Moments, Skewness and Kurtosis
Symmetry (समता )
A frequency distribution is symmetric about ‘a’,
if the corresponding frequency curve is symmetric about ‘a’. That is, the
ordinate at X = a divides frequency curve into two equal parts. These two parts
are mirror images of each other.
For example:
Class :
0-10 10-20 20-30 30-40 40-50
Frequency : 5 12 20 12 5
Properties
1.
For unimodal symmetric distribution,
Mean
= Median = Mode. It is bell shaped.
2.
The quartiles are equidistant.
Q3-Q2 = Q2-Q1
3.
The odd order central moments are equal to zero. i.e.
µ1 = µ3= µ5 =
….. = 0.
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Skewness
The
lack of symmetry or departure from symmetry of tails of a frequency curve is
called ‘skewness’. The shape of the curve is decided on the basis of the length
of its tails and its peakedness.
If both the tails of the curve are
equally distributed then the curve is symmetric and if both the tails of the
curve (left and right) are not equally distributed then the curve is asymmetric,
and is called a ‘skewed curve’.
For
a symmetric curve,
Mean = Median =
Mode.
and
for an asymmetric curve,
Mean ≠
Median ≠ Mode
If the left tail of a frequency curve is longer than the right tail, the curve is said to be negatively skewed. In this case mean and median are pulled away from mode to the left. Similarly if the right tail is longer than the left tail the curve is positively skewed and the mean and median are pulled away from mode to the right side.
In short, if the curve rises
gradually, reaches a maximum and then falls equally gradually it is called
symmetric curve. It look likes a bell. If the curve increases rapidly and falls
slowly it is called positively skewed and if it rises gradually and falls rapidly
it is called negatively skewed.
It is measured by the constant β1 and is given by
If β1 = 0 , distribution is symmetric.
β1 < 0 , distribution is negatively skew.
β1 > 0 , distribution is positively skew.
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Kurtosis (वशिंडता/ शिखरदोष )
Kurtosis
is the property of a distribution which expresses its relative peakedness.
As already mentioned, the shape of the curve is a decided on the basis of length of its tails and its peakedness. It is clear from the figure in which all the curves A, B and C are symmetrical and hence we define a new measure known as measure of kurtosis. Kurtosis gives us an idea about the flatness, peakedness of the curve. It is measured by the constant β2 and is given by
If β2 = 3 , distribution is mesokurtic. सामान्यशिखरी
β2 < 3 , distribution is platykurtic.सपाटशिखरी
β2 > 3 , distribution is leptokurtic.उंचशिखरी
The curve of type B, which is neither flat nor peak, is called mesokurtic curve or normal curve. The curve of type C, which is less peak than normal (more flat than normal) is called platykurtic curve. The curve of type A which is more peak than normal (less flat than normal) is called leptokurtic curve.
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