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Standard Deviation

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Standard Deviation - Lesson Summary

Standard deviation is the positive square root of the variance. It is denoted by s.
 
The standard deviation of the observations x 1, x 2, x 3,..., x n =   σ = 1 n ∑ i = 1 n ( x i - x _ ) 2
 
Variance =  σ 2 = ∑ i = 1 n ( x i - x _ ) 2 n

                      = ( x 1 - x _ ) 2 + ( x 2 - x _ ) 2 + .... + ( x n - x _ ) 2 n

                      =  ( x 1 2 + x 2 2 + .... + x n 2 ) - 2 x _ ( x 1 + x 2 + .... + x n ) + n ( x _ ) 2 n

                      = ( x 1 2 + x 2 2 + .... + x n 2 ) n - 2 x _ ( x 1 + x 2 + .... + x n ) n + n ( x _ ) 2 n
 
Sum of the observations divided by the number of observations is the mean of the observations.
 
=  ( x 1 2 + x 2 2 + .... + x n 2 ) n -  2 ( x _ ) 2 + ( x _ ) 2                                            {∵  x _ = ( x 1 + x 2 + ... + x n ) n }

=  ( x 1 2 + x 2 2 + .... + x n 2 ) n -  ( x _ ) 2         

σ 2 = ∑ i = 1 n x i 2 n - ∑ i = 1 n x i n 2

σ = ∑ i = 1 n x i 2 n - ∑ i = 1 n x i n 2
 
Therefore, standard deviation of an ungrouped data can also be calculated by using this formula.
 
Variance of a discrete frequency distribution: σ2 = 1 N   ∑ i = 1 n ( x i - x _ ) 2    
x i’s are the n observations.

 f i’s are the frequencies of the observations.

N is the total frequency.

x _  is the mean of the n observations.

Standard deviation of a discrete frequency distribution:

σ = 1 N ∑ i = 1 n f i ( x i - x _ ) 2

Alternate formula:

σ = 1 N   N ∑ f i x i 2 - ( ∑ f i x i ) 2   
Variance of a continuous frequency distribution: σ2 = 1 N ∑ i = 1 n f i ( x i - x _ ) 2

x i’s are the mid-values of the class intervals.

f i’s are the frequencies of the class intervals.

x _  is the mean of the frequency distribution observation.

N is the total frequency.
Standard deviation of a continuous frequency distribution: σ = 1 N ∑ i = 1 n f i ( x i - x _ ) 2

Alternative formula for the standard deviation of a continuous frequency distribution:
 
Variance of a continuous distribution = σ 2

= 1 N ∑ i = 1 n f i ( x i - x _ ) 2

= 1 N ∑ i = 1 n f i ( x i 2 - 2 x i x _ + x _ 2 )

= 1 N [ ∑ i = 1 n f i x i 2 - ∑ i = 1 n 2 x _ f i x i + ∑ f i i = 1 n x _ 2 ]

= 1 N [ ∑ i = 1 n f i x i 2 - 2 x _ ∑ i = 1 n f i x i + ∑ f i i = 1 n x _ 2 ]

= 1 N [ ∑ i = 1 n f i x i 2 - 2 x _ N x _ + x _ 2 N]            (Since, ∑ i = 1 n f i x i   = N  x _ )

=  1 n ∑ i = 1 n ( f i x i 2 - x _ 2 N)

= 1 N ∑ i = 1 n f i x i 2 - ∑ i = 1 n f i x i N 2 N                (Since, ∑ i = 1 n f i x i N = x _ )

σ 2 = 1 N 2 N ∑ i= 1 n f i x i 2 - ∑ i = 1 n f i x i 2

Standard derivation (σ) = 1 N N ∑ i= 1 n f i x i 2 - ∑ i = 1 n f i x i 2

This is the alternative formula for the standard deviation. Note that there is no need to find the mean and deviations of the frequency distribution.
 
Note:
  • The standard deviation of an arithmetic series a, a+d, a+2d, a+3d,.....,a+(n-1)d is n 2 - 1 12
where c.d. is the common difference of the series and n is the number of terms in the series.
  • If the standard deviation of the observations, x1, x2, x3,....,xn is σ and k is some constant, then:
    • The standard deviation of the observations x1 ± k, x2 ± k, x3 ± k,..., xn ± k is σ.
    • The standard deviation of the observations kx1, kx2, kx3,....,kxn is kσ.
    • The standard deviation of the observations x1/k, x2/k, x3/k,....,xn/k is σ/k. 

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