This question addresses calculating a p value from the mean and standard deviation statistics of a sample. I understand that the -general- philosophy is to divide the sample standard deviation by the root of the population size to get the standard deviation of the sampling distribution, using the assumption that the standard deviation of the sample is roughly equal to the standard deviation of the hypothetical larger population. Then one calculates a z-score from the number of standard deviations of the sampling distribution to calculate what percent of the time the observed result would have occurred by random chance. I understand that the particular formula for the sampling distribution standard deviation depends on the particular statistic, say difference of means is a different formula.
The texts and videos that I've looked at use language like "the sample standard deviation is the best number we have available to estimate the population standard deviation." I just don't find that explanation satisfying.
This approach hinges on the validity of estimating the standard deviation of the entire population as being approximately equal to the standard deviation of the representative sample. However, we don't make the same assumption that the mean of the population is the approximately equal to the mean of the sample. At some level, it feels like the final result of significance or non-significance is only self-validating or checking for self-consistency of an assumption that is baked into the methodology.
So to restate, why is the sample standard deviation a good approximation of the population standard deviation, but the sample mean is not a good approximation of the population mean? I found online an equation for standard deviation of the sampling distribution of standard deviations:
standard error of standard deviation = .71 sample standard deviation / root N.
Does the relative narrowness of the standard error compared to standard deviation play a role in justifying the approximation?