= How do I obtain the standard deviation from the standard error of the mean (s.e.m.)? =
$$\frac{\mbox{The standard deviation}}{\sqrt{\mbox{sample size}}}$$= standard error of the mean
i.e.
standard deviation = $$\sqrt{\mbox{sample size}} \mbox{multiplied by the standard error of the mean}$$
It follows from this (e.g. p.218 of Babbie) that the standard error fo the mean decreases with sample size, N.
This follows since:
The variance of the mean = 1/N x variance of the response = 1/N x The variance of N randomly sampled responses from a parent population = 1/ N^2 x (Nsigma^2)
= sigma^2/N where sigma^2 is the (unobserved true) variance of the (parent population of the) response. Since this involves a term 1/N the variance (or its square root = standard error) of the mean decreases with sample size.
The mean (= to the midpoint or median in a Normal distribution) on the other hand is not proportional to sample size so is uneffected by N. One can see this easily by considering an example: suppose we have a sample of size 3 of a responses = 1 2 3 then the mean is 2. Suppose I take a sample of size 7 say of the same response and get values of 1 1 2 2 2 3 3 then the mean = 2 there since it is symmetric about the (hypothesised true) mean of 2 (which
follows from sampling from a response following a normal distribution).
So it follows that the one sample z-test statistic = mean / s.e.(mean) will increase with increasing sample size because the mean stays the same and the s.e.(mean) decreases.
__Reference__
Babbie, E. (2008). The Basics of Social Research. Fourth Edition. Thomson Wadsworth: Belmont.CA.