ICAI Notes- Unit 1: Measures of Central Tendency and Dispersion | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


MEASURES OF CENTRAL
TENDENCY AND DISPERSION
14
After reading this chapter, students will be able to understand:
? To understand different measures of central tendency, i.e. Arithmetic Mean, Median, Mode,
Geometric Mean and Harmonic Mean, and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
CHAPTER
© The Institute of Chartered Accountants of India
Page 2


MEASURES OF CENTRAL
TENDENCY AND DISPERSION
14
After reading this chapter, students will be able to understand:
? To understand different measures of central tendency, i.e. Arithmetic Mean, Median, Mode,
Geometric Mean and Harmonic Mean, and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
CHAPTER
© The Institute of Chartered Accountants of India
14.2
STATISTICS
In many a case, like the distributions of height, weight, marks, profit, wage and so on, it has been
noted that starting with rather low frequency, the class frequency gradually increases till it reaches
its maximum somewhere near the central part of the distribution and after which the class frequency
steadily falls to its minimum value towards the end. Thus, central tendency may be defined as the
tendency of a given set of observations to cluster around a single central or middle value and the
single value that represents the given set of observations is described as a measure of central
tendency or, location, or average. Hence, it is possible to condense a vast mass of data by a single
representative value. The computation of a measure of central tendency plays a very important
part in many a sphere. A company is recognized by its high average profit, an educational
institution is judged on the basis of average marks obtained by its students and so on. Furthermore,
the central tendency also facilitates us in providing a basis for comparison between different
distribution. Following are the different measures of central tendency:
(i) Mean
(ii) Median (Me)
(iii) Mode (Mo)
Geometric Mean (GM)
Harmonic Mean (HM)
Following are the criteria for an ideal measure of central tendency:
(i) It should be properly and unambiguously defined.
(ii) It should be easy to comprehend.
(iii) It should be simple to compute.
(iv) It should be based on all the observations.
(v) It should have certain desirable mathematical properties.
(vi) It should be least affected by the presence of extreme observations.
For a given set of observations, the AM may be defined as the sum of all the observations  divided
by the number of observations. Thus, if a variable x assumes n values x
1
, x
2
, x
3
,………..x
n
, then the
AM of x, to be denoted by 
X
, is given by,
       
n
x ...... .......... x x x
X
n 32 1
? ? ? ?
?
           = 
n
x
n
1i
i ?
?
Arithmetic Mean (AM) ?
?
?
© The Institute of Chartered Accountants of India
Page 3


MEASURES OF CENTRAL
TENDENCY AND DISPERSION
14
After reading this chapter, students will be able to understand:
? To understand different measures of central tendency, i.e. Arithmetic Mean, Median, Mode,
Geometric Mean and Harmonic Mean, and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
CHAPTER
© The Institute of Chartered Accountants of India
14.2
STATISTICS
In many a case, like the distributions of height, weight, marks, profit, wage and so on, it has been
noted that starting with rather low frequency, the class frequency gradually increases till it reaches
its maximum somewhere near the central part of the distribution and after which the class frequency
steadily falls to its minimum value towards the end. Thus, central tendency may be defined as the
tendency of a given set of observations to cluster around a single central or middle value and the
single value that represents the given set of observations is described as a measure of central
tendency or, location, or average. Hence, it is possible to condense a vast mass of data by a single
representative value. The computation of a measure of central tendency plays a very important
part in many a sphere. A company is recognized by its high average profit, an educational
institution is judged on the basis of average marks obtained by its students and so on. Furthermore,
the central tendency also facilitates us in providing a basis for comparison between different
distribution. Following are the different measures of central tendency:
(i) Mean
(ii) Median (Me)
(iii) Mode (Mo)
Geometric Mean (GM)
Harmonic Mean (HM)
Following are the criteria for an ideal measure of central tendency:
(i) It should be properly and unambiguously defined.
(ii) It should be easy to comprehend.
(iii) It should be simple to compute.
(iv) It should be based on all the observations.
(v) It should have certain desirable mathematical properties.
(vi) It should be least affected by the presence of extreme observations.
For a given set of observations, the AM may be defined as the sum of all the observations  divided
by the number of observations. Thus, if a variable x assumes n values x
1
, x
2
, x
3
,………..x
n
, then the
AM of x, to be denoted by 
X
, is given by,
       
