The standard error (SE) is an amazingly useful statistical device for defining confidence intervals. In layman terms standard error is measure of how far a sample statistic is from it’s true value. This post will go through the process of deriving the SE of the mean.

I have always wanted to dig deeper into where the comes from.

The mathematical derivation is pretty straight forward. First, is to note that the mean is a sample mean of a population.

We know that the variance is equal to the expected value for the square difference from the mean.

Replacing with yields.

since

And there we have it.

Most commonly, standard error is a calculated from a sample for a sample mean .

For example the SE for a 95% confidence interval (alpha of 5%) from a normal distribution would be.

I found this a helpful exercise in gaining confidence if the formula’s I am using. The key was to understand that the SE is looking at the sample mean and not an individual sample value.

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