Normal Distribution

Normal distribution with population mean and standard deviation labeled.

The normal distribution is a continuous probability distribution that appears naturally in statistics and probability. The shape of the normal distribution is a “bell curve” whose center is equal to the mean of the distribution. The area under the curve is equal to and can be used to calculate the probability of an event occurring in a range of values.

General Equation

The function above gives the general form of the normal distribution in terms of the standard deviation and mean of the distribution. This function describes a family of probability density functions that can be used to calculate probability. Note, that the probability density function is often abbreviated as PDF.

Variable Description
The standard deviation is denoted with the symbol (sigma) and is calculated with this formula. The standard deviation value describes how far the population is distributed around the mean of the population.
The circle constant (tau) appears as a scaling factor that ensures the area under the distribution is equal to .
Euler’s Number is shorthand for the exponential function where and gives the expression useful properties in addition to making the values of the other variables more meaningful.
The mean of the population, denoted with the symbol (mu), describes the center of the distribution. The bell-curve is symmetrical around the mean.
The input .

Given the input the function returns the relative likelihood of the event of occurring. The area under the function can be used to calculate the probability of an event occurring for a range of values. This is discussed below.

Note, the standard normal distribution is a special case of the normal distribution where the mean is and the standard deviation is . This distribution has historical significance because it allows values to be referenced in a lookup table rather than calculated by hand. Of course, computers make computing values on and areas under the variations of the distribution trivial.