# possible use case of central limit theorem for analysts

This is a bit of a long shot but I would appreciate any help please.

I have to do a basic stats course for our analysts, which I try to make as applicable and useful as possible using our data (e.g. showing skewed distributions and promoting the median over the arithmetic mean - basic stuff). Our analysts deal mainly with observational data but AFIK they never have to estimate the population mean of some external process. This is one of the main applications of the CLT. I am looking for any other use case for our analysts and would appreciate a possible example. We will later on also talk about bootstrapping, which is somewhat related but I want to keep it simple for now.

PS:

I think my main issue is, that we do not have a population and samples from it. Internally we have processes and we may want to compare process A vs process B. However, I would use bootstrapping for this. I just cannot see the direct application for CLT as we never intentionally collect several samples from a population ....

• Could you please give a concrete example of your work. It is a widespread misconception especially in the e-commerce field that you cannot use t-tests etc if your data is not normally distributed. However, companies are typically not interested in single value performance, but in totals ( total revenue etc). In this case averages are what you are interested in testing, and assuming your sample is large enough the central limit theorem basically covers you and enables you to use standard z-test/t-test/linear regression tests and no bootstrapping is necessary. So what is your sample size? Commented Dec 22, 2020 at 20:34
• I believe you have a misunderstanding about the CLT. It is used to justify that the average of a single sample ( of say 100 widgets) is approximately normally distributed. Commented Dec 22, 2020 at 20:39
• @seanv507 My understanding of the CLT is that you take several samples from a population. Each sample contains let us say 30 data points/rows. You then calculate your statistics for each sample (e.g. mean). The distribution of this statistics is approximately normal. So the mean of the means is a good estimate of the population mean. Not sure what you think I misunderstand? The issue I have is that we do not deliberately obtain several samples from some population/process. So I am struggling to find an application in my setting. Commented Dec 23, 2020 at 9:15
• @seanv507 I understand what you say about the t-test etc. However, if your data is just ONE sample of an external population and you then pretend that it is a population and do CLT on let us say 2 levels of your ONE sample is the result useful? I know we deal with observational data so any insights are potentially dodgy anyway. Commented Dec 23, 2020 at 9:26
• What I am saying is that you take a single sample of 30 data points, and calculate the sample mean. Because of CLT, that sample average can be approximately normal, so you can use z/t test. You don't need mean of means. You can investigate how close the (single) sample mean is to being normal by running bootstraps,calculating sample mean from each bootstrap sample. (You do that once, and then happily use standard t test until process changes dramatically) Commented Dec 23, 2020 at 10:52

• I'm not clear on why it is useful to discuss a method that works in the limit, when our datasets are finite. The CLT can produce arbitrarily bad confidence intervals (tail non-coverage probabilities far from $\alpha/2$ for at least one of the confidence limits) when the raw data distribution is very asymmetric (e.g., when you didn't know to take logs). Commented Dec 22, 2020 at 12:49