# Standard Deviation

In statistics, the standard deviation is a measure of the spread or variation in a data set. It is often referenced using “s” which represents the standard deviation of a sample. It is used to estimate the standard deviation of a population (σ or sigma).

The formula for a standard deviation can be done in multiple ways, but the most traditional formula is as follows:

To calculate the standard deviation, let’s look at a data set using 5 data points (15, 22, 27, 28, 31)

The first section of the equation is to calculate the average

This is also called x-bar (x with a line over the top). This is the mean (also known as the average).

The average of the 5 data points is 24.6, when you add up the data points, and divide by n (the number of data points, which is 5).

Next, let’s look at this part of the equation

This tells us to subtract each X value from the average.

15 – 24.6 = -9.6

22 – 24.6 = -2.6

27 – 24.6 = 2.4

28 – 24.6 = 3.4

31 – 24.6 = 6.4

Next in the equation, we square those differences, and sum up (Σ) those numbers

(-9.6)^{2} + (-2.6)^{2} + (2.4)^{2} + (3.4)^{2} + (6.4)^{2} = 92.16 + 6.76 + 5.76 + 11.56 + 40.96 = 157.2

Next in the equation, we divide by n-1, which is 5-1 = 4

157.2 / 4 = 39.3

Finally, we take the square root of the value to compute the standard deviation

Square root of 39.3 = 6.27 (you can also compute this by taking 39.3 to the 0.5 power, 39.3^{0.5})