# Why the random variable $Z_n$ in the central limit theorem?

I understood more or less the central limit theorem, but I think not enough, because I am still wondering why we define a new random variable $Z_n$, which we conclude at the end to be standard normally distributed.

I have also seen that in some cases the CLT simply states that the sample mean $\overline{X}_n$ (the mean of a sample of random variables) is normally distributed (not standard normally distributed, i.e. with $\mu = 0$ and $\sigma = 1$).

In my case, the random variable $Z_n$ is defined like that

$$Z_n = \frac{\overline{X}_n - \mu}{\frac{\sigma}{\sqrt{n}}}$$

I noticed that $Z_n$ is mostly used in practice, i.e. to find probabilities.

I have a few questions:

1. Why the random variable is defined as it is defined?

2. What's the relation between the sample mean being normally distributed and $Z_n$ being standard normally distributed?

3. Could $Z_n$ have been defined differently, if yes, how, and why?

• (Informal) If $n$ is large, the random variable $\bar{X}_n$ has a close to normal distribution. But not close to standard normal. Subtracting $\mu$ is done because then $\bar{X}_n-\mu$ is close to a normal with mean $0$. The standard deviation of $\bar{X}_n$, and therefore of $\bar{X}_n-\mu$, is $\sigma/\sqrt{n}$. Dividing $\bar{X}_n-\mu$ by $\sigma/\sqrt{n}$ gives us a random variable which is nearly normal, has mean $0$, and standard deviation $1$, so a rv $Z_n$ which is close to standard normal. – André Nicolas Nov 16 '15 at 18:08

Recall that the CLT is about the limit, as $n$ increases arbitrarily towards infinity. The primary problem with formally looking at the variable $\overline X$ is that the standard deviation of the variable (standard error) is decreasing towards zero -- graphically, the distribution is becoming arbitrarily skinny and in the limit is not normally distributed, but an infinitely skinny vertical line at the mean.