The standard deviation

The standard deviation is the sum of squared deviations (or errors) of a data set with respect to the mean divided by the number of data. 

By far the most common formula for computing variance in a sample is:

standard-deviation-formula
Where:

  • s² = standard deviation
  • X = value
  • M = mean
  • N = sample size

Since samples are usually used to estimate parameters, s² is the most commonly used measure of variance. Calculating the variance is an important part of many statistical applications and analyses. It is the first step in calculating the standard deviation.

Example:

if your set is 1, 2, 3, 4, 6, 8 we can compute the standard deviation as follow:

X Freq. M (X-M) (X-M)^2
1 1 4 3 9
2 1 4 2 4
3 1 4 1 1
4 1 4 0 0
6 1 4 2 4
8 1 4 4 16


   
24 6     34

 

  • Sample size N = 6 (frequency)
  • The mean : 24/6 = 4
  • The sum of (X-M)^2 = 34
  • 34/6 = 5.67

The standard deviation is 5.67