The average and measure of
dispersion can describe the distribution but they are not sufficient to
describe the nature of the distribution. For this purpose we use other concepts
known as Skewness and Kurtosis. The symmetrical and skewed distributions are shown by curves as
Skewness
Skewness means lack of
symmetry. A distribution is said to be symmetrical when the values are
uniformly distributed around the mean. For example, the following distribution
is symmetrical about its mean 3.
x : 1 2
3 4 5
frequency (f ) : 5 9 12 9 5
In a symmetrical
distribution the mean, median and mode coincide, that is, mean = median = mode.
Several measures are used to
express the direction and extent of skewness of a dispersion. The important
measures are that given by Pearson. The first one is the Coefficient of
Skewness:
For a symmetric distribution
Sk = 0. If the
distribution is negatively skewed then Sk
is negative and if it is positively skewed then Sk is positive. The range for Sk is from -3 to 3.
The other measure uses the b (read ‘beta’) coefficient
which is given by, 
where, m2 and m3 are the second
and third central moments. The second central moment m2 is nothing but
the variance. The sample estimate of this coefficient is 
where m2
and m3 are the sample central
moments given by
where, m2 and m3 are the second
and third central moments. The second central moment m2 is nothing but
the variance. The sample estimate of this coefficient is where m2
and m3 are the sample central
moments given by 
For a symmetrical
distribution b1 = 0. Skewness is positive or negative depending upon
whether m3 is positive or negative.
Kurtosis
A measure of the peakness or convexity of a
curve is known as Kurtosis.
It is clear from the above figure that all
the three curves, (1), (2) and (3) are symmetrical about the mean. Still they
are not of the same type. One has different peak as compared to that of others.
Curve (1) is known as mesokurtic (normal curve); Curve (2) is known as leptocurtic (leading curve) and
Curve (3) is known as platykurtic (flat curve). Kurtosis is measured by Pearson’s
coefficient, b2 (read ‘beta -
two’).It is given by 
.
The sample estimate of this coefficient is
where, m4
is the fourth central moment given by m4 = 
. The sample estimate of this coefficient is
where, m4
is the fourth central moment given by m4 =
The distribution is called normal if b2
= 3. When b2 is more than 3 the distribution is said to be
leptokurtic. If b2 is less than 3 the distribution is said to be
platykurtic.