The standard deviation is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

Example:Two data sets: A1 {0, 5, 9, 14} and A2 {5, 6, 8, 9}. Both of their mean are 7, but the standard deviation of A2 is small than A1.

The standard deviation of a population is defined by the following formula:

where  is the population standard deviation,  is the population mean,  is the th element from the population, and  is the number of elements in the population.

Example: To calculate the population standard deviation of A {5, 6, 8, 9}.

Step1:

Step2:

The standard deviation of a sample is defined by slightly different formula:

Where  is the sample standard deviation,  is the sample mean,  is the th element from the sample, and  is the number of elements in the sample. Using this equation, the standard deviation of the sample is an unbiased estimate of the standard deviation of the population.

Difference between population standard deviation and sample standard deviation:

All variable values are used, it is called a population; when only a subset of available values is used, it is called a sample.

Things to remember:

  1. is the population standard deviation which is usually unknown
  2. is the sample standard deviation which is an estimate of the unknown standard deviation.
  3. The sum of squares is divided by 1 less than the sample size to account for the error in estimation from the sample standard deviation (called the degrees of freedom).

References & Resources

  • Wikipedia - http://en.wikipedia.org/wiki/Standard_deviation