Statical Theory -Chapter 4 Measure Of Dispersion, Moments and Skewness
LESSON 4 MEASURES OF DISPERSION
LESSON 4
MEASURES OF DISPERSION
Why dispersion?
Measures of central tendency, Mean, Median, Mode, etc., indicate the central position of a series. They indicate the general magnitude of the data but fail to reveal all the peculiarities and characteristics of theseries. In other words, they fail to reveal the degree of the spread out or the extent of the variability inindividual items of the distribution. This can be explained by certain other measures, known as ‘Measures ofDispersion’ or Variation.
We can understand variation with the help of the following example :
----------------------------------------------
Series 1 Series 11 Series III
---------------------------------------------
10 2 10
10 8 12
10 20 8
---------------------------------------------
∑X = 30 30 30
----------------------------------------------
In all three series, the value of arithmetic mean is 10. On the basis of this average, we can say that the series are alike. If we carefully examine the composition of three series, we find the following differences:
(i) In case of 1st series, three items are equal; but in 2nd and 3rd series, the items are unequal and do not follow any specific order.
(ii) The magnitude of deviation, item-wise, is different for the 1st, 2nd and 3rd series. But all these deviations cannot be ascertained if the value of simple mean is taken into consideration.
(iii) In these three series, it is quite possible that the value of arithmetic mean is 10; but the value of median may differ from each other. This can be understood as follows ;
I II III
10 2 8
10 Median 8 Median 10 Median
10 20 12
The value of Median’ in 1st series is 10, in 2nd series = 8 and in 3rd series = 10. Therefore, the value of the Mean and Median are not identical.
(iv) Even though the average remains the same, the nature and extent of the distribution of the size of the items may vary. In other words, the structure of the frequency distributions may differ even (though their means are identical.
What is Dispersion?
Simplest meaning that can be attached to the word ‘dispersion’ is a lack of uniformity in the sizes or quantities of the items of a group or series. According to Reiglemen, “Dispersion is the extent to which the magnitudes or quantities of the items differ, the degree of diversity.” The word dispersion may also be used to indicate the spread of the data.
In all these definitions, we can find the basic property of dispersion as a value that indicates the extent to which all other values are dispersed about the central value in a particular distribution.
Properties of a good measure of Dispersion
There are certain pre-requisites for a good measure of dispersion:
1. It should be simple to understand.
2. It should be easy to compute.
3. It should be rigidly defined.
4. It should be based on each individual item of the distribution.
5. It should be capable of further algebraic treatment.
6. It should have sampling stability.
7. It should not be unduly affected by the extreme items.
Types of Dispersion
The measures of dispersion can be either ‘absolute’ or “relative”. Absolute measures of dispersion are expressed in the same units in which the original data are expressed. For example, if the series is expressed as Marks of the students in a particular subject; the absolute dispersion will provide the value in Marks. The only difficulty is that if two or more series are expressed in different units, the series cannot be compared on the basis of dispersion.
‘Relative’ or ‘Coefficient’ of dispersion is the ratio or the percentage of a measure of absolute dispersion to an appropriate average. The basic advantage of this measure is that two or more series can be compared with each other.
Methods of Dispersion
Methods of studying dispersion are divided into two types :
(i) Mathematical Methods: We can study the ‘degree’ and ‘extent’ of variation by these methods. In this category, commonly used measures of dispersion are :
(a) Range
(b) Quartile Deviation
(c) Average Deviation
(d) Standard deviation and coefficient of variation.
(ii) Graphic Methods: Where we want to study only the extent of variation, whether it is higher or lesser a Lorenz-curve is used.
Mathematical Methods
(a) Range
It is the simplest method of studying dispersion. Range is the difference between the smallest value and the largest value of a series. While computing range, we do not take into account frequencies of different groups.
Formula: Absolute Range = L – S
Coefficient of Range =
where, L represents largest value in a distribution
S represents smallest value in a distribution
We can understand the computation of range with the help of examples of different series,
(i) Raw Data: Marks out of 50 in a subject of 12 students, in a class are given as follows:
12, 18, 20, 12, 16, 14, 30, 32, 28, 12, 12 and 35.
