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Unit 8: Skewness and Kurtosis: Karl Pearson, Bowley, Kelly's Methods

In this case skewness is present. It is important to note that if Q is farther away from median Notes
1
than the Q , then skewness will be negative and if the case is opposite the skewness will be
3
positive. Dr. Bowley has given the following method of skewness.
S k = (Q − M − ) ( 3 − 1 = Q + 3 Q − 1 2M
)M Q
Coefficient of Skewness is
(Q − M ) (M Q− − )
Coefficient of S k = ( 3 M + )Q − ( 3 − 1 1 )M Q

Q + Q − 2M
1
Coefficient of S k = 3 Q – Q 1
3
The calculation of median and quartiles in the case of individual, discrete and continuous series
is already explained in Unit - 4.
Example 4: If sum and difference of two quartiles are 22 and 8 respectively. Find the co-efficient
of skewness when the median is 10.5.
Solution: Given Q − Q 1 = 8; Q + Q = 22 and M = 10.5
3
1
3
−
Q + Q − 2M − ( 22 2 )10.5 22 21 1
1
Now, Coefficient of S = 3 Q – Q 1 = 8 = 8 = 8 = 0.125
k
3
Example 5: If Bowley’s co-efficient of skewness is – 0.36, Q = 8.6 and median = 12.3. What is the
1
quartile co-efficient of dispersion ?
Solution: Given, Bowley’s Coefficient of S = – 0.36, Q = 8.6, M = 12.3 Coefficient of Q.D = ?
k
1
×
−
Q + Q − 2M Q + 8.6 2 12.3
1
Coefficient of S k = 3 Q – Q 1 or – 0.36 = 3 Q − 8.6
3
3
−
or − 0.36 (Q − 3 8.6 ) = Q + 3 8.6 24.6
or − 0.36Q + 3 3.096 = Q − 16
3
or − 0.36Q − 3 Q 3 = – 16 – 3.096
or − 1.36Q 3 = – 19.096

19.095
or Q 3 = 1.36 = 14.04

−
Q – Q 14.04 8.6 5.44
Coefficient of Q.D. = 3 1 = = = 0.24
+
Q + Q 1 14.04 8.6 22.64
3
(3) Kelly’s Method
Prof. Kelly has given a formula which is based on deciles or percentiles. It is defined as
P +
or S k = 90 P − 10 2P 50
or S k = D + 9 D − 1 2D 5
Coefficient of skewness is defined as

LOVELY PROFESSIONAL UNIVERSITY 121