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I'm calculating standard deviation for a normally distributed variable. I'm receiving data samples online, and I've noticed that right after the start (for a small sample size) standard deviation is far too away from a real. I need somehow to select the number of samples I need to wait, until I will be able to use the calculated deviation.

So, I've started my research in Excel.

I am generating normally distributed variable with $\sigma = 15$:

=NORM.INV(RAND(),0,15)

Measuring the population standard deviation:

=STDEV.P(A2:A14553)

And for each new sample I'm measuring the sample standard deviation for all previous samples:

=STDEV.S($A$2:A3)

And then dividing sample standard deviation by population standard deviation and obtaining this ratio for each sample size:

STDEV.S/STDEV.P

After refreshing it multiple times I am able to see that I need to wait something like 3000 samples before starting using sample standard deviation. But I don't really want to hardcode it, since I don't fully understand how this number can change for different dataset.

What is the right way to calculate this number?

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I've found this paragraph in wiki, which directly answers my question.

For example, for a 95% CI and sample size N=100, sample SD lies from 0.88 × SD to 1.16 × SD

So, to find N all I need is to choose what difference from real SD I can let and how confident I want to be.

Here's a full answer, if the wiki will be updated:

The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the confidence interval or CI. To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of N = 2 has only 1 degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows:

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  • $\begingroup$ You need to put the answer in the answer, not just a link to somewhere else. While you can link elsewhere for support or for additional information, answers must be able to stand on their own - external links may move or even disappear altogether; your answer must still be fully understandable if that occurs. $\endgroup$
    – Glen_b
    Commented Aug 2, 2019 at 9:52
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There is no “right” standard deviation that comes from the “correct” number of samples. There is only the law of large numbers and the rule of thumb that is 30 samples.

Your experiment is a great way to think through this and shows you what you need to know:

The sample distribution (standard deviation included) approaches the population distribution as it becomes closer in count to the population, and so sample heavily when possible and do not rely too heavily on any sample that is not quite large.

Another way to understand this is to research the closed form calculation of the sample standard deviation. You will see that an argument of this function is the sample size.

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    $\begingroup$ What does the actual rule of thumb you refer to claim to be true, and how does it help decide what the standard error of the standard deviation would be? $\endgroup$
    – Glen_b
    Commented Aug 2, 2019 at 9:54

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