ICAI Notes- Unit 2: Measures of Central Tendency and Dispersion | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


14.30
STATISTICS
The second important characteristic of a distribution is given by dispersion. Two distributions
may be identical in respect of its first important characteristic i.e. central tendency and yet they
may differ on account of scatterness. The following figure shows a number of distributions having
identical measure of central tendency and yet varying measure of scatterness. Obviously,
distribution is having the maximum amount of dispersion.
After reading this chapter, students will be able to understand:
? To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
Deviation and Standard Deviation and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
? To understand a set of observation, it is equally important to have knowledge of dispersion
which indicates the volatility. In advanced stage of chartered accountancy course, volatility
measures will be useful in understanding risk involved in financial decision making.
Absolute Measure of Dispersion
Overview of
Dispersion
Relative Measure of Dispersion
Range
Standard
Deviation
Mean
Deviation
Quartile
Deviation Coefficient of
Mean Deviation
Coefficient
of Variation
Coefficient
of Range
Coefficient of
Quartile Deviation
UNIT
OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


14.30
STATISTICS
The second important characteristic of a distribution is given by dispersion. Two distributions
may be identical in respect of its first important characteristic i.e. central tendency and yet they
may differ on account of scatterness. The following figure shows a number of distributions having
identical measure of central tendency and yet varying measure of scatterness. Obviously,
distribution is having the maximum amount of dispersion.
After reading this chapter, students will be able to understand:
? To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
Deviation and Standard Deviation and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
? To understand a set of observation, it is equally important to have knowledge of dispersion
which indicates the volatility. In advanced stage of chartered accountancy course, volatility
measures will be useful in understanding risk involved in financial decision making.
Absolute Measure of Dispersion
Overview of
Dispersion
Relative Measure of Dispersion
Range
Standard
Deviation
Mean
Deviation
Quartile
Deviation Coefficient of
Mean Deviation
Coefficient
of Variation
Coefficient
of Range
Coefficient of
Quartile Deviation
UNIT
OVERVIEW
© The Institute of Chartered Accountants of India
14.31 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Figure 14.2.1
Showing distributions with identical measure of central tendency
and varying amount of dispersion.
Dispersion for a given set of observations may be defined as the amount of deviation of the
observations, usually, from an appropriate measure of central tendency. Measures of dispersion
may be broadly classified into
1. Absolute measures of dispersion. 2. Relative measures of dispersion.
Absolute measures of dispersion are classified into
(i) Range (ii) Mean Deviation
(iii) Standard Deviation (iv) Quartile Deviation
Likewise, we have the following relative measures of dispersion :
(i) Coefficient of Range. (ii) Coefficient of Mean Deviation
(iii) Coefficient of Variation (iv) Coefficient of Quartile Deviation.
We may note the following points of distinction between the absolute and relative measures of
dispersion :
I Absolute measures are dependent on the unit of the variable under consideration whereas
the relative measures of dispersion are unit free.
II For comparing two or more distributions, relative measures and not absolute measures of
dispersion are considered.
III Compared to absolute measures of dispersion, relative measures of dispersion are difficult
to compute and comprehend.
Characteristics for an ideal measure of dispersion
As discussed in section 14.2.1 an ideal measure of dispersion should be properly defined, easy to
comprehend, simple to compute, based on all the observations, unaffected by sampling
fluctuations and amenable to some desirable mathematical treatment.
© The Institute of Chartered Accountants of India
Page 3


