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.
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.