assignment

1. by
Dr. Rehan Ahmad Khan

2. MEASURES OF DISPERSION
DEFINITIONS, SCOPE AND LIMITATIONS
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7. Measures of Dispersion
The measures of central tendencies i.e. mean, median, mode etc. condense
the series or frequency distribution into a single figure which is used to
describe a distribution. It is quite possible that several series may have the
same average, but their individual observations may highly differ from the
average. In such cases an average may not be the most typical or
representation. We therefore, require some more information regarding the
spread of the data about the average. This is done by measuring the
spread/dispersion and a quantity that measures this characteristic is called a
measure of dispersion, spread or variability.
Dispersion
The degree of scatterness or variation of the numerical data about center (i.e.
mean or median) is known as dispersion. Hence the dispersion measures to
which the individual values vary from the particular average value.
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9. Objectives of Computing Dispersion
 Comparative study
• Measures of dispersion give a single value indicating the degree of
consistency or uniformity of distribution. This single value helps us in
making comparisons of various distributions.
• The smaller the magnitude (value) of dispersion, higher is the consistency
or uniformity and vice-versa.
 Reliability of an average
• A small value of dispersion means low variation between observations and
average. It means that the average is a good representative of observation
and very reliable.
• A higher value of dispersion means greater deviation among the
observations. In this case, the average is not a good representative and it
cannot be considered reliable.
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10. Objectives of Computing Dispersion
 Control the variability
• Different measures of dispersion provide us data of variability from different
angles, and this knowledge can prove helpful in controlling the variation.
• Especially in the financial analysis of business and medicine, these
measures of dispersion can prove very useful.
 Basis for further statistical analysis
• Measures of dispersion provide the basis for further statistical analysis like
computing correlation, regression, test of hypothesis, etc.
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11. Types of Dispersion
There are two types of dispersions viz: Absolute dispersion and relative
dispersion.
1. Absolute Dispersion
If we measure the dispersion and express it in terms of the original data, it is
known as absolute dispersion, e.g. the average salaries of 6 individuals is
Rs.7000 and average dispersion of salaries from average is Rs.1290. Then
Rs.1290 is called absolute dispersion.
2. Relative Dispersion
If the objective to compare the dispersion for more than one series for the
comparison purposes, then absolute dispersion cannot be used. For example,
if we have heights and weights of students in centimeters and kilograms
respectively. Then we cannot compare the variability of these two series
through absolute measure of dispersion as they are in two different units.
Hence, two series has to express as a ratio or percentage of the average and
such a measure is known as relative measure of dispersion. In short relative
measures of dispersion are used for comparing more than one series whereas
absolute dispersion are calculated for simple series. 11

12. Measures of Absolute Dispersion
Following are the various measures of absolute dispersion:
• Range
• Semi-Inter quartile range or Quartile Deviation
• Mean Deviation
• Variance/Standard deviation
Range
The range R is defined as the difference between the two extreme values, i.e.
the largest and the smallest of the distribution. In symbols:
Range = XL – XS
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where XL stands for the largest value, and
XS stands for the smallest value.
Example: The following marks were obtained by 15 students. Find the range.
51, 90, 40, 25, 7, 14, 28, 72, 44, 23, 65, 85, 3, 59, 67
Here XL (the largest value) = 90, and
XS (the smallest value) = 3
Range = XL – XS = 90 – 3 = 87 marks

13. Advantages of Range
• It is a relatively simple to determine the range, even for a large set of
numbers.
• It is easy to understand.
• It gives a rough and quick picture of variability of data.
Limitation of the Range
• It takes into consideration only the two extreme values in a set and does
not tell us anything at all about the other values in the set.
• It is highly unstable measure because it is based on only two extreme
values.
• It cannot be computed in case of open and classes.
• In spite of these limitations it is widely used in daily temperatures, Quality
Control Charts, Share Market, Precious metal markets etc.
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14. The Variance and the Standard Deviation
Variance
Variance is defined as: the mean of the squares of the deviation taken
from the arithmetic mean. When it is computed from the population,
the variance is denoted by , while computed from the sample denoted
by S2. In symbols
(for Population Data)
i
N
2

(X  )2
(forSampleData)
i
S2

(X  X )2
n n
S2
  
 
n
x2
 x 
2
n
14
S2
 x
x2
2

15. The Variance and the Standard Deviation
Standard Deviation
• It is the most important measure of dispersion and is widely used. Standard
deviation is defined as “the positively square-root of the arithmetic mean of
the squares of the deviations of the observations from their mean” or
simply as the positively square –root of variance. When it is computed from
the population, the standard deviation is denoted by , while from the
sample is denoted by S. In symbols
(for Population Data)
i
N
 
(X  )2
(forSampleData)
i
n
S 
(X  X )2
n n
S   
 
x2
 x 
2
x2
15
n
S  x 2


16. Advantages and disadvantages of Standard Deviation
Advantages
• It is simple to understand and easy to calculate
• It is rigorously defined and always give a definite value
• It is based on all the observations
• It is possible for further algebraic treatment
• It is less affected by the fluctuations of sampling and is a stable measure
• It is the basis for measuring the completion coefficient sampling and
statistical inferences.
• It is used to compare the variability of two or more distributions
Disadvantages
• It gives more weight to extreme values, because the values are squared
up.
• It is affected by the change in item in the series as is based on all the
observations
• It is not popular measure with the economists where the most of the data is
positively skewed
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Example: A random sample of 10 automobile parts companies gave the
following information about profit (in thousands of rupees):
2 4 1 5 9 7 1 1
1 9 2 0 5 2 9 1 5
Profits
Xi
Xi  X (X  X )2
i Direct Method
24 8.6 73.96 Step 1. Find the mean of the series ( X )
Step 2. Calculate the deviations of each value
from the mean i.e. Xi  X
Step 3. Square the deviations compute in Step
(2) and add them to get (X  X )2
i
Step 4. Divide this sum i.e. (X  X )2
by the
i
number of observations to obtain
variance
Step 5. Take the positive square root of
variance to obtain Standard Deviation
15 -0.4 0.16
9 -6.4 40.96
7 -8.4 70.56
11 -4.4 19.36
19 3.6 12.96
20 4.6 21.16
5 -10.4 108.16
29 13.6 184.96
15 -0.4 0.16
0 532.4
154
n
X   15.4
10
Xi 532.4
i
n 10
S2
   53.24
(X  X )2
532.4
10
i
n
S    53.24  Rs.7.30
(X  X )2

18. Example: The following table gives the weights of 9 students in a statistics
class. Calculate the Variance and Standard Deviation:
Weights (kg): 45, 52, 56, 67, 59, 70, 40, 58, 67.
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X i X 2
i
4 5 2 0 2 5
5 2 2 7 0 4
5 6 3 1 3 6
6 7 4 4 8 9
5 9 3 4 8 1
7 0 4 9 0 0
4 0 1 6 0 0
5 8 3 3 6 4
6 7 4 4 8 9
 X  514  30188
 X 2
9 9
n
S2
  
 
 
n
30188  514 
2
  
 
 3354.2222  (57.11)2
 3354.2222  3261.5521
 92.6701
X 2
 X 
2
S2
S   92.6701  9.63 kg.