n
x ...... .......... x x x
X
n 32 1
? ? ? ?
?
           = 
n
x
n
1i
i ?
?
Arithmetic Mean (AM) ?
?
?
© The Institute of Chartered Accountants of India
14.3 MEASURES OF CENTRAL TENDENCY AND DISPERSION
       X = 
n
x
i ?
              ……………………..(14.1.1)
In case of a simple frequency distribution relating to an attribute, we have
n 3 2 1
n n 3 3 2 2 1 1
f ....... .......... f f f
x f ....... .......... x f x f x f
x
? ? ? ?
? ? ? ?
?
= 
?
?
i
i i
f
x f
       X =  
N
x f
i i ?
……………………..(14.1.2)
assuming the observation x
i
 occurs f
i
 times, i=1,2,3,……..n and N= f
i
.
In case of grouped frequency distribution also we may use formula (14.1.2) with x
i  
as the mid
value of the i-th class interval, on the assumption that all the values belonging to the i-th
class interval are equal to x
i
.
However, in most cases, if the classification is uniform, we consider the following formula
for the computation of AM from grouped frequency distribution:
   
N
A x
d f
i i ?
C ? ? ?
    …………………………..(14.1.3)
Where,  
C
A x
d
i
i
?
?
  A = Assumed Mean
  C  = Class Length
 ILLUSTRATIONS:
Example 14.1.1: Following are the daily wages in Rupees of a sample of 9 workers: 58, 62, 48, 53,
70, 52, 60, 84, 75. Compute the mean wage.
Solution: Let x denote the daily wage in rupees.
Then as given, x
1
=58, x
2
=62, x
3
=
 
48, x
4
=53, x
5
=70
, 
x
6
=52, x
7
=60, x
8
=84 and x
9
=75.
Applying (14.1.1) the mean wage is given by,
9
i
i=1
x
x=  
9
?
= ` 
9
) 75 84 60 52 70 53 48 62 58 ( ? ? ? ? ? ? ? ?
= ` 
9
562
= ` 62.44.
?
© The Institute of Chartered Accountants of India
Page 4


MEASURES OF CENTRAL
TENDENCY AND DISPERSION
14
After reading this chapter, students will be able to understand:
? To understand different measures of central tendency, i.e. Arithmetic Mean, Median, Mode,
Geometric Mean and Harmonic Mean, and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
CHAPTER
© The Institute of Chartered Accountants of India
14.2
STATISTICS
In many a case, like the distributions of height, weight, marks, profit, wage and so on, it has been
noted that starting with rather low frequency, the class frequency gradually increases till it reaches
its maximum somewhere near the central part of the distribution and after which the class frequency
steadily falls to its minimum value towards the end. Thus, central tendency may be defined as the
tendency of a given set of observations to cluster around a single central or middle value and the
single value that represents the given set of observations is described as a measure of central
tendency or, location, or average. Hence, it is possible to condense a vast mass of data by a single
representative value. The computation of a measure of central tendency plays a very important
part in many a sphere. A company is recognized by its high average profit, an educational
institution is judged on the basis of average marks obtained by its students and so on. Furthermore,
the central tendency also facilitates us in providing a basis for comparison between different
distribution. Following are the different measures of central tendency:
(i) Mean
(ii) Median (Me)
(iii) Mode (Mo)
Geometric Mean (GM)
Harmonic Mean (HM)
Following are the criteria for an ideal measure of central tendency:
(i) It should be properly and unambiguously defined.
(ii) It should be easy to comprehend.
(iii) It should be simple to compute.
(iv) It should be based on all the observations.
(v) It should have certain desirable mathematical properties.
(vi) It should be least affected by the presence of extreme observations.
For a given set of observations, the AM may be defined as the sum of all the observations  divided
by the number of observations. Thus, if a variable x assumes n values x
1
, x
2
, x
3
,………..x
n
, then the
AM of x, to be denoted by 
X
, is given by,
       
n
x ...... .......... x x x
X
n 32 1
? ? ? ?
?
           = 
n
x
n
1i
i ?
?
Arithmetic Mean (AM) ?
?
?
© The Institute of Chartered Accountants of India
14.3 MEASURES OF CENTRAL TENDENCY AND DISPERSION
       X = 
n
x
i ?
              ……………………..(14.1.1)
In case of a simple frequency distribution relating to an attribute, we have
n 3 2 1
n n 3 3 2 2 1 1
f ....... .......... f f f
x f ....... .......... x f x f x f
x
? ? ? ?
? ? ? ?
?
= 
?
?
i
i i
f
x f
       X =  
N
x f
i i ?
……………………..(14.1.2)
assuming the observation x
i
 occurs f
i
 times, i=1,2,3,……..n and N= f
i
.
In case of grouped frequency distribution also we may use formula (14.1.2) with x
i  
as the mid
value of the i-th class interval, on the assumption that all the values belonging to the i-th
class interval are equal to x
i
.
However, in most cases, if the classification is uniform, we consider the following formula
for the computation of AM from grouped frequency distribution:
   
