The Central Limit Theorem

Basically, the Central Limit Theorem states that if certain assumptions are satisfied, then no matter what distribution a population has, means of data from that population tend to be normally distributed. Mathematically, if $Y_1, Y_2, \ldots$ are independent random variables from the same distribution with mean $\mu$ and variance $\sigma^2$ and if

$$\overline{Y}_n=\frac{1}{n}(Y_1+Y_2+\cdots+Y_n),$$

then for large $n$, $\frac{\overline{Y}_n-\mu}{\sigma/\sqrt{n}}= \frac{\sum_{i=1}^nY_i-n \mu}{\sigma\sqrt{n}}$ has approximately a $N(0,1)$ distribution.

Normal Quantile Plots

A normal quantile plot is a scatterplot in which the data are plotted against their associated normal quantiles. If the data are normal, the plot should resemble a straight line.

Cauchy Distribution

The Cauchy distribution is an example of a distribution for which a variance is not defined (In fact, a mean is not defined for it, either.) The probability density function of the Cauchy distribution model is

$$p_Y(y)=\frac{1}{\pi(1+y^2)},~-\infty<y<\infty.$$