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MEASURES OF DISPERSION
Made by :
BhanwarIshaan Sood
Jasmine Singh
Apandeep Singh
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Definition of Dispersion
Dispersion indicates the measure of the extent to
which individual items differ. It indicates lack of
uniformity in the size of items.
Dispersion or spread is the degree of the scatter or
variation of the variables about central value
OR
The degree to which numerical data tend to spreadabout an average value is called the Variation or
dispersion .
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Objectives of
Measuring Dispersion:
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It should be rigidly defined.
It should be simple to understand & easy
to calculate.
It should be based upon all values of given
data.It should be capable of further mathematical
treatment.
It should have sampling stability. It should be not be unduly affected by
extreme values.
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Absolute Measures of Dispersion:The measures of dispersion which are expressed in
terms of original units of a data are termed as
Absolute Measures.
Relative Measures of Dispersion:Relative measures of dispersion, are also known as
coefficients of dispersion, are obtained as ratios or
percentages.
These are pure numbers independent of the units
of measurement.
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Methods of Studying Dispersion:
1. Range
2. Quartile Deviation or Semi-inter quartile Range.
3. Mean Deviation.
4. Standard Deviation.
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RANGE
Definition:For Unclassified data: Range is defined as the
difference between the largest and the smallest values
of the data,
Symbolically,R = L S
Where L = Largest value, S = Smallest value, R = Range
The relative measure of range is defined as,
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RANGE
Definition:
For Classified data: Range is defined as the differencebetween the upper boundary of last class interval and
the lower boundary of first class boundary of the
distribution.
Symbolically,R = ULI LFI
Where ULI = upper boundary of last class interval,
LFI = lower boundary of first class interval, R = Range
The relative measure of range is defined as,
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MERITS OF RANGE :
1. Range is rigidly defined.2. Range is simple to understand and easy to calculate.
DEMERITS OF RANGE :
1. Range is not based upon all observation of given data.
2. Range is not capable for further mathematical treatment.
3. Range is much affected by extreme values.
4. Range is much affected by sampling variation.
5. Range can not be calculated for open end classes without
any assumptions.
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What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
Soln: The largest score (L) is 9;
The smallest score (S) is 1;
Range= R =L - S = 9 - 1 = 8.
Coefficient of Range = R = = = = 0.8
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QUARTILE
DEVIATION
Definition:Quartile Deviation(Q.D.) is defined as
Q.D. =
Where Q3 = Upper (Third) quartile,
Q1 = Lower (First) quartile
The relative measure of quartile deviation is defined
as,
Coefficient of Q.D. =
13
13
2
13QQ
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MERITS OF QUARTILE DEVIATION :
1. Quartile deviation is rigidly defined.
2. Quartile deviation is simple to understand and easy tocalculate.
2. Quartile deviation is not affected by extreme values.
3. Quartile deviation can be calculated for open end classes
without assumptions
DEMERITS OF QUARTILE DEVIATION :
1. Quartile deviation is not based upon all observations of
data.
2. Quartile deviation is not capable of further mathematical
treatment.
3. Quartile deviation is much affected by sampling
fluctuations
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Range and Quartile deviation are not
based upon all observations. They are
positional measures of dispersion. They do not
show any scatter of the observations from an
average. The mean deviation based upon all
the observations.
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MEAN DEVIATION :
Definition:For Unclassified data: Let x1, x2,., xn are n observations of given
data. If n values x1, x2,., xn have an Arithmetic mean thenare the deviations of values from mean. Mean deviation about
mean is defined as follow,
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Similarly, If Me is median of given data, Then
Men deviation about median is given by,
If Mo is the mode of given data. Than Mean
Deviation about mode is,
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For Classified data: Let the variable X has values x1, x2,.,
xn with frequencies f1, f2,., fn
If n values x1, x2,., xn have an Arithmetic mean thenMean deviation about mean is defined as follow,
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Similarly, If Me is median of given data, Then Mean
deviation about median is given by,
If Mo is the mode of given data. Than Mean Deviation
about mode is,
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Merits of Mean Deviation
1. Mean deviation is rigidly defined.
2. Mean deviation is simple to understand and easy to calculate.3. Mean deviation is based upon all observations,
Demerits of Mean Deviation
1. The greatest drawback of Mean deviation is that algebraic signs
are ignored while taking deviations from items.
2. Mean deviation is not capable of further mathematical treatment.
3. Mean deviation is much affected by sampling variation.
4. Mean deviation is much affected by extreme values.5. There no hard & fast rule in the selection of particular average,
with respect to which the deviation are computed.
6. Mean deviation can not be calculated for open end classes
without any assumptions.
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Definition:
For Classified data: Let variables X has values x1,
x2,., xn with frequencies f1, f2,., fn . If n values x1,
x2,., xn have an Arithmetic mean Than Standarddeviation is given by
Where N = Total frequency
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STANDERD DEVIATION
Definition:
For Unclassified data: Let x1, x2,., xn are n
observations of given data. If n values x1, x2,., xn have
an Arithmetic mean Than
Standard deviation is given by
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Coefficient of variation (C.V.)
When this is expressed as percentage, that is multiplied by 100, it is
called Coefficient of variation. The coefficient of variation is the
ratio of standard deviation to the arithmetic mean expressed as
percentage.
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Merits of Standard deviation
1. Standard deviation is rigidly defined.
2. Standard deviation is based upon all observations.3. Standard deviation is capable of further mathematical
treatment.
4. Standard deviation is less affected by sampling
variations
Demerits of Standard deviation:
1. Standard deviation is not simple to understand
and not easy to calculate.
1. Standard deviation is much affected by extremevalues.
3. Standard deviation can not be calculated for
open end classes without any assumptions.
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