In the example, the maximum or the highest marks obtained by a candidate is ‘35’ and the lowest marks obtained by a candidate is ‘12’. Therefore, we can calculate range;
L = 35 and S = 12
Absolute Range = L – S = 35 – 12 = 23 marks
Coefficient of Range =
(ii) Discrete Series
----------------------------------------------------------
Marks of the Students in No. of students
Statistics (out of 50)
(X) (f)
-----------------------------------------------------------
Smallest 10 4
12 10
18 16
Largest 20 15
-----------------------------------------------------------
Total = 45
-----------------------------------------------------------
Absolute Range = 20 – 10 = 10 marks
Coefficient of Range =
(iii) Continuous Series
------------------------------------------
X Frequencies
------------------------------------------
10 – 15 4
S = 10 15 – 20 10
L = 30 20 – 25 26
25 – 30 8
-------------------------------------------
Absolute Range = L – S = 30 – 10 = 20 marks
Coefficient of Range =
Range is a simplest method of studying dispersion. It takes lesser time to compute the ‘absolute’ and ‘relative’ range. Range does not take into account all the values of a series, i.e. it considers only the extreme items and middle items are not given any importance. Therefore, Range cannot tell us anything about the character of the distribution. Range cannot be computed in the case of “open ends’ distribution i.e., a distribution where the lower limit of the first group and upper limit of the higher group is not given.
The concept of range is useful in the field of quality control and to study the variations in the prices of the shares etc.
(b) Quartile Deviations (Q.D.)
The concept of ‘Quartile Deviation does take into account only the values of the ‘Upper quartile (Q3) and the ‘Lower quartile’ (Q1). Quartile Deviation is also called ‘inter-quartile range’. It is a better method when we are interested in knowing the range within which certain proportion of the items fall.
‘Quartile Deviation’ can be obtained as :
(i) Inter-quartile range = Q3 – Q1
(ii) Semi-quartile range =
(iii) Coefficient of Quartile Deviation =
Calculation of Inter-quartile Range, semi-quartile Range and Coefficient of Quartile Deviation in case of Raw Data
Suppose the values of X are : 20, 12, 18, 25, 32, 10
In case of quartile-deviation, it is necessary to calculate the values of Q1 and Q3 by arranging the given data in ascending of descending order.
For Complete Excerise click the image link 🔗 below :
LESSON 4
MEASURES OF DISPERSION
Why dispersion?
Measures of central tendency, Mean, Median, Mode, etc., indicate the central position of a series. They indicate the general magnitude of the data but fail to reveal all the peculiarities and characteristics of theseries. In other words, they fail to reveal the degree of the spread out or the extent of the variability inindividual items of the distribution. This can be explained by certain other measures, known as ‘Measures ofDispersion’ or Variation.
We can understand variation with the help of the following example :
----------------------------------------------
Series 1 Series 11 Series III
---------------------------------------------
10 2 10
10 8 12
10 20 8
---------------------------------------------
∑X = 30 30 30
----------------------------------------------
In all three series, the value of arithmetic mean is 10. On the basis of this average, we can say that the series are alike. If we carefully examine the composition of three series, we find the following differences:
(i) In case of 1st series, three items are equal; but in 2nd and 3rd series, the items are unequal and do not follow any specific order.
(ii) The magnitude of deviation, item-wise, is different for the 1st, 2nd and 3rd series. But all these deviations cannot be ascertained if the value of simple mean is taken into consideration.
(iii) In these three series, it is quite possible that the value of arithmetic mean is 10; but the value of median may differ from each other. This can be understood as follows ;
I II III
10 2 8
10 Median 8 Median 10 Median
10 20 12
The value of Median’ in 1st series is 10, in 2nd series = 8 and in 3rd series = 10. Therefore, the value of the Mean and Median are not identical.
(iv) Even though the average remains the same, the nature and extent of the distribution of the size of the items may vary. In other words, the structure of the frequency distributions may differ even (though their means are identical.
What is Dispersion?
Simplest meaning that can be attached to the word ‘dispersion’ is a lack of uniformity in the sizes or quantities of the items of a group or series. According to Reiglemen, “Dispersion is the extent to which the magnitudes or quantities of the items differ, the degree of diversity.” The word dispersion may also be used to indicate the spread of the data.
In all these definitions, we can find the basic property of dispersion as a value that indicates the extent to which all other values are dispersed about the central value in a particular distribution.
