Standard Deviation Concept:
The standard deviation is one number that is used to express how far (on average) the data points are from the mean value of the data set. (The specifics of how this average distance from the mean is measured is found in the definition below.)  If you have only a sample of data, use the formula for sample standard deviation, and if you have data from the entire population, then you can use the population standard deviation formula.  

A related term is Variance.  The variance is the standard deviation squared, and the variance is often called the mean squared error.  

Sample Standard Deviation Definition:          
The sample standard deviation is the square root of the sum of the squared differences around the arithmetic mean divided by the sample size minus 1.

Formula: 

Where:

                  = Sample arithmetic mean

                   = Sample size

                  =   Observation of the  random variable

                  = Summation of all  values in the sample

 Population Standard Deviation Definition:
The population standard deviation is the square root of the sum of the squared differences around the arithmetic mean divided by the size of the population. 

Formula:

(Note: The denominator is different in the population and sample standard deviation formulas.)