# Determining the Sample Standard Deviation

Suppose the brain volume for adult women is about 1,100 cc, with a standard deviation of 75 cc. Then consider a sample of 144 random women taken from this population, with an unspecified sample mean. Determine the approximate 10th percentile of the distribution of the distribution of sample means of the 144 women?

The solution to this problem hinges on the determination that the standard deviation of the sample mean is $$75/12 = 6.25$$. But why is this true? How could we possibly justify this? Where does the number, 12, come from?

The number 12 appears because there was a sample of 144 women, and $$\sqrt{144}=12$$.
This is explained by the Central Limit Theorem. In general, if $$X$$ is a random variable then, if you take $$n$$ samples of $$X$$ a bunch of times, the mean of those samples, $$\bar{X}$$ will be normally distributed with $$\bar{X} \sim \mathcal{N}\left(\mu_{_X},\dfrac{\sigma_{_X}}{\sqrt{n}}\right)$$ where $$\mu_{_X}$$ and $$\sigma_{_X}$$ are the population mean and standard deviation, respectively.
• @StanleyYu You don't need to take multiple samples for the CLT to apply, and you can take samples of any, sufficiently large, size. The theorem says that if you take a sample of size $n$ from a population with known mean and st. dev., then whatever the average of that sample turns out to be, it will fall under the normal distribution that I gave in my answer. When I said "take $n$ sample a bunch of times" I meant that, in order to see the normal distribution to take shape, you would need to take many samples of that same size. Apr 12, 2023 at 16:45