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Statistical Methods in Economics

Notes (b) Negative skewness: A distribution in which more than half of the area under the distribution
curve is to the left side of the mode, it is said to be a negatively skewed distribution. In this case,
the elongated tail is to the left and mean is less than the median which is less than mode and
Q − 3 M M Q .
<
−
1
Diagrammatically, the negative skewness can be explained as below:

XMM 0
Figure: 7

8.2 Karl Pearson’s, Bowley and Kelly’s Methods

The following are the main methods of measuring Skewness of data:
(1) Karl Pearson’s Method
(2) Bowley’s Method
(3) Kelly’s Method
(1) Karl Pearson’s Method
The method of skewness given by Karl Pearson is also called as First Measure of Skewness.
This method is based on the difference between the ‘mean’ and ‘mode’. Thus,

S k = X – Z , [where S = skewness; X = arithmetic mean; Z = mode]
k
In a symmetrical distribution, mean and mode coincide, so skewness will be zero. If X > Z , the
skewness will be positive and will have positive sign. If X < Z , the skewness will be negative
and will have negative sign.
• Karl Pearson’s Co-efficient of Skewness
Karl Pearson has given a formula for relative measure of skewness. It is also known as
Karl Pearson’s of coefficient Skewness of Pearsonian Coefficient of Skewness. The formula
is that the difference between the mean and mode is divided by the standard deviation.
Mean–Mode X – Z
Coefficient of S = StandardDeviation = σ ... (1)
k
Mode = 3 Median – 2 Mean

Or Z = 3M – 2 X
Substituting the value of modes in equation (1)
X – ( )3M – 2X ( ) 3X – M
S
Coefficient of k = =
σ σ
In a distribution, we have more than one mode, i.e., mode is ill-defined, we cannot apply
the above state formula. Then we have the following alternative formula:
( ) 3X – M
Coefficient of S = σ
k

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