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}, … are independent random variables from the same distribution with mean μ and variance σ^{2} and if

\overline{Y}_{n} = \frac{1}{n}(Y_{1} + Y_{2} + ⋯ + Y_{n}),

then for large n, \frac{\overline{Y}_{n}−μ}{σ/\sqrt{n}} = ^{\frac{∑_{i=1}^{n} Y_{i}−nμ}{σ\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}{π(1 + y^{2})}, − ∞ < y < ∞.