I think only sample standard deviation should be used. But I am not sure.
Solved – Should I use sample standard deviation or population standard deviation in hypothesis testing
hypothesis testingpopulationsamplestandard deviationt-test
Related Solutions
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
The two forms of standard deviation are relevant to two different types of variability. One is the variability of values within a set of numbers and one is an estimate of the variability of a population from which a sample of numbers has been drawn.
The population standard deviation is relevant where the numbers that you have in hand are the entire population, and the sample standard deviation is relevant where the numbers are a sample of a much larger population.
For any given set of numbers the sample standard deviation is larger than the population standard deviation because there is extra uncertainty involved: the uncertainty that results from sampling. See this for a bit more information: Intuitive explanation for dividing by $n-1$ when calculating standard deviation?
For an example, the population standard deviation of 1,2,3,4,5 is about 1.41 and the sample standard deviation is about 1.58.
Best Answer
It really depends on what you mean. It would be a very rare situation that you actually know the population standard deviation but don't know anything else; more typically you would have an assumed population standard deviation, or in some cases a bound on the standard deviation.
If you're interested in a hypothesis about the standard deviation (e.g. $H_0:\sigma=15$ vs $H_1:\sigma\neq 15$), then the hypothesis will certainly be about a population standard deviation, but of course you won't know the population standard deviation in that case and you'd estimate its value from the sample.
If you're talking about the case when you're testing a hypothesis about a population mean (choosing between a z-test and a t-test say), then if you really do know the population standard deviation it would often make sense to use the additional information; it's like getting extra data for free.
However, that may make the distribution of your test statistic more sensitive to the other assumptions. For this reason it may sometimes make sense to ignore the information in order to have a somewhat more robust test (in large samples the additional information from knowing the standard deviation doesn't add so much in any case).
If you mean something other than those possibilities, you'll have to clarify further what situation you're asking about.