Why is the population standard deviation approximated as the sample standard deviation? 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 sample 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?
Thank you
 A: I'm answering my own question but I think that I have an intuitive justification for why the sample standard deviation reflects the population standard deviation.
As our sample size becomes larger and larger, it better reflects the population, and thus the variation of the population.  I realize that we could say the same for the sample mean, but at least for the case of flipping coins, the sample s.d. seems to tend adhere more towards the pop s.d. compared to sample mean.
Suppose we have a true 50 50 coin: p=0.5 q=0.5.  Half of all flips in the coin's history (q = 0.5) give tails = 0, and half of all flips in the coin's history (p = 0.5) give heads = 1.  The population mean of all flips in coin's history = 0.5, and Standard Deviation = root (0.5*0.5) = 0.5.
if flip 10 times and get q=0.6, p=0.4 for 1 sample
then sample mean = 0.4, SD = root (0.6 x 0.4) = 0.490
sample mean has decreased 20% from pop mean, but SD decreased 2%
if flip 10 times and get q=0.7, p=0.3 for 1 sample
then sample mean = 0.3, SD = root (0.7 x 0.3) = 0.458
sample mean has decreased 40% from pop mean, but SD decreased 10%
if flip 10 times and get q=0.8, p=0.2 for 1 sample
then sample mean = 0.2, SD = root (0.8 x 0.2) = 0.4
sample mean has decreased 60% from pop mean, but SD decreased 20%
suppose flip 10 times and get q=0.9 p = 0.1 for 1 sample
then sample mean = 0.1, SD = root (0.9 * 0.1) = 0.3
sample mean has decreased 80% from pop mean, but SD decreased 40%
This scenario shows that over the vast majority of the sampling distribution of possible sample means of 10 flip outcomes, the sample standard deviation does not stray too far from 0.5, even though the sample mean can vary much more
A: No matter the shape of the population distribution, the Central Limit Theorem holds that the sampling distribution of the sample means (and sample variances) approaches a normal distribution as the sample size $T$ (number of observations or data points) gets larger, especially for sample sizes over 30. As you take more samples, the sample distribution graph will look more like a normal distribution, and so on average, the sample moments will be the population moments as $T\uparrow$. (If you add up the means or standard deviations from all of your samples and find the average, that average will be close to the actual moments for the population.)
