Page 158 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 7: Measures of Dispersion

Kelly's absolute measure of Skewness = (P - P ) - (P - P ) and Notes
90 50 50 10
P 90 P 50 P 50 P 10 P 90 P 10 2P 50
Kelly's Coefficient of Skewness S P
P P P P P P
90 50 50 10 90 10
( We note that P = M ).
50 d
3. Measure of Skewness based on Moments
This measure is based on the property that all odd ordered moments of a symmetrical
distribution are zero. Therefore, a suitable -coefficient can be taken as a relative measure
of skewness.
Since = 0 and = 1 for every distribution, these do not provide any information about
1 2
the nature of a distribution. The third -coefficient, i.e., can be taken as a measure of the
3
coefficient of skewness. The skewness will be positive, negative or zero (i.e. symmetrical
distribution) depending upon whether > 0, < 0 or = 0. Thus, the coefficient of Skewness
3
based on moments is given as

S M 3 3 3 1 1

Alternatively, the skewness is expressed in terms of . Since is always a non-negative
1 1
number, the sign of skewness is given by the sign of µ .
3
Example: Calculate the Karl Pearson's coefficient of skewness from the following data:
Size : 1 2 3 4 5 6 7

Frequency : 10 18 30 25 12 3 2
Solution:
To calculate Karl Pearson's coefficient of skewness, we first find X , M and s from the given
o
distribution.
Size (X) Frequency (f ) d X 4 fd fd 2
1 10 3 30 90
2 18 2 36 72
3 30 1 30 30
4 25 0 0 0
5 12 1 12 12
6 3 2 6 12
7 2 3 6 18
Total 100 72 234

fd 72
X A 4 3.28
N 100

2 2
2
fd fd 234 72
1.35
N N 100 100
Also, (by inspection) = 3.00

LOVELY PROFESSIONAL UNIVERSITY 153