Page 157 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes In a symmetrical distribution, mean, median and mode are equal and the ordinate at mean
divides the frequency curve into two parts such that one part is the mirror image of the other,
positive skewness results if some observations of high magnitude are added to a symmetrical
distribution so that the right hand tail of the frequency curve gets elongated. In such a situation,
we have Mode < Median < Mean. Similarly, negative skewness results when some observations
of low magnitude are added to the distribution so that left hand tail of the frequency curve gets
elongated and we have Mode > Median > Mean. As shown in Figure 7.1).

Measures of Skewness

A measure of skewness gives the extent and direction of skewness of a distribution. As in case of
dispersion, we can define the absolute and the relative measures of skewness. Various measures
of skewness can be divided into three broad categories : Measures of Skewness based on (i) X ,
M and M , (ii) quartiles or percentiles and (iii) moments.
d o
1. Measure of Skewness based on X , M and M : This measure was suggested by Karl Pearson.
d o
According to this method, the difference between X and M can be taken as an absolute
o
measure of skewness in a distribution, i.e., absolute measure of skewness = X – M .
o
Alternatively, when mode is ill defined and the distribution is moderately skewed, the
above measure can also be approximately expressed as 3(X – M ).
d
A relative measure, known as Karl Pearson's Coefficient of Skewness, is given by

X M Mean Mode 3 X M
S k o or S d
Standard deviation k

We note that if S > 0, the distribution is positively skewed,
k
if S < 0, the distribution is negatively skewed and
k
if S = 0, the distribution is symmetrical.
k
2. Measure of Skewness based on Quartiles or Percentiles
(a) Using Quartiles

This measure, suggested by Bowley, is based upon the fact that Q and Q are
1 3
equidistant from median of a symmetrical distribution, i.e., Q - M = M - Q .
3 d d 1
Therefore, (Q - M ) - (M - Q ) can be taken as an absolute measure of skewness.
3 d d 1
A relative measure, known as Bowley's Coefficient of Skewness, is defined as
Q M M Q Q 2M Q Q Q 2M
S 3 d d 1 3 d 1 3 1 d
Q
Q M M Q Q Q Q Q
3 d d 1 3 1 3 1
The value of S will lie between – 1 and + 1.
Q
It may be noted here that S and S are not comparable, though, in the absence of
k Q
skewness, both of them are equal to zero.
(b) Using Percentiles
Bowley's measure of skewness leaves 25% observations on each extreme of the
distribution and hence is based only on the middle 50% of the observations. As an
improvement to this, Kelly suggested a measure based on the middle 80% of the
observations.

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