# Large sample size changing standard deviation

I've got a few questions about the following prompt. I'll just copypasta it so you can read for yourself. The context is malaria in Africa and a drug developed to fight it.

"Before the drug trial, the average lenvgth of time before a child in the population got malaria was 89 days with standard deviation 41 days. During the experiment [the drug experiment], 724 children were treated with the drug and their average length of time until infection was 97.5 days."

We talked a bit about hypothesis testing, one-tail and two-tailed intervals, etc., and I get all that. Here are some of the thoughts that bother me:

1) Since our sample size is so big, the probability density function should be normal, my instructor claims. But that usually involves taking many, many samples from the same population, no? We've only taken one sample of 724 children and lumped them into one test, so we can't really call this a "normal distribution" unless perhaps the original population had a normal distribution anyway. Am I correct here?

2) Assuming our sample is represented by a normal distribution, the standard deviation of our sample is $\frac{\sigma}{\sqrt{n}}$. This greatly decreases the population's standard deviation of 41 to a much smaller value. My thought is this: what if our "sample size" was the entire population. Then, wouldn't our sample's standard deviation be almost zero as the population increases. (Law of large numbers, I think?) Then, if that's true, since our sample was the whole population, the poplulation mean would in fact not be 41. Am I incorrect here?

3) Assuming that the medicine did not work and the mean value of the treated population would be 89, we found that the probability of our sample having a mean value of 97.5 or greater was less than .0002, or something like that. Since our critical value was 5%, we deemed that the medicine must indeed have worked, since that would be such an unlikely outcome. Is it safe to assume at this point that the medicine in fact increased the length of time until malaria contraction for the whole group? I would think so, but then we proceeded to do a two-tailed test in order to account for a possible decrease in time, as well. Since staying at the same mean time was so improbable, wouldn't lowering the amount of time until infection be even less probable?

I think that's about it for now. This isn't any sort of homework. It was an in-class discussion exercise we did, and I asked my professor about it, but she didn't quite satisfy my questions.

Thanks for your time!

## 1 Answer

1. You don't need multiple separate samples, you just need one large sample. However, "large" in mathematical terms is entirely a function of the problem. One can construct situations where there is enough tailing to make the central limit theorem convergence arbitrarily slow, without so much tailing that the central limit theorem fails entirely. These situations are unusual in applications, however: usually the convergence is either fairly quick or it fails entirely (because the limit is actually a Pareto distribution or something like this).

2. You may misunderstand which standard deviation is which. The number $\frac{\sigma}{\sqrt{n}}$ is the standard deviation of the sample means. This is nonzero because there is still some variability between the sample means of different samples. This formula breaks down in the real world when $n$ approaches the size of the entire population, call it $N$. That's because the sample mean of the entire population is not random at all so it does not have standard deviation $\frac{\sigma}{\sqrt{N}}$, it has standard deviation $0$. So there is not really a contradiction here, there is just a breakdown of the model when $N$ is finite and $n$ is extremely large.

3. I think it would make more sense to do a one-tailed test in this context, but the conclusion for a one-tailed test would be the same anyway.