Standard deviation
The standard deviation is defined as the square root of the arithmetic mean of the
squares of the deviations of the items from their arithmetic mean.
Calculation of Standard deviation - Individual series:
If x1 , x 2 , x3 ,............x n are ‘n’ individual observed values of a variable X, with mean x then
the Standard deviation is denoted by  and is defined as
1 n
xi  x 2

n i1
 S .D ( ) =
Where x is mean of the series.
The formula can also extended as below
S .D ( ) =
hence
1 n
xi  x 2

n i1

1 n
2
( xi  2 xi x  x 2 )

n i 1

1 n 2
1 n
1 n 2
xi   x
 xi  2 x n 
n i 1
n i 1
i 1

1 n 2
xi  2 x 2  x 2

n i 1

1 n 2
xi  x 2

n i 1
1 n
xi  x 2

n i1
=
1 n 2
xi  x 2

n i 1
Calculation of Standard deviation -Discrete series:
Direct method:
If x1 , x 2 , x3 ,............x n are ‘n’ individual observed values of a variable X occurred with
frequencies f 1 , f 2 , f 3 ,.........., f n and with mean x , then Standard deviation is defined as
n
 f x
i 1
 S .D ( ) =
Where x 
i
i
 x
2
n
f
fx
f
i
i 1
i
i
is mean of the series.
i
The above formula can be extended to
n
 f x
i 1
 S .D ( ) =
i
i
 x
2
n
f
i 1
n
fx
i
i 1
=
2
i
n
f
i
i 1
Where x 
fx
f
i
 x2
i
i
i
Short cut method:
In this method the formula for standard deviation is given by
n
fd
S .D ( ) 
i
i 1
 n
  fi di
  i 1n

  fi
 i 1
2
i
n
f
i
i 1
2



 where d i  xi  A


A = assumed mean
Continuous series (or) Grouped frequency distribution:
The calculation of standard deviation in case of grouped frequency distribution is same as the
calculation of mean deviation in case of discrete frequency distribution. The only difference is to use
mid points of the classes in place of values of the variable.
Step deviation method (or) change of origin and scale:
In this method the formula for standard deviation is given by
n
S .D ( )  C 
fu
i
i 1
2
i
n
f
i 1
i
 n
  f i ui
  i 1n

  fi
 i 1






2
where d i 
xi  A
C
A = assumed mean
C = class interval
Variance: The square of the standard deviation is called variance and is denoted by  2 .
Coefficient of variation:
The relative measure of standard deviation is called coefficient of standard deviation and is
given by

Coeff of S.D =
x

Coefficient of variation is defined as 100
x