Standard Deviation Percentage

 The standard deviation is  most commonly used term in statistics. The relative standard deviations to consider the accuracy of compute the standard deviation of given analytical data. The Standard deviation is the square root of average squared deviation from the mean.

Standard Deviation  :

Standard  deviation is the arithmetic mean of all the deviation of observations taken about their mean

Standard equation When each of the given terms has frequency 1  
Let x₁, x₂, …, x_n be the n given observations and let M be their mean. Then, the variance σ² is given by
σ² =[ (x₁ – M)² + (x₂ – M)² + … + (x_n – M)² ] / n = Σ d²_i /n,
where the deviation from the mean, d_i = (x_i – M)
And, therefore, the standard deviation σ is given by
σ = + √{Σ(x_i – M)²/n} = √Σ d_i²/n, where d_i = (x_i – M)

Formula to Calculate Percentage of Standard Deviation:

When frequencies of the variable are given

In this case, the variance is given by

σ² = (Σ f_i d_i²/Σ f_i) & S.D.= σ = √(Σ f_i d_i²)/n

we proceed in same way  as we have done earlier

but here each d_i²  is multiplied by correponding f_i

and apply the above formula

and we get standard deviation for frequency distribution

Ex :  Find the variance and standard deviation from the following frequency distribution table:

Variable (xi)	2	4	6	8	10	12	14	16
Frequency (fi)	4	4	5	15	8	5	4	5

Sol :  We have

Variable xi	Frequency fi	fi xi	_  di = (xi – M)	di2	fi di2
2	4	8	– 7	49	196
4	4`	16	–     5	25	100
6	5	30	–  3	9	45
8	15	120	–  1	1	15
10	8	80	1	1	8
12	5	60	3	9	45
14	4	56	5	25	100
16	5	80	7	49	245
	Σ fi = 50	Σ fi xi = 450			Σ fi di2 = 754
		... M = 450/50 = 9

.^.. Variance, σ² = Σ f_i d_i²/ Σ f_i = 754/50 = 15.08

And, standard deviation, σ = √15.08 = 3.88

Percentage of standard deviation:

Percentage of standard deviation or relative standard deviation = (standard deviation / mean)  x 100.

Calculating Percentage of Standard Deviation:

Calculate the variance as well as the standard deviation percentage of the given table of the data:

xi	7	10	12	16	18	25	28
fi	2	5	13	10	6	4	1

Solution:

Presenting the data in tabular form, we get

xi	fi	fixi	(xi - mean)	(xi - mean)2	fi(xi - mean)2
7	2	14	-8	64	128
10	5	50	-5	25	125
12	13	156	-3	9	117
16	10	160	1	1	10
18	6	108	3	9	54
25	4	100	10	100	400
28	1	28	13	169	169
	41	616			1003

Here, N= 41 and `sum_(i=1)^7` fixi  = 616.

Therefore, mean = (`sum_(i=1)^7` fixi ) `-:` N = (1/41) x 616   =  15



and                `sum_(i=1)^7`fi(xi - mean)2   = 1003.

Hence, Variance(σ2)  =  (1/ N) `sum_(i=1)^7` fi(xi - mean)2  =  (1/41) x 1003   =  24.46

and

Standard deviation (σ) = `sqrt(24.46)`   =  4.94

Relative standard deviation or standard deviation percentage = (σ / mean) x 100  =  (4.94 / 15 ) x 100     =   32.9%