# How To Deal With the Central Limit Theorem, and is it Related to the Normal Distribution?

$f\left( x \right)=\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}\exp \left( -\frac{{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}} \right)$

### Manipulating the Normal Distribution

$\int\limits_{-\infty }^{\infty }{\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}\exp \left( -\frac{{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}} \right)dx}=1$

$\int\limits_{-\infty }^{\infty }{\frac{x}{\sqrt{2\pi {{\sigma }^{2}}}}\exp \left( -\frac{{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}} \right)dx}=\mu$

and

$\int\limits_{-\infty }^{\infty }{\frac{{{x}^{2}}}{\sqrt{2\pi {{\sigma }^{2}}}}\exp \left( -\frac{{{\left( x-\mu \right)}^{2}}}{2{{\sigma }^{2}}} \right)dx}={{\mu }^{2}}+{{\sigma }^{2}}$

### Standard Normal Distribution and Z-scores

$Z=\frac{X-\mu }{\sigma }$

$Z=\frac{X-\mu }{\sigma}$

Have you even wondered why the back of the Stats textbooks come with normal distribution tables ONLY for the standard normal distribution? It is because all normal distributions can be reduced to the standard normal distributions, via z-scores, and it would be really impractical, or impossible, to print out ALL possible tables for all possible normal distributions.

$X-72<75.5-72$

$\frac{X-72}{8}<\frac{75.5-72}{8}$

### The Central Limit Theorem (CLT)

But, if you take repetitions of a random variable, from ANY distribution, and you compute their average, those averages will be (what you think?) dangerously resembling to a normal distribution, especially when the sample size (number of repetitions) is large.

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