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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?

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  • $\begingroup$ (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. $\endgroup$ – André Nicolas Nov 16 '15 at 18:08
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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.

There's a great animation of this in the Wikipedia article on the Dirac delta function.

In order to avoid this complication, the sampling distribution is standardized at each step, that is, we divide back out by the standard error, and in so doing are looking at a distribution that does in fact retain an interesting shape in the limit, namely the standard normal curve. This has the additional side-benefit that most numerical calculations (whether by table or computer algorithm) are practically done in terms of the standard normal curve as a simplifying conversion, so it's useful to have a theorem about that transformation.

Generally the relation between this and the original distribution is that we infer the original distribution is "approximately normally distributed" for large sample size, and the degree to which that's usefully correct has been investigated empirically. For example, see: Boos and Hughes-Oliver, "How Large Does n Have to Be for Z and t Intervals?" (American Statistician, Vol. 54, No. 2, pp. 121-128).

There have been alternate definitions proposed for the "standard normal distribution", and CLT could be reformulated in those terms if desired. See the article at Wikipedia.

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