How to calculate standard deviation

Standard Deviation of a probability distribution measures the distribution of data from the mean value. Probability distribution maps the probability of a variable within a particular range of values.

For Example: Suppose we toss a coin two times

All possible outcomes: HH, HT, TH, TT
Total number of outcomes = 4
Let a variable, X measures the number of heads in this experiment
Probability of n heads = (Outcomess with n heads)/ (Total number of outcomes)

Number of Heads                  Probability
 (X)                                 (Probability of n heads)             
 
    0                                  0.25                                                
    1                                  0.5
    2                                  0.25

For a probability distribution, plot number of Heads(X) on the x-axis and the probability of n heads on the y-axis.

Range of values in this case = 0-2

Consider a data set with following value

2, 2, 4, 5, 3, 6, 13

Step 1: Count the number of elements in the data set
N = 7

Step 2: Find the mean or average of the data set

Arithmetic Mean (average) = (2 + 2 + 4 + 5 + 3 + 6 + 13)/7  
                                         = 35/7 = 5

Step 3: Subtract Mean from each element in the data set and square the result

(2 – 5) = -3   :   (-3) ^ 2 = 9
(2 – 5) = -3   :   (-3) ^ 2 = 9
(4 – 5) = -1   :   (-1) ^ 2 = 1
(5 – 5) = 0    :   (0) ^ 2 = 0
(3 – 5) = -2   :   (-2) ^ 2 = 4
(6 – 5) = 1    :    (1) ^ 2 = 1
(13 –5) = 8   :    (8) ^ 2 = 64


For Sample Set

Step 4: Divide the sum of the resulting values with one less than the number of elements in the data set and find the square root

SD(Standard Deviation) = Sqrt ((9 + 9 + 1 + 0 + 4 + 1 + 64)/6)
      = 3.82

Variance is the square of standard deviation = SD ^ 2 = 14.59

For Complete Set

Step 4: Divide the sum of the resulting values with the number of elements in the data set and find the square root

SD(Population Standard Deviation) = Sqrt ((9 + 9 + 1 + 0 + 4 + 1 + 64)/7)
      = 3.54

Population variance is the square of standard deviation = SD ^ 2
                                                                             = 12.25

Example of Sample Set: Sample voters in an exit poll
Example of Complete Set: Set of all the voters in an election