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 =
Variance = σ
2 =
=
=
=
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 - ( 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.