Seminar on
MEASURES OF DISPERSION
Presented by
JAYAKUMARA
Research Scholar
1
INTRODUCTION
The Measures of central tendency gives us a birds
eye view of the entire data they are called averages
of the first order, it serve to locate the centre of the
distribution but they do not reveal how the items
are spread out on either side of the central value.
The measure of the scattering of items in a
distribution about the average is called dispersion.
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INTRODUCTION
Dispersion measures the extent to which the items
vary from some central value. It may be noted that
the measures of dispersion or variation measure
only the degree but not the direction of the
variation. The measures of dispersion are also
called averages of the second order because they
are based on the deviations of the different values
from the mean or other measures of central
tendency which are called averages of the first
order.
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DEFINITION
In the words of Bowley “Dispersion is the
measure of the variation of the items”
According to Conar “Dispersion is a
measure of the extent to which the
individual items vary”
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METHODS OF MEASURING
DISPERSION
Range
Quartile Deviation
Mean Deviation
Standard Deviation
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RANGE
It is defined as the difference between the
smallest and the largest observations in a
given set of data.
Formula is R = L – S
Ex. Find out the range of the given
distribution: 1, 3, 5, 9, 11
The range is 11 – 1 = 10.
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QUARTILE DEVIATION
It is the second measure of dispersion, no doubt
improved version over the range. It is based on the
quartiles so while calculating this may require
upper quartile (Q3) and lower quartile (Q1) and
then is divided by 2. Hence it is half of the
deference between two quartiles it is also a semi
inter quartile range.
The formula of Quartile Deviation is
(Q D) = Q3 - Q1
2
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MEAN DEVIATION
Mean Deviation is also known as average
deviation. In this case deviation taken from
any average especially Mean, Median or
Mode. While taking deviation we have to
ignore negative items and consider all of
them as positive. The formula is given
below
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MEAN DEVIATION
The formula of MD is given below
MD = d
N (deviation taken from mean)
MD = m
N (deviation taken from median)
MD = z
N (deviation taken from mode)
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STANDARD DEVIATION
The concept of standard deviation was first
introduced by Karl Pearson in 1893. The
standard deviation is the most useful and the
most popular measure of dispersion. Just as
the arithmetic mean is the most of all the
averages, the standard deviation is the best
of all measures of dispersion.
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STANDARD DEVIATION
The standard deviation is represented by the Greek
letter (sigma). It is always calculated from the
arithmetic mean, median and mode is not
considered. While looking at the earlier measures
of dispersion all of them suffer from one or the
other demerit i.e.
Range –it suffer from a serious drawback
considers only 2 values and neglects all the other
values of the series.
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STANDARD DEVIATION
Quartile deviation considers only 50% of the item and
ignores the other 50% of items in the series.
Mean deviation no doubt an improved measure but ignores
negative signs without any basis.
Karl Pearson after observing all these things has given us a
more scientific formula for calculating or measuring
dispersion. While calculating SD we take deviations of
individual observations from their AM and then each
squares. The sum of the squares is divided by the number
of observations. The square root of this sum is knows as
standard deviation.
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MERITS OF STANDARD
DEVIATION
Very popular scientific measure of dispersion
From SD we can calculate Skewness, Correlation
etc
It considers all the items of the series
The squaring of deviations make them positive
and the difficulty about algebraic signs which was
expressed in case of mean deviation is not found
here.
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DEMERITS OF STANDARD
DEVIATION
Calculation is difficult not as easier as Range and
QD
It always depends on AM
Extreme items gain great importance
The formula of SD is = √∑d2
N
Problem: Calculate Standard Deviation of the following
series
X – 40, 44, 54, 60, 62, 64, 70, 80, 90, 96
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Solution :
NO OF YOUNG ADULTS
VISIT TO THE LIBRARY
IN 10 DAYS (X)
d=X - A.M
d2
40
-26
676
44
-22
484
54
-12
144
60
-6
36
62
-4
16
64
-2
4
70
4
16
80
14
196
90
24
596
96
30
900
N=10
X=660
d2 = 3048
Standard deviation
AM = X
N
= 660 = 66 AM
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SD = √∑d2
N
SD =√3048
10
= 17.46
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Bibliography
Sony, R.S.(2009). Essential business
mathematics and statistics. New Delhi: Ane
books.
Thank you
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