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Standard Deviation

The standard deviation of a set of values is the root mean square deviation, when deviations are taken from the A.M. of observations. Root mean square deviation means that we take deviations of observations from the A.M., then square those deviations followed by arithmetic mean of the squared values and finally taking its square root.



If a variable x assumes n values x1, x2, x3, … xn, then its standard deviation is given by Description: 96352.png


The above relation can be simplified as Description: 96346.png 


For grouped frequency distribution, the standard deviation is given by Description: 96340.png 



The above relation can be simplified as Description: 96334.png



In the above relations Description: 96328.png

What is the standard deviation of 5, 5, 9, 9, 9, 10, 5, 10, 10?
x 5 9 10
f 3 3 3
Description: 96314.png 
x f Description: 96304.png  Description: 96297.png  Description: 96291.png 
5 3 3 9 27
9 3 –1 1 3
10 3 –2 4 12
Description: 96278.png 
The square of standard deviation known as variance is also sometimes regarded as a measure of dispersion.
Variance = S 2
The ratio of standard deviation to the corresponding mean expressed as percentage is called coefficient of variation.
Description: 96272.png 
What is the coefficient of variation of the following numbers?
53, 52, 61, 60, 64
 Description: 96266.png
Description: 96259.png 
Description: 96253.png 

Properties of Standard Deviation

  1. If all the observations assumed by a variable are constant, i.e., equal, then S = 0. This means that if all the values taken by a variable x are k, then S = 0. This result applies to range as well as mean deviation.
  2. Standard deviation remains unaffected due to a change of origin, but is affected in the same ratio due to a change of scale.
    If there are two variables x and y related as y = a + bx for any two constants a and b, then S of y is given by
    Sy = | b | Sx
  3. If there are two groups containing n1 and n2 observations, Description: 105500.png as respective A.M.s, S1 and S2 as respective S.D.s, then the combined S.D. is given by
    Description: 96247.png
    and Description: 96241.png is the combined A.M. 


  1. It is based on all the observations.
  2. It is rigidly defined.
  3. It has a greater mathematical significance and is capable of further mathematical treatments.
  4. It represents the true measurements of dispersion of a series.
  5. It is least affected by fluctuation of sampling.
  6. It is not reliable and dependable measure of dispersion.
  7. It is extremely useful in correlation, etc.


  1. It is difficult to compute unlike other measures of dispersion.
  2. It is not simple to understand.
  3. It gives more weighted to extreme values.
  4. It consumes much time and labour while computing it.

Note: The standard deviation of first n natural numbers is Description: 98528.png
When comparing the coefficient of variation of two or more objects, the one having the lesser coefficient of variation is said to be more consistent.

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