Properties of a good measure of Dispersion
There are certain pre-requisites for a good measure of dispersion:
1. It should be simple to understand.
2. It should be easy to compute.
3. It should be rigidly defined.
4. It should be based on each individual item of the distribution.
5. It should be capable of further algebraic treatment.
6. It should have sampling stability.
7. It should not be unduly affected by the extreme items.
Types of Dispersion
The measures of dispersion can be either ‘absolute’ or “relative”. Absolute measures of dispersion are expressed in the same units in which the original data are expressed. For example, if the series is expressed as Marks of the students in a particular subject; the absolute dispersion will provide the value in Marks. The only difficulty is that if two or more series are expressed in different units, the series cannot be compared on the basis of dispersion.
‘Relative’ or ‘Coefficient’ of dispersion is the ratio or the percentage of a measure of absolute dispersion to an appropriate average. The basic advantage of this measure is that two or more series can be compared with each other.
Methods of Dispersion
Methods of studying dispersion are divided into two types :
(i) Mathematical Methods: We can study the ‘degree’ and ‘extent’ of variation by these methods. In this category, commonly used measures of dispersion are :
(a) Range
(b) Quartile Deviation
(c) Average Deviation
(d) Standard deviation and coefficient of variation.
(ii) Graphic Methods: Where we want to study only the extent of variation, whether it is higher or lesser a Lorenz-curve is used.
Mathematical Methods
(a) Range
It is the simplest method of studying dispersion. Range is the difference between the smallest value and the largest value of a series. While computing range, we do not take into account frequencies of different groups.
Formula: Absolute Range = L – S
Coefficient of Range =
where, L represents largest value in a distribution
S represents smallest value in a distribution
We can understand the computation of range with the help of examples of different series,
(i) Raw Data: Marks out of 50 in a subject of 12 students, in a class are given as follows:
12, 18, 20, 12, 16, 14, 30, 32, 28, 12, 12 and 35.
In the example, the maximum or the highest marks obtained by a candidate is ‘35’ and the lowest marks obtained by a candidate is ‘12’. Therefore, we can calculate range;
L = 35 and S = 12
Absolute Range = L – S = 35 – 12 = 23 marks
Coefficient of Range =
(ii) Discrete Series
----------------------------------------------------------
Marks of the Students in No. of students
Statistics (out of 50)
(X) (f)
-----------------------------------------------------------
Smallest 10 4
12 10
18 16
Largest 20 15
-----------------------------------------------------------
Total = 45
-----------------------------------------------------------
Absolute Range = 20 – 10 = 10 marks
Coefficient of Range =
(iii) Continuous Series
------------------------------------------
X Frequencies
------------------------------------------
10 – 15 4
S = 10 15 – 20 10
L = 30 20 – 25 26
25 – 30 8
-------------------------------------------
Absolute Range = L – S = 30 – 10 = 20 marks
Coefficient of Range =
Range is a simplest method of studying dispersion. It takes lesser time to compute the ‘absolute’ and ‘relative’ range. Range does not take into account all the values of a series, i.e. it considers only the extreme items and middle items are not given any importance. Therefore, Range cannot tell us anything about the character of the distribution. Range cannot be computed in the case of “open ends’ distribution i.e., a distribution where the lower limit of the first group and upper limit of the higher group is not given.
The concept of range is useful in the field of quality control and to study the variations in the prices of the shares etc.
(b) Quartile Deviations (Q.D.)
The concept of ‘Quartile Deviation does take into account only the values of the ‘Upper quartile (Q3) and the ‘Lower quartile’ (Q1). Quartile Deviation is also called ‘inter-quartile range’. It is a better method when we are interested in knowing the range within which certain proportion of the items fall.
‘Quartile Deviation’ can be obtained as :
(i) Inter-quartile range = Q3 – Q1
(ii) Semi-quartile range =
(iii) Coefficient of Quartile Deviation =
Calculation of Inter-quartile Range, semi-quartile Range and Coefficient of Quartile Deviation in case of Raw Data
Suppose the values of X are : 20, 12, 18, 25, 32, 10
In case of quartile-deviation, it is necessary to calculate the values of Q1 and Q3 by arranging the given data in ascending of descending order.
For Complete Excerise click the image link 🔗 below :
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