## What is Standard Error of the Mean?

Aliases: *SEM*

The standard error of the mean of a distribution is the (population) standard deviation of the arithmetic mean, usually estimated through the sample standard deviation using the formula SEM = SD_{X} / √N where N is the sample size. The standard error of the mean has a normal distribution regardless of the underlying distribution of the variable of interest (e.g. revenue per user, order value, session duration, page load speed) due to the Central Limit Theorem (CLT).

In A/B testing the standard error of the mean plays a central role since most of the statistics used are arithmetic averages. When reporting a confidence interval around a reported average one is just reporting the value that lies within x standard deviations of the mean so that 1-CDF(x) equals the confidence level. A p-value for an arithmetic mean is calculated in a similar manner by measuring the distance in standard errors between an observed mean and the hypothetical mean posited by the null hypothesis.