14.30
STATISTICS
The second important characteristic of a distribution is given by dispersion. Two distributions
may be identical in respect of its first important characteristic i.e. central tendency and yet they
may differ on account of scatterness. The following figure shows a number of distributions having
identical measure of central tendency and yet varying measure of scatterness. Obviously,
distribution is having the maximum amount of dispersion.
After reading this chapter, students will be able to understand:
? To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
Deviation and Standard Deviation and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
? To understand a set of observation, it is equally important to have knowledge of dispersion
which indicates the volatility. In advanced stage of chartered accountancy course, volatility
measures will be useful in understanding risk involved in financial decision making.
Absolute Measure of Dispersion
Overview of
Dispersion
Relative Measure of Dispersion
Range
Standard
Deviation
Mean
Deviation
Quartile
Deviation Coefficient of
Mean Deviation
Coefficient
of Variation
Coefficient
of Range
Coefficient of
Quartile Deviation
UNIT
OVERVIEW
© The Institute of Chartered Accountants of India
14.31 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Figure 14.2.1
Showing distributions with identical measure of central tendency
and varying amount of dispersion.
Dispersion for a given set of observations may be defined as the amount of deviation of the
observations, usually, from an appropriate measure of central tendency. Measures of dispersion
may be broadly classified into
1. Absolute measures of dispersion. 2. Relative measures of dispersion.
Absolute measures of dispersion are classified into
(i) Range (ii) Mean Deviation
(iii) Standard Deviation (iv) Quartile Deviation
Likewise, we have the following relative measures of dispersion :
(i) Coefficient of Range. (ii) Coefficient of Mean Deviation
(iii) Coefficient of Variation (iv) Coefficient of Quartile Deviation.
We may note the following points of distinction between the absolute and relative measures of
dispersion :
I Absolute measures are dependent on the unit of the variable under consideration whereas
the relative measures of dispersion are unit free.
II For comparing two or more distributions, relative measures and not absolute measures of
dispersion are considered.
III Compared to absolute measures of dispersion, relative measures of dispersion are difficult
to compute and comprehend.
Characteristics for an ideal measure of dispersion
As discussed in section 14.2.1 an ideal measure of dispersion should be properly defined, easy to
comprehend, simple to compute, based on all the observations, unaffected by sampling
fluctuations and amenable to some desirable mathematical treatment.
© The Institute of Chartered Accountants of India
14.32
STATISTICS
For a given set of observations, range may be defined as the difference between the largest and
smallest of observations. Thus if L and S  denote the largest and smallest observations respectively
then we have
Range = L – S
The corresponding relative measure of dispersion, known as coefficient of range, is given by
Coefficient of range = 100
S L
S L
?
?
?
For a grouped frequency distribution, range is defined as the difference between the two extreme
class boundaries. The corresponding relative measure of dispersion is given by the ratio of the
difference between the two extreme class boundaries to the total of these class boundaries,
expressed as a percentage.
We may note the following important result in connection with range:
Result:
Range remains unaffected due to a change of origin but affected in the same ratio due to
a change in scale i.e., if for any two constants a and b, two variables x and y are related by y =
a + bx,
Then the range of y is given by
  
x y
R b  R ? ? …………………………………………… (14.2.1)
Example 14.2.1: Following are the wages of 8 workers expressed in Rupees.
82, 96, 52, 75, 70, 65, 50, 70. Find the range and also its coefficient.
Solution: The largest and the smallest wages are L = ` 96 and S= ` 50
Thus range = ` 96 – ` 50 = ` 46
Coefficient of range = 100
50 96
50 96
?
?
?
 = 31.51
Example 14.2.2: What is the range and its coefficient for the following distribution of weights?
Weights in kgs. : 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74
No. of Students : 12 18 23 10 3
Solution: The lowest class boundary is 49.50 kgs. and the highest class boundary is 74.50 kgs.
Thus we have
Range = 74.50 kgs. – 49.50 kgs.
= 25 kgs.
© The Institute of Chartered Accountants of India
Page 4


