# Clarification on central limit theorem

I found this definition of central limit theorem in the book: Intro to probability and statistics using R:

I thought the sample mean $\bar X$ itself followed the normal distribution and not the calculated quantity as shown.

• (1) $\bar X$ never follows a normal distribution unless the population distribution is normal. (2) The limiting distribution of $\bar X$ is a constant (the value $\mu$).
– whuber
Sep 29 '15 at 23:40

Well the sample mean $\bar{X}$ approaches a normal distribution with mean $\mu$ and variance $\frac{\sigma^2}{n}$, i.e. $\bar{X}\sim N(\mu,\frac{\sigma^2}{n})$ however if you standardize the sample mean (by writing it in the form in the question), you find that approaches a Normal distribution with mean $0$ and variance $1$, i.e. $Z \sim N(0,1)$.

• Welcome to CV! I suggest you replace "follows a normal distribution" with "approaches a normal distribution", since the CLT does not claim that the distribution is exact. Sep 29 '15 at 21:24
• Thanks for that. (+1) Hope you continue contributing to our site. Sep 29 '15 at 21:29
• In the context of "approaches", "$n$" is meaningless.
– whuber
Sep 29 '15 at 23:41
• This answer is problematic, e.g. it's not true that $\bar X\sim\mathcal{N}\left(\mu,\frac{\sigma^2}{n}\right)$. The distribution of $\bar X$ looks kinda like normal when $n$ is large. Sep 30 '15 at 1:16

It's a very common mistake - which I make sometimes myself - to have the non-constant on the right hand side of the arrow: $$f_n\to \mathcal{N}\left(\mu,\frac{\sigma^2}{n}\right)$$, where $f_n$ is the probability distribution of $\bar X$.

Hence, we want to have the constant right hand side to apply our convergence proving tricks, e.g. $\mathcal{N}(0,1)$ would be a non-changing distribution. So, we formulate the convergence statement using it: $$Z=\frac{\bar X-\mu}{\sigma/\sqrt n}\sim\tilde f_n$$, where $\tilde f_n$ - sequence of distributions (functions), which converges to the standard normal: $$\tilde f_n\to\mathcal{N}(0,1)$$
It's still tempting to think about the distribution of $\bar X$ as one that looks more and more like normal, i.e. taking a shape of the normal distribution. However I wouldn't use the term converges, or even approaches, as they are loaded in mathematics. You'll run into people who will be confused with this language for they will expect to see the non changing right hand side.