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.
To see why this makes sense, I would recommend watching this fantastic video by 3Blue1Brown on the topic: But what is the Central Limit Theorem?