Central Limit Theorem

The central limit theorem establishes that when independent events are added the sum tends toward a normal distribution (a bell curve) even if the original variables themselves are not normally distributed.

This provides a way to understand characteristics of a population of data points using only samples taken from that population.

In mathematical terms, this can be understood in terms of the:

$\\X=\left [ x_{1},x_{2},...,x_{n} \right ] \\\\Z=\left ( \frac{\bar{X_{n}}-\mu }{\frac{\sigma}{ \sqrt{n}}} \right ) \,as\,n\rightarrow \infty \\\\\mathrm{\textup{where:}} \\\\X\,\,\,\,\mathrm{\textup{a group of random samples}} \\\bar{X}\,\,\,\,\mathrm{\textup{sample mean}} \\n\,\,\,\,\,\,\mathrm{\textup{number of random samples}} \\Z\,\,\,\,\,\mathrm{\textup{standard normal distribution}} \\\sigma\,\,\,\,\,\,\mathrm{\textup{standard deviation}} \\\mu\,\,\,\,\,\,\mathrm{\textup{mean}} \\\infty\,\,\,\,\mathrm{\textup{infinity}}$

An Example

To see this visually, look at the example below. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 512. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.

Central Limit Theorem

An Example

References