Standard error of the mean - derivation

Can anyone point me in the right direction? I'm trying to understand how the equation for the standard error of the mean is derived

Where Standard error of the mean = standard deviation of the the sample divided by th3 square root of the # of samples.


Ok if anyone is reading this old thread I'd love some clarification. In thislink above from JohnM I am getting lost at the step underlined in red in the attached image. Very similarly, I get lost following this thread when someone provides the explanation that "We know in general that Var(kY)=k^2Var(Y)". Sorry, but I don't know that, in general :) Can anyone explain why there is all of a sudden a squared (1/n) term in the math? i.e. can you elaborate on why "Var(kY)=k^2Var(Y)" and not"Var(kY)=kVar(Y)"?


Last edited:


Ambassador to the humans
Start with Var(kY). Do you know the mathematical definition of variance? Writing that out is the next step. The rest follows from properties of the expectation operator.

If you're still having trouble show what you've done and we can try to help you get to the next step.
Hi Dason, thanks. I've attached something that I think makes sense and explains the answer to my question, but I don't have formal training in this and am not certain. Can you take a look? Where can I read and intro to the "properties of the expectation operator".