that "$E[X]$" usually refers to a number--the expectation of $X$--and that your formula is the sample mean, which is a realization of a random variable. It is often written $\bar{X}$ and its realization (as given by the formula) as $\bar{x}$. This question seeks information about computing the standard error of the sample mean qua estimator of the expectation. – whuber Aug 20 '14 at 21:31