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I'm given $n$ samples from a random number generator that produces normally distributed values. Can I calculate the probability that the standard deviation that the RNG uses is $\sigma > \sigma_{max}$? Also, can I calculate the probability that it's $\sigma < \sigma_{min}$?

I don't really have a strong mathematical or statistical background. I have no idea whether this is even doable, or maybe it's trivial. In any case, I thought I'd ask the community here.

EDIT:

If it makes any difference, I don't really need to calculate the probability. I just need to be able to check whether it's above a certain value (for instance, 0.01).

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    $\begingroup$ Of course you can! You can use hypothesis testing for variances just as you would do for means! Quick reference: itl.nist.gov/div898/handbook/eda/section3/eda358.htm $\endgroup$
    – David
    Commented Mar 8, 2019 at 11:36
  • $\begingroup$ @asdf If I understood the article you linked, the one-trailed test can tell me if the numbers I get are unlikely to happen (probability at most $\alpha$) if the RNG's standard deviation were actually below (or above) the required value? $\endgroup$
    – relatively_random
    Commented Mar 8, 2019 at 22:33
  • $\begingroup$ Yes, that is right. Choose an $\alpha$, calculate the threshold value for the one-sided test and compare with the one from the sample! $\endgroup$
    – David
    Commented Mar 9, 2019 at 14:25

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When doing probability especially with pseudo-random # generators (unknown how they're formed by us, but technically deterministic), we usually need to run a large # of n samples through, generating many samples independently. from there, we calculate the Standard Deviation of each and check it against some sigma that you're interested in. The probability in this case would be P(sd > sigma | randomly generated samples), read as "The Probability the standard deviation is greater than sigma, given these independently generated samples. That's written as a conditional probability. We say we'd like a large # of independent samples, in the hope that the Law of Large #s will converge to a probability for this pseudo-RNG.

I don't have time at the moment to post code from R showing this, but I can come back later and do that.

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    $\begingroup$ Could you explain how this answers the question about the distribution of the sample SD from a Normal population? $\endgroup$
    – whuber ♦
    Commented Mar 8, 2019 at 12:08

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