usually standard error is: standard deviations of the sampling mean

in essence it gives us how much variations in our “means” if we took a bunch of independent measurement samples.

formula: $$SE(\hat{x} ) = \frac{\sigma}{\sqrt{n}} $$ $\hat{x}$ is sample mean $\sigma$ is population std $n$ is number of sample

derivation:

$$SE(\hat{x}) = Std(samplemean) = \sqrt{VAR[\hat{X}]}$$ $$\sqrt{VAR[\frac{T}{n}]} = \sqrt{\frac{1}{n^2}VAR[T]} = \sqrt{\frac{1}{n}\sigma^2} = \frac{\sigma}{\sqrt{n}}$$ $T$ is sum of the mean from random variable $\hat{X}$ notice that $T$ consist of sum of iid ( independent and identically distributed) RV because we get T by sampling from the same distribution over and over again (bootstraping).

References

https://en.wikipedia.org/wiki/Standard_error

https://math.stackexchange.com/questions/1372761/trying-to-understand-bienaym%C3%A9-formula