Approximating sample distribution standard deviation

Question

I've seen in video lessons that if the sample size is big enough (n>30) sample distribution standard deviation can be approximated by sample standard deviation. How do we get the sample distribution standard deviation if sample size is small (n=10)?

Answer 1

I'll first respond to the question, and then talk about the information you're basing your question on, some of which I regard as potentially misguided.

1) If we assume that the sample values are independently drawn from the same normal distribution, we can compute the distribution of the variance, and from that compute the distribution of the standard deviation.

As it turns out for the normal case, the sample variance is the population variance ($\sigma^2$) times a chi-squared random variable divided by its degrees of freedom (one less than the sample size). From there, the distribution of the standard deviation is a scaled chi-distribution.

The green curve shows the distribution shape when dealing with the standard deviation of two observations. The blue curve is for 5 observations. The red curve 11 observations, and the purple curve is for 41. As you see, even at n=41 it can still fairly easily be below 0.8 or above 1.2 times $\sigma$ (roughly 7% chance of one or the other happening).

However, if you're using that standard deviation in say a test of means, then it's the distribution of your test statistic (e.g. $\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$) that you have to worry about. For some intuition on that, you may want to look at this recent answer.

2) I'd regard any advice that 30 is large enough to consider that the sample standard deviation has converged to the population standard deviation (at least without some strong conditions) as nonsensical. Sometimes, that may be reasonable, but in other circumstances, I wouldn't think it's even close, and the advice has the potential to lead to some badly mistaken actions.

Even if we're dealing with normal data, for a t-statistic with 29 df, a statistic of 3 have a p-value of 0.55% (and, for example, if you're doing multiple comparisons, you might well be using a significance level below 0.5%), but if you treat it as if the standard deviation was known, you'd get a p-value less than half what it should be.

With normal data in some circumstances I'd look for eighty - or in other situations, perhaps even hundreds of observations - before I'd be content to treat the sample standard deviation like it was the population standard deviation.

If we're not dealing with normal data, it might have to be much, much more -- or in some cases you might never get there.