N
A x
d f
i i ?
C ? ? ?
    …………………………..(14.1.3)
Where,  
C
A x
d
i
i
?
?
  A = Assumed Mean
  C  = Class Length
 ILLUSTRATIONS:
Example 14.1.1: Following are the daily wages in Rupees of a sample of 9 workers: 58, 62, 48, 53,
70, 52, 60, 84, 75. Compute the mean wage.
Solution: Let x denote the daily wage in rupees.
Then as given, x
1
=58, x
2
=62, x
3
=
 
48, x
4
=53, x
5
=70
, 
x
6
=52, x
7
=60, x
8
=84 and x
9
=75.
Applying (14.1.1) the mean wage is given by,
9
i
i=1
x
x=  
9
?
= ` 
9
) 75 84 60 52 70 53 48 62 58 ( ? ? ? ? ? ? ? ?
= ` 
9
562
= ` 62.44.
?
© The Institute of Chartered Accountants of India
14.4
STATISTICS
Example 14.1.2: Compute the mean weight of a group of BBA students of St. Xavier’s College
from the following data:
Weight in kgs. 44 – 48 49 – 53 54 – 58 59 – 63 64 – 68 69 – 73
No. of Students 3 4 5 7 9 8
Solution: Computation of mean weight of 36 BBA students
Table 14.1.1
No. of
Weight in kgs. Student (f
i
) Mid-Value (x
i
) f
i
x
i
(1) (2) (3) (4) = (2) x (3)
44 – 48                                  3 46 138
49 – 53                                  4 51 204
54 – 58                                  5 56 280
59 – 63                                  7 61 427
64 – 68                                  9 66 594
69 – 73                                  8 71 568
Total 36 – 2211
Applying (14.1.2), we get the average weight as
N
x f
x
i i ?
?
  = 
36
2211
 kgs.
  = 61.42 kgs.
Example 14.1.3: Find the AM for the following distribution:
Class Interval 350 – 369 370 – 389 390 – 409 410 – 429 430 – 449 450 – 469 470 – 489
     Frequency 23 38 58 82 65 31 11
Solution: We apply formula (14.1.3) since the amount of computation involved in finding the
AM is much more compared to Example 14.1.2. Any mid value can be taken as A. However,
usually A is taken as the middle most mid-value for an odd number of class intervals and any
one of the two middle most mid-values for an even number of class intervals. The class length is
taken as C.
                                    
© The Institute of Chartered Accountants of India
Page 5


MEASURES OF CENTRAL
TENDENCY AND DISPERSION
14
After reading this chapter, students will be able to understand:
? To understand different measures of central tendency, i.e. Arithmetic Mean, Median, Mode,
Geometric Mean and Harmonic Mean, and computational techniques of these measures.
? To learn comparative advantages and disadvantages of these measures and therefore, which
measures to use in which circumstance.
CHAPTER
© The Institute of Chartered Accountants of India
14.2
STATISTICS
In many a case, like the distributions of height, weight, marks, profit, wage and so on, it has been
noted that starting with rather low frequency, the class frequency gradually increases till it reaches
its maximum somewhere near the central part of the distribution and after which the class frequency
steadily falls to its minimum value towards the end. Thus, central tendency may be defined as the
tendency of a given set of observations to cluster around a single central or middle value and the
single value that represents the given set of observations is described as a measure of central
tendency or, location, or average. Hence, it is possible to condense a vast mass of data by a single
representative value. The computation of a measure of central tendency plays a very important
part in many a sphere. A company is recognized by its high average profit, an educational
institution is judged on the basis of average marks obtained by its students and so on. Furthermore,
the central tendency also facilitates us in providing a basis for comparison between different
distribution. Following are the different measures of central tendency:
(i) Mean
(ii) Median (Me)
(iii) Mode (Mo)
Geometric Mean (GM)
Harmonic Mean (HM)
Following are the criteria for an ideal measure of central tendency:
(i) It should be properly and unambiguously defined.
(ii) It should be easy to comprehend.
(iii) It should be simple to compute.
(iv) It should be based on all the observations.
(v) It should have certain desirable mathematical properties.
(vi) It should be least affected by the presence of extreme observations.
For a given set of observations, the AM may be defined as the sum of all the observations  divided
by the number of observations. Thus, if a variable x assumes n values x
1
, x
2
, x
3
,………..x
n
, then the
AM of x, to be denoted by 
X
, is given by,
       