14.30
STATISTICS
The second important characteristic of a distribution is given by dispersion. Two distributions
may be identical in respect of its first important characteristic i.e. central tendency and yet they
may differ on account of scatterness. The following figure shows a number of distributions having
identical measure of central tendency and yet varying measure of scatterness. Obviously,
distribution is having the maximum amount of dispersion.
After reading this chapter, students will be able to understand:
? To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
Deviation and Standard Deviation and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
? To understand a set of observation, it is equally important to have knowledge of dispersion
which indicates the volatility. In advanced stage of chartered accountancy course, volatility
measures will be useful in understanding risk involved in financial decision making.
Absolute Measure of Dispersion
Overview of
Dispersion
Relative Measure of Dispersion
Range
Standard
Deviation
Mean
Deviation
Quartile
Deviation Coefficient of
Mean Deviation
Coefficient
of Variation
Coefficient
of Range
Coefficient of
Quartile Deviation
UNIT
OVERVIEW
© The Institute of Chartered Accountants of India
14.31 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Figure 14.2.1
Showing distributions with identical measure of central tendency
and varying amount of dispersion.
Dispersion for a given set of observations may be defined as the amount of deviation of the
observations, usually, from an appropriate measure of central tendency. Measures of dispersion
may be broadly classified into
1. Absolute measures of dispersion. 2. Relative measures of dispersion.
Absolute measures of dispersion are classified into
(i) Range (ii) Mean Deviation
(iii) Standard Deviation (iv) Quartile Deviation
Likewise, we have the following relative measures of dispersion :
(i) Coefficient of Range. (ii) Coefficient of Mean Deviation
(iii) Coefficient of Variation (iv) Coefficient of Quartile Deviation.
We may note the following points of distinction between the absolute and relative measures of
dispersion :
I Absolute measures are dependent on the unit of the variable under consideration whereas
the relative measures of dispersion are unit free.
II For comparing two or more distributions, relative measures and not absolute measures of
dispersion are considered.
III Compared to absolute measures of dispersion, relative measures of dispersion are difficult
to compute and comprehend.
Characteristics for an ideal measure of dispersion
As discussed in section 14.2.1 an ideal measure of dispersion should be properly defined, easy to
comprehend, simple to compute, based on all the observations, unaffected by sampling
fluctuations and amenable to some desirable mathematical treatment.
© The Institute of Chartered Accountants of India
14.32
STATISTICS
For a given set of observations, range may be defined as the difference between the largest and
smallest of observations. Thus if L and S  denote the largest and smallest observations respectively
then we have
Range = L – S
The corresponding relative measure of dispersion, known as coefficient of range, is given by
Coefficient of range = 100
S L
S L
?
?
?
For a grouped frequency distribution, range is defined as the difference between the two extreme
class boundaries. The corresponding relative measure of dispersion is given by the ratio of the
difference between the two extreme class boundaries to the total of these class boundaries,
expressed as a percentage.
We may note the following important result in connection with range:
Result:
Range remains unaffected due to a change of origin but affected in the same ratio due to
a change in scale i.e., if for any two constants a and b, two variables x and y are related by y =
a + bx,
Then the range of y is given by
  
x y
R b  R ? ? …………………………………………… (14.2.1)
Example 14.2.1: Following are the wages of 8 workers expressed in Rupees.
82, 96, 52, 75, 70, 65, 50, 70. Find the range and also its coefficient.
Solution: The largest and the smallest wages are L = ` 96 and S= ` 50
Thus range = ` 96 – ` 50 = ` 46
Coefficient of range = 100
50 96
50 96
?
?
?
 = 31.51
Example 14.2.2: What is the range and its coefficient for the following distribution of weights?
Weights in kgs. : 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74
No. of Students : 12 18 23 10 3
Solution: The lowest class boundary is 49.50 kgs. and the highest class boundary is 74.50 kgs.
Thus we have
Range = 74.50 kgs. – 49.50 kgs.
= 25 kgs.
© The Institute of Chartered Accountants of India
14.33 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Also, coefficient of range = 
74.50 49.50
100
74.50 49.50
?
?
?
= 
25
100
124
?
= 20.16
Example 14.2.3 : If the relationship between x and y is given by 2x+3y=10 and the range of x
is ` 15, what would be the range of y?
Solution: Since  2x+3y=10
Therefore, y = x
3
2
–
3
10
Applying (14.2.1) , the range of y is given by
x y
R b R ? ? = 2/3 × ` 15
= ` 10.
Since range is based on only two observations, it is not regarded as an ideal measure of dispersion.
A better measure of dispersion is provided by mean deviation which, unlike range, is based on
all the observations. For a given set of observation, mean deviation is defined as the arithmetic
mean of the absolute deviations of the observations from an appropriate measure of central
tendency. Hence if a variable x assumes n values x
1
, x
2
, x
3
…x
n
, then the mean deviation of x about
an average A is given by
? ?
A i
1
MD = x A
n
……………………………………….(14.2.2)
For a grouped frequency distribution, mean deviation about A is given by
? ? ?
A ii
1
MD x A f
n
 …………………………………....(14.2.2)
Where x
i
 and f
i 
denote the mid value and frequency of the i-th class interval and
   
