# What is the standard error of the mean? [duplicate]

I don't know anything about this topic. Can you explain and describe standard error of the mean?

• From wikipedia article on standard error: "different samples drawn from a population would have different values of the sample mean, so there is a distribution of sampled means. The standard error of the mean is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population". Which part do you not understand? – TrynnaDoStat Feb 27 '15 at 19:05

## 1 Answer

In a simple example suppose we observe some measurement $n$ times and we know that that measurement is random, its's underlying distribution does not change, and that each measurement is independent of those taken before and after it. We call this an iid sample. It may be expressed as $X_1,X_2,...,X_n$ independently and identically distributed random variables with some underlying distribution $f_X(x)$. The sample mean can be calculated as

$$\frac{1}{n}\sum_{i=1}^{n} x_i = \bar x$$

$\bar x$ is an approximation of the population mean. The population mean or expected value of the random variables is a function of $f_X(x)$ which I will call $\mu$. The population mean is not a random variable but rather a fixed constant so it has no variance. However, because sample mean $\bar x$, is a function of the random variables $X_1...X_n$, $\bar x$ is, itself to, a random variable and does have variance $${Var}[\bar x]=\sigma^2_{\bar x}$$

So $\sigma_{\bar x}$ is the standard deviation or standard error of the sample mean.

The equation for $\sigma^2_{\bar x}$ and its sample approximation change with $f_X(x)$. Further if we do not assume the sample above is iid, $\mu$ becomes a function of multiple distributions and the formulas get complicated...but this is the general idea.