n
x ...... .......... x x x
X
n 32 1
? ? ? ?
?
           = 
n
x
n
1i
i ?
?
Arithmetic Mean (AM) ?
?
?
© The Institute of Chartered Accountants of India
14.3 MEASURES OF CENTRAL TENDENCY AND DISPERSION
       X = 
n
x
i ?
              ……………………..(14.1.1)
In case of a simple frequency distribution relating to an attribute, we have
n 3 2 1
n n 3 3 2 2 1 1
f ....... .......... f f f
x f ....... .......... x f x f x f
x
? ? ? ?
? ? ? ?
?
= 
?
?
i
i i
f
x f
       X =  
N
x f
i i ?
……………………..(14.1.2)
assuming the observation x
i
 occurs f
i
 times, i=1,2,3,……..n and N= f
i
.
In case of grouped frequency distribution also we may use formula (14.1.2) with x
i  
as the mid
value of the i-th class interval, on the assumption that all the values belonging to the i-th
class interval are equal to x
i
.
However, in most cases, if the classification is uniform, we consider the following formula
for the computation of AM from grouped frequency distribution:
   
N
A x
d f
i i ?
C ? ? ?
    …………………………..(14.1.3)
Where,  
C
A x
d
i
i
?
?
  A = Assumed Mean
  C  = Class Length
 ILLUSTRATIONS:
Example 14.1.1: Following are the daily wages in Rupees of a sample of 9 workers: 58, 62, 48, 53,
70, 52, 60, 84, 75. Compute the mean wage.
Solution: Let x denote the daily wage in rupees.
Then as given, x
1
=58, x
2
=62, x
3
=
 
48, x
4
=53, x
5
=70
, 
x
6
=52, x
7
=60, x
8
=84 and x
9
=75.
Applying (14.1.1) the mean wage is given by,
9
i
i=1
x
x=  
9
?
= ` 
9
) 75 84 60 52 70 53 48 62 58 ( ? ? ? ? ? ? ? ?
= ` 
9
562
= ` 62.44.
?
© The Institute of Chartered Accountants of India
14.4
STATISTICS
Example 14.1.2: Compute the mean weight of a group of BBA students of St. Xavier’s College
from the following data:
Weight in kgs. 44 – 48 49 – 53 54 – 58 59 – 63 64 – 68 69 – 73
No. of Students 3 4 5 7 9 8
Solution: Computation of mean weight of 36 BBA students
Table 14.1.1
No. of
Weight in kgs. Student (f
i
) Mid-Value (x
i
) f
i
x
i
(1) (2) (3) (4) = (2) x (3)
44 – 48                                  3 46 138
49 – 53                                  4 51 204
54 – 58                                  5 56 280
59 – 63                                  7 61 427
64 – 68                                  9 66 594
69 – 73                                  8 71 568
Total 36 – 2211
Applying (14.1.2), we get the average weight as
N
x f
x
i i ?
?
  = 
36
2211
 kgs.
  = 61.42 kgs.
Example 14.1.3: Find the AM for the following distribution:
Class Interval 350 – 369 370 – 389 390 – 409 410 – 429 430 – 449 450 – 469 470 – 489
     Frequency 23 38 58 82 65 31 11
Solution: We apply formula (14.1.3) since the amount of computation involved in finding the
AM is much more compared to Example 14.1.2. Any mid value can be taken as A. However,
usually A is taken as the middle most mid-value for an odd number of class intervals and any
one of the two middle most mid-values for an even number of class intervals. The class length is
taken as C.
                                    
© The Institute of Chartered Accountants of India
14.5 MEASURES OF CENTRAL TENDENCY AND DISPERSION
Table 14.1.2 Computation of AM
Class Interval Frequency(f
i
) Mid-Value(x
i
)
c
A x
d
i
i
?
? f
i
d
i
=
20
50 . 419 x
i
?
(1) (2) (3) (4) (5) = (2)X(4)
350 – 369 23 359.50 – 3 – 69
370 – 389 38 379.50 – 2 – 76
390 – 409 58 399.50 – 1 – 58
410 – 429 82      419.50 (A)   0     0
430 – 449 65 439.50   1   65
450 – 469 31 459.50   2   62
470 – 489 11 479.50   3   33
Total 308 –   – – 43
The required AM is given by
 C
N
d f
A x
i i
? ? ?
?
= 419.50 + 
? ?
308
43 –
×20
= 419.50 – 2.79
= 416.71
Example 14.1.4: Given that the mean height of a group of students is 67.45 inches. Find the
missing frequencies for the following incomplete distribution of height of 100 students.
Height in inches 60 – 62 63 – 65 66 – 68 69 – 71 72 – 74
No. of Students 5 18 – – 8
Solution: Let x denote the height and f
3
 and f
4
 as the two missing frequencies.
© The Institute of Chartered Accountants of India