7.
4 Kurtosis 207
7.4 Kurtosis
Two or more distributions may have identical
average, variation and skewness, but they show
different degrees of concentration of values
of observation around mode and hence they
may show different degrees of peakedness
of the distributions. Kurtosis is the measure
of Peakedness or flatness of the frequency
distribution.
Types of Kurtosis
Look at the following frequency curves
All of them have same centre of location, dispersion and are symmetrical, but
they differ in peakedness. In figure 1 the curve is more peaked than others and
is called as Lepto Kurtic. In figure 2 the curve is less peaked than others or it
is more flat topped, is called as Platty Kurtic. In figure 3 curve is neither more
peaked nor more flat topped or it is moderately peaked, is called Meso Kurtic.
Meso Kurtic is also known as Natural Curve or Normal Curve.
208 Skewness and Kurtosis
The word “Kurtosis” is derived from the Greek word meaning “Humped” or
“Bulginess”. Famous British Statistician William S Gosset ( Student) has very
humorously pointed out the nature of kurtosis in his research paper “Errors
of Routine Analysis” that platy kurtic curves, like the platypus, are squat with
short tails; lepto kurtic curves are high with long tails like the Kangaroos
noted for lepping. Gossets little sketch is reproduced as shown in the figure.
7.5 Measures of Kurtosis
Kurtosis is measured by coefficient β2 (pronounced as beeta two) is defined by
µ4
β2 =
µ22
µ4
= 4
σ
Karl Pearson defined one more coefficient of kurtosis as γ2 = β2 − 3
If β2 = 3 ( ie,γ2 = 0) the curve is meso kurtic.
If β2 > 3 ( ie, γ2 > 0) the curve is lepto kurtic.
If β2 < 3 ( ie, γ2 < 0) the curve is platy kurtic.
Illustration 7.14
The first four central moments of a distribution are 0, 2.5, 0.7 and 18.75. Test
the kurtosis of the distribution
7.5 Measures of Kurtosis 209
Solution. Given µ1 = 0, µ2 = 2.5, µ3 = 0.7, µ4 = 18.75
µ4
Coefficient of Kurtosis,β2 =
µ22
18.75
=
2.52
=3
Since β2 = 3 the distribution is meso kurtic.
Illustration 7.15
For the following data calculate coefficient of skewness and coefficient of
kurtosis and comment on it 2, 3, 7, 8, 10
Solution.
30
x= =6
X (x − 6) (x − 6)2 (x − 6)3 (x − 6)4 P5
(x − x)
2 -4 16 -64 256 µ1 = = 0(Always)
n
3 -3 9 -27 81
(x − x)2 46
P
7 1 2 1 1 µ2 = = = 9.2
P n 3 5
8 4 4 8 16 (x − x) −18
µ3 = = = −3.6
10 4 16 64 256 5
P n 4
30 0 46 -18 610 (x − x) 610
µ4 = = = 122
n 5
µ23 (−3.6)2
Coefficient of skewness, β1 = = = 0.0166
µ32 9.23
Since µ3 < 0, the distribution is negatively skewed
µ4 122
Coefficient of kurtosis, β2 = = = 1.44
µ22 9.22
Since β2 < 3, the distribution is platy kurtic
210 Skewness and Kurtosis
Let us sum up
In this chapter the concept of skewness and kurtosis were introduced. Skewness
means lack of symmetry where as kurtosis is the measure of peakedness. There are
two types of skewness, positive skewness and negative skewness. If a frequency curve
has longer tail towards right side of mode, then it is said to be positively skewed. If a
frequency curve has longer tail towards left side of mode, it is said to be negatively
skewed. If a curve is relatively narrow and peaked at the top, it is called leptokurtic.
The curve which is more flat topped is called platy kurtic. The curve which is neither
more peaked nor more flat topped is called meso kurtic. Various measures of skewness
and kurtosis were also discussed here. Measures of skewness indicate to what extend
and in what direction the distribution of variable differs from symmetry of a frequency
curve. Measures of kurtosis denote the shape of top of a frequency curve. For a
symmetric curve Karl Persons coefficient of skewness , Bowleys coefficient of skewness
and moment coefficient of skewness are all equal to zero.
Learning outcomes
After transaction of this unit, the learner:-
• distinguishes symmetric and asymmetric distributions.
• recognises skewness of distributions.
• evaluates and interprets nature of skewness.
• explains kurtosis of distributions.
• evaluates and interprets types of kurtosis.
