Standard Deviation

If is the frequency distribution, then standard deviation, denoted by Greek letter σ (sigma) is given by

where is the arithmetic mean of the distribution and ∑f =N .

Mean Deviation	Standard Deviation
Uses absolute value to assure that final result is positive	Squares to make all values positive, then takes positive square root.
Using absolute value makes it less suitable for further mathematical treatment.	Most suited measure of dispersion for further mathematical treatment.
Makes use of all values in the distribution.	Makes use of all values in the distribution.
Can be calculated about any measure of central tendency	Calculated using arithmetic mean.

Standard Deviation is the most used measure of dispersion – and most suitable for further mathematical treatment.

Variance

The square of standard deviation is called variance, denoted by

Example 1: Find the Standard Deviation and Variance

x:	1	2	3	4
f:	2	1	4	1

Solution:

STEP 1: To find Mean

x	f	fx
1	2	2
2	1	2
3	4	12
4	1	4
Total	∑f = 8	∑fx = 20

STEP 2: To find Standard Deviation

x
1	2	-1,5	2.25	4.50
2	1	-0.5	0.25	0.25
3	4	0.5	0.25	1
4	1	1.5	2.25	2.25

Example 2: Find the Standard Deviation and Variation

x:	8	11	12	14	16
f:	2	2	5	2	2

(Try it yourself!)

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In fact, if the numbers are big, you can simplify the calculation!

1. Choose any A (preferably median of observations or the observation with highest frequency) and h
2. Put
3. Calculate 𝜎_d
4. Then 𝜎_𝑥 = h 𝜎_d

Example 3

x:	6	10	12	14	16
f:	2	2	5	2	2

(Try it yourself with A = 12, h = 2)

x	f		fd
	∑f=		∑fd=

d	f
-3	1
-1	2
0	3
1	2
3	2

𝜎_𝑑² =

𝜎_𝑑 =

Coefficient of Variation

Coefficient of Variation =

Example 1: Calculate the coefficient of variation

x:	1	2	3	4	5
f:	1	2	2	2	1

(Try it yourself)

Moments

The r^th moment of a distribution centred at any number A is given by

About 0,

1. 𝜇₀′ = 1
2. 𝜇₁′ is the arithmetic mean.

The r^th  moment about (i.e., ) is called r^th central moment

1. 𝜇₀ = 1
2. 𝜇₁ = 0
3. 𝜇₂ = 𝜎²