### Normal Distribution

The normal distribution is by far the most commonly occurring distribution. If you have no good reason for suspecting that your data is not normally distributed, do not use another distribution. The normal distribution class has two parameters, `mu` and `sigma` which specify the location and scaling of the distribution.

- The value `mu` is the average or mean value.

- The value `sigma` is the standard deviation, which is a measure of variability.

If X1, X2, . . . Xn is the set of sample measurement values, normal distribution parameters are calculated using the following formulae:

Mean:

`mu = (sum x_i)/n`

Sigma:

`sigma = sqrt((sum(x_i - mu)^2) /(n-1)`

The density function for the normal distribution is given by:

`f(x) = 2/(sigma sqrt pi)e^((-(x-a)^2)/(2sigma^2)`

The normal distribution function cannot be expressed as a combination of common functions. The distribution function is given by:

`F(x) =1/(sigma sqrt pi)int_-oo^x e^((-(x-mu)^2)/(2sigma^2)` `dx`

For the special case where `mu = 0` and `sigma = 0`, the distribution function is called the standard normal distribution and is denoted `Phi(x)`. That is:

`Phi(x) =1/(sqrt 2pi)int_-oo^x e^((-x^2)/(2)` `dx`