Definition of dispersion

Contents
1. Introduction Of measures of dispersion.
1. Definition of Dispersion.
2. Purpose of Dispersion
3. Properties of a Good Measures of Dispersion
4. Measures of Dispersion
5. Range
6. Quartile deviation.
7. Mean deviation.
8. Variance
9. Standard deviation.
10. Coefficient of variance.
11. Conclusion
12. References.

Definition of Dispersion
i. Dispersion measures the variability of a set of
observations among themselves or about some
central values.
ii. According to Brooks & Dicks, “Dispersion or Spread
is the degree of the scatter or variations of the
variables about some central values.”

Purposes of Dispersion
i. To determine the reliability of average;
ii. To serve as a basis for the control of the
variability;
iii. To compare to or more series with regard to
their variability; and
iv. To facilitate the computation of other
statistical measures.

Properties of a Good Measures of Dispersion
 It should be simple to understand.
 It is should be easy to compute.
 It is should rigidly defined.
 It should be based on all observations.
 It should have sampling fluctuation.
 It should be suitable for further algebraic treatment.
 It should be not be affected by extreme observations.

Measures of Dispersion
• The numerical values by which we measure the
dispersion or variability of a set of data or a frequency
distribution are called measures of dispersion.
• There are two kinds of Measures of Dispersion:
1. Absolute measures of dispersion
2. Relative measures of dispersion

The Absolute measures of dispersion are:
1. Range
2. Quartile deviation
3. Mean deviation
4. Variance & standard deviation
The Relative measures of dispersion are:
1. Coefficient of range
2. Coefficient of quartile deviation
3. Coefficient of mean deviation
4. Coefficient of variation

1. Range
• Range is the difference between the largest & smallest
observation in set of data.
• In symbols, Range = L – S.
Where,
L = Largest value.
S = Smallest value.
In individual observations and discrete series, L and S are
easily identified.

2. Coefficient of Range
• The percentage ratio of range & sum of maximum &
minimum observation is known as Coefficient of Range.
• Coefficient of Range =
Xmax
– Xmin
Xmax
+ X𝑚𝑖𝑛×100

• The monthly incomes in taka of seven employees of a firm
are 5500,5750,6500,67 50,7000 & 8500. Compute Range
& Coefficient of Range.
•Solution
The range of the income of the employees is
Range = 8500-5500
= TK 3000

• Coefficient of Range =
Xmax
– Xmin
Xmax
+ Xmin
×100
=
8500−5500
8500+5500
× 100
=
3000
14000
× 100
= 21.43%

Merits and Demerits of Range
Merits
1. The range measure the total spread in the set of data.
2. It is rigidly defined.
3. It is the simplest measure of dispersion.
4. It is easiest to compute.
5. It takes the minimum time to compute.
6. It is based on only maximum and minimum values.

Demerits
1. It is not based on all the observations of a set of data.
2. It is affected by sampling fluctuation.
3. It is cannot be computed in case of open-end distribution.
4. It is highly affected by extreme values.

When To Use the Range
• The range is used when you have ordinal data or you are
presenting your results to people with little or no knowledge
of statistics.
• The range is rarely used in scientific work as it is fairly
insensitive.
• It depends on only two scores in the set of data, XL and XS
Two very different sets of data can have the same range:
1 1 1 1 9 vs 1 3 5 7 9

3. Quartile Deviation
•The average difference of 3rd & 1st Quartile is known
as Quartile Deviation.
•Quartile Deviation =
Q3−Q1
2

4. Coefficient of Quartile Deviation
• The relative measure corresponding to this measure,
called the coefficient of quartile deviation, is defied by
• Coefficient of Quartile Deviation= 𝑥 =
𝑄3_𝑄1
𝑄3+𝑄1
∗ 100

Advantage of Quartile Deviation
1. It is superior to range as a measure of variation.
2. It is useful in case of open-end distribution.
3. It is not affected by the presence of extreme values.
4. It is useful in case of highly skewed distribution.

Disadvantage of Quartile Deviation
1. It is not based on all observations.
2. It ignores the first 25% and last 25% of the observation.
3. It is not capable of mathematical manipulation.
4. It is very much affected by sampling fluctuation.
5. It is not a good measure of dispersion since it depends on
two position measure.

3. Mean Deviation
The difference of mean from their observation & their
mean is known as mean deviation

Mean Deviation for ungrouped data
Suppose X1, X2,……….,Xn are n values of variable, and is
the mean and the mean deviation (M.D.) about mean is
defined by

Mean Deviation for grouped data

4. Coefficient of Mean Deviation
•The Percentage ratio of mean deviation & mean is as
known as Coefficient of Mean Deviation(C.M.D.).

For Grouped & Ungrouped Data C.M.D.

Merits of Mean Deviation:
1. It is easy to understand and to compute.
2. It is less affected by the extreme values.
3. It is based on all observations.
Limitations of Mean Deviation:
1. This method may not give us accurate results.
2. It is not capable of further algebraic treatment.
3. It is rarely used in sociological and business studies.

5. Variance
•The square deviation of mean from their observation
and their mean is as known as variance.

6.Standard Deviation
•The square deviation of mean from their
observation & the square root variance is as
known as Standard Deviation

Merits of Standard Deviation
1. It is rigidly defied.
2. It is based on all observations of the distribution.
3. It is amenable to algebraic treatment.
4. It is less affected by the sampling fluctuation.
5. It is possible to calculate the combined standard deviation

Demerits of Standard Deviation
1. As compared to other measures it is difficult to
compute.
2. It is affected by the extreme values.
3. It is not useful to compare two sets of data when the
observations are measured in different ways.

6. Coefficient of Variation
• The percentage ratio of Standard deviation and mean
is as known as coefficient of variation

• Consider the measurement on yield and plant height of a paddy
variety. The mean and standard deviation for yield are 50 kg and
10 kg respectively. The mean and standard deviation for plant
height are 55 am and 5 cm respectively.
• Here the measurements for yield and plant height are in
different units. Hence the variabilities can be compared only by
using coefficient of variation.
• For yield, CV=
10
50
× 100
= 20%

Conclusion
• The measures of variations are useful for further treatment
of the Data collected during the study.
• The study of Measures of Dispersion can serve as the
foundation for comparison between two or more frequency
distributions.
• Standard deviation or variance is never negative.
• When all observations are equal, standard deviation is zero.
• when all observations in the data are increased or
decreased by constant, standard deviation remains the
same.

Reference
•Business Statistics, by Manindra Kumar Roy & Jiban
Chandra Paul, first edition,2012, page no 162-195.
•https://www.slideshare.net/sachinshekde9/measures-
of-dispersion-38120163.

Definition of dispersion

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