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Introduction:
Measures of central tendency tells us only about the center of the distribution. It measures a single
value, which is not a sufficient information and does not describe all features of data. Another
feature of the observations is as to how the observations are spread about the center. The
observation may be close to the center or they may be spread away from the center. If the
observation are close to the center (usually the arithmetic mean or median), we say that dispersion,
scatter or variation is small. If the observations are spread away from the center, we say dispersion
is large. Suppose we have three groups of students who have obtained the following marks in a
test. The arithmetic means of the three groups are also given below:
(i) Group A: 46, 48, 50, 52, 54 ?̅
?=50
(ii) Group B: 30, 40, 50, 60, 70 ?̅
?=50
(iii) Group C: 40, 50, 60, 70, 80 ?̅
?=60
In a group A and B arithmetic means are equal i.e. ?̅
?=?̅
?=50. But in group A the observations
are concentrated on the center. All students of group A have almost the same level of performance.
We say that there is consistence in the observations in group A. In group B the mean is 50 but the
observations are not closed to the center. One observation is as small as 30 and one observation is
as large as 70. Thus there is greater dispersion in group B. In group C the mean is 60 but the spread
of the observations with respect to the center 60 is the same as the spread of the observations in
group B with respect to their own center which is 50. Thus in group B and C the means are different
but their dispersion is the same. In group A and C the means are different and their dispersions are
also different. Dispersion is an important feature of the observations and it is measured with the
help of the measures of dispersion, scatter or variation. The word variability is also used for this
idea of dispersion.
Measure of dispersion
Some brief definitions of dispersion are:
1. By dispersion we mean the extent to which the values are spread out from the average. The
measures used for this purpose are called measures of dispersion or measures of variation.
2. The degree to which numerical data tend to spread about an average value is called the
dispersion or variation of the data.
3. Dispersion or variation may be defined as a statistics signifying the extent of the
scatteredness of items around a measure of central tendency.
4. Dispersion or variation is the measurement of the scatter of the size of the items of a series
about the average.
Types of measure of dispersion:
There are two types of measure of dispersion which are:
(a) Absolute Measure of Dispersion
(b) Relative Measure of Dispersion
Absolute Measures of Dispersion:
These measures give us an idea about the amount of dispersion in a set of observations. They
give the answers in the same units as the units of the original observations. When the
observations are in kilograms, the absolute measure is also in kilograms. If we have two sets
of observations, we cannot always use the absolute measures to compare their dispersion. We
shall explain later as to when the absolute measures can be used for comparison of dispersion
in two or more than two sets of data. The absolute measures which are commonly used are:
1. The Range
2. The Quartile Deviation
3. The Mean Deviation
4. The Variance and Standard deviation
Relative Measure of Dispersion:
These measures are calculated for the comparison of dispersion in two or more than two sets of
observations. These measures are free of the units in which the original data is measured. If the
original data is in dollar or kilometers, we do not use these units with relative measure of
dispersion. These measures are a sort of ratio and are called coefficients. Each absolute measure
of dispersion can be converted into its relative measure. Thus the relative measures of dispersion
are:
1. Coefficient of Range or Coefficient of Dispersion.
2. Coefficient of Quartile Deviation or Quartile Coefficient of Dispersion.
3. Coefficient of Mean Deviation or Mean Deviation of Dispersion.
4. Coefficient of Standard Deviation or Standard Coefficient of Dispersion.
5. Coefficient of Variation (a special case of Standard Coefficient of Dispersion)
Range and Coefficient of Range
The Range:
Range is defined as the difference between the maximum and the minimum observation of the
given data. If ? denotes the maximum observation ?0 denotes the minimum observation then the
range is defined as.
? = ? − ?0
In case of grouped data, the range is the difference between the upper boundary of the highest class
and the lower boundary of the lowest class. It is also calculated by using the difference between
the mid points of the highest class and the lowest class. It is the simplest measure of dispersion. It
gives a general idea about the total spread of the observations. It does not enjoy any prominent
place in statistical theory. But it has its application and utility in quality control methods which are
used to maintain the quality of the products produced in factories. The quality of products is to be
kept within certain range of values. The range is based on the two extreme observations. It gives
no weight to the central values of the data. It is a poor measure of dispersion and does not give a
good picture of the overall spread of the observations with respect to the center of the observations.
Coefficient of Range:
It is relative measure of dispersion and is based on the value of range. It is also called range
coefficient of dispersion. It is defined as:
? ? ? =
? − ?0
? + ?0
Example:
Set A: 10, 15, 18, 20, 20.
Set B: 30, 35, 40, 45, 50.
The values of range and coefficient of range are calculated as:
Range Coefficient of Range
Set A: (Mathematics) 20−10=10 20 − 10
20 + 10 = 0.33
Set B: (English) 50−30=20 50 − 30
50 + 30 = 0.25
In set A the range is 10 and in set B the range is 20. Apparently it seems as if there is greater
dispersion in set B. But this is not true. The range of 20 in set B is for large observations and the
Course Title: Introduction to Statistics Instructor: Muhammad Fawad
4
range of 10 in set A is for small observations. Thus 20 and 10 cannot be compared directly. Their
base is not the same. Marks in Mathematics are out of 25 and marks of English are out of 100.
Thus, it makes no sense to compare 10 with 20. When we convert these two values into coefficient
of range, we see that coefficient of range for set A is greater than that of set B. Thus there is greater
dispersion or variation in set A. The marks of students in English are more stable than their marks
in Mathematics.
Example:
Following are the wages of 8 workers of a factory. Find the range and the coefficient of range.
Wages in Dollars 1400, 1450, 1520, 1380, 1485, 1495, 1575, 1440.
Solution:
Here Largest value =?=1575 and Smallest Value =?0=1380
Range =? − ? = 1575 − 1380 = 195
? ? ? =
? − ?0
? + ?0
=
1575 − 1380
1575 + 1380 =
195
2955 = 0.66
Example:
The following distribution gives the numbers of houses and the number of persons per house.
Number of Persons 1 2 3 4 5 6 7 8 9 10
Number of Houses 26 113 120 95 60 42 21 14 5 4
Calculate the range and coefficient of range.
Solution:
Here Largest value =? = 10 and Smallest Value =?0 = 1
? = ? − ?0 = 10 − 1 = 9
? ? ? =
? − ?0
? + ?0
=
10 − 1
10 + 1
=
9
11 = 0.818
Example:
Find the range of the weight of the students of a university.
Weights (Kg) 60−62 63−65 66−68 69−71 72−74
Number of Students 5 18 42 27 8
Calculate the range and coefficient of range.