?
i
N= f
In most cases we take A as mean or median and accordingly, we get mean deviation about mean
or mean deviation about median.
A relative measure of dispersion applying mean deviation is given by
© The Institute of Chartered Accountants of India
Page 5


14.30
STATISTICS
The second important characteristic of a distribution is given by dispersion. Two distributions
may be identical in respect of its first important characteristic i.e. central tendency and yet they
may differ on account of scatterness. The following figure shows a number of distributions having
identical measure of central tendency and yet varying measure of scatterness. Obviously,
distribution is having the maximum amount of dispersion.
After reading this chapter, students will be able to understand:
? To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
Deviation and Standard Deviation and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
? To understand a set of observation, it is equally important to have knowledge of dispersion
which indicates the volatility. In advanced stage of chartered accountancy course, volatility
measures will be useful in understanding risk involved in financial decision making.
Absolute Measure of Dispersion
Overview of
Dispersion
Relative Measure of Dispersion
Range
Standard
Deviation
Mean
Deviation
Quartile
Deviation Coefficient of
Mean Deviation
Coefficient
of Variation
Coefficient
of Range
Coefficient of
Quartile Deviation
UNIT
OVERVIEW
© The Institute of Chartered Accountants of India
14.31 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Figure 14.2.1
Showing distributions with identical measure of central tendency
and varying amount of dispersion.
Dispersion for a given set of observations may be defined as the amount of deviation of the
observations, usually, from an appropriate measure of central tendency. Measures of dispersion
may be broadly classified into
1. Absolute measures of dispersion. 2. Relative measures of dispersion.
Absolute measures of dispersion are classified into
(i) Range (ii) Mean Deviation
(iii) Standard Deviation (iv) Quartile Deviation
Likewise, we have the following relative measures of dispersion :
(i) Coefficient of Range. (ii) Coefficient of Mean Deviation
(iii) Coefficient of Variation (iv) Coefficient of Quartile Deviation.
We may note the following points of distinction between the absolute and relative measures of
dispersion :
I Absolute measures are dependent on the unit of the variable under consideration whereas
the relative measures of dispersion are unit free.
II For comparing two or more distributions, relative measures and not absolute measures of
dispersion are considered.
III Compared to absolute measures of dispersion, relative measures of dispersion are difficult
to compute and comprehend.
Characteristics for an ideal measure of dispersion
As discussed in section 14.2.1 an ideal measure of dispersion should be properly defined, easy to
comprehend, simple to compute, based on all the observations, unaffected by sampling
fluctuations and amenable to some desirable mathematical treatment.
© The Institute of Chartered Accountants of India
14.32
STATISTICS
For a given set of observations, range may be defined as the difference between the largest and
smallest of observations. Thus if L and S  denote the largest and smallest observations respectively
then we have
Range = L – S
The corresponding relative measure of dispersion, known as coefficient of range, is given by
Coefficient of range = 100
S L
S L
?
?
?
For a grouped frequency distribution, range is defined as the difference between the two extreme
class boundaries. The corresponding relative measure of dispersion is given by the ratio of the
difference between the two extreme class boundaries to the total of these class boundaries,
expressed as a percentage.
We may note the following important result in connection with range:
Result:
Range remains unaffected due to a change of origin but affected in the same ratio due to
a change in scale i.e., if for any two constants a and b, two variables x and y are related by y =
a + bx,
Then the range of y is given by
  
