Gaussian (Normal) Distribution

The Gaussian distribution, (also known as the Normal distribution) is a probability distribution. Its bell-shaped curve is dependent on \( \mu \), the mean, and \( \sigma \), the standard deviation (\(\sigma^2\) being the variance).

\[ f(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^\frac{-(x-\mu)^2}{2\sigma^2} \]

The peak of the graph is always located at the mean and the area under the curve is always exactly equal to 1. 68% of all the values lie within one standard deviation of the mean. At \(2\sigma\), this increases to 95%, and 99.7% of the values lie within \(3\sigma\) of the mean.