x y
R b  R ? ? …………………………………………… (14.2.1)
Example 14.2.1: Following are the wages of 8 workers expressed in Rupees.
82, 96, 52, 75, 70, 65, 50, 70. Find the range and also its coefficient.
Solution: The largest and the smallest wages are L = ` 96 and S= ` 50
Thus range = ` 96 – ` 50 = ` 46
Coefficient of range = 100
50 96
50 96
?
?
?
 = 31.51
Example 14.2.2: What is the range and its coefficient for the following distribution of weights?
Weights in kgs. : 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74
No. of Students : 12 18 23 10 3
Solution: The lowest class boundary is 49.50 kgs. and the highest class boundary is 74.50 kgs.
Thus we have
Range = 74.50 kgs. – 49.50 kgs.
= 25 kgs.
© The Institute of Chartered Accountants of India
14.33 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Also, coefficient of range = 
74.50 49.50
100
74.50 49.50
?
?
?
= 
25
100
124
?
= 20.16
Example 14.2.3 : If the relationship between x and y is given by 2x+3y=10 and the range of x
is ` 15, what would be the range of y?
Solution: Since  2x+3y=10
Therefore, y = x
3
2
–
3
10
Applying (14.2.1) , the range of y is given by
x y
R b R ? ? = 2/3 × ` 15
= ` 10.
Since range is based on only two observations, it is not regarded as an ideal measure of dispersion.
A better measure of dispersion is provided by mean deviation which, unlike range, is based on
all the observations. For a given set of observation, mean deviation is defined as the arithmetic
mean of the absolute deviations of the observations from an appropriate measure of central
tendency. Hence if a variable x assumes n values x
1
, x
2
, x
3
…x
n
, then the mean deviation of x about
an average A is given by
? ?
A i
1
MD = x A
n
……………………………………….(14.2.2)
For a grouped frequency distribution, mean deviation about A is given by
? ? ?
A ii
1
MD x A f
n
 …………………………………....(14.2.2)
Where x
i
 and f
i 
denote the mid value and frequency of the i-th class interval and
   
?
i
N= f
In most cases we take A as mean or median and accordingly, we get mean deviation about mean
or mean deviation about median.
A relative measure of dispersion applying mean deviation is given by
© The Institute of Chartered Accountants of India
14.34
STATISTICS
Coefficient of mean deviation = 100
A
A  about deviation Mean  
? …………….(14.2.3)
Mean deviation takes its minimum value when the deviations are taken from the median.
Also mean deviation remains unchanged due to a change of origin but changes in the same
ratio due to a change in scale i.e. if y = a + bx, a and b being constants,
then MD of y = |b| × MD of x ………………………(14.2.4)
Example 14.2.4: What is the mean deviation about mean for the following numbers?
5, 8, 10, 10, 12, 9.
Solution:
The mean is given by
6
9 12 10 10 8 5
X
? ? ? ? ?
?  = 9
Table 14.2.1
Computation of MD about AM
x
i x x
i
?
  5   4
  8  1
10  1
10  1
12  3
  9  0
Total 10
Thus mean deviation about mean is given by
? ?
i
x x
10
= = 1.67
n 6
Example. 14.2.5: Find mean deviations about median and also the corresponding coefficient for
the following profits (‘000 `) of a firm during a week.
82, 56, 75, 70, 52, 80, 68.
Solution:
The profits in thousand rupees is denoted by x. Arranging the values of x in an ascending order,
we get
© The Institute of Chartered Accountants of India