Detailed Notes: Statistics | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
 
44 Measures of Central Tendency  
 
 
 
 
 
 2.1.1 Introduction. 
An average or a central value of a statistical series in the value of the variable which 
describes the characteristics of the entire distribution. 
The following are the five measures of central tendency. 
(1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) 
Mode 
 2.1.2 Arithmetic Mean. 
Arithmetic mean is the most important among the mathematical mean. 
According to Horace Secrist, 
“The arithmetic mean is the amount secured by dividing the sum of values of the items in a 
series by their number.” 
(1) Simple arithmetic mean in individual series (Ungrouped data) 
(i) Direct method : If the series in this case be 
n
x x x x ......, , , ,
3 2 1
 then the arithmetic mean x 
is given by 
of terms Number 
series of the Sum
? x ,  i.e., 
?
?
?
? ? ? ?
?
n
i
i
n
x
n n
x x x x
x
1
3 2 1
1 ....
 
(ii) Short cut method 
Arithmetic mean 
n
d
A x
?
? ? ) ( , 
where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the 
individual item, 
?d = sum of deviations and n = number of items. 
(2) Simple arithmetic mean in continuous series (Grouped data) 
(i) Direct method : If the terms of the given series be 
n
x x x ...., , ,
2 1
 and the corresponding 
frequencies be 
n
f f f .... , ,
2 1
,  then the arithmetic mean x is given by,  
?
?
?
?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
n
n n
f
x f
f f f
x f x f x f
x
1
1
2 1
2 2 1 1
....
....
. 
(ii) Short cut method : Arithmetic mean 
f
A x f
A x
?
? ?
? ?
) (
) ( 
Where A = assumed mean, f = frequency and x – A = deviation of each item from the 
assumed mean. 
Page 2


 
 
 
 
44 Measures of Central Tendency  
 
 
 
 
 
 2.1.1 Introduction. 
An average or a central value of a statistical series in the value of the variable which 
describes the characteristics of the entire distribution. 
The following are the five measures of central tendency. 
(1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) 
Mode 
 2.1.2 Arithmetic Mean. 
Arithmetic mean is the most important among the mathematical mean. 
According to Horace Secrist, 
“The arithmetic mean is the amount secured by dividing the sum of values of the items in a 
series by their number.” 
(1) Simple arithmetic mean in individual series (Ungrouped data) 
(i) Direct method : If the series in this case be 
n
x x x x ......, , , ,
3 2 1
 then the arithmetic mean x 
is given by 
of terms Number 
series of the Sum
? x ,  i.e., 
?
?
?
? ? ? ?
?
n
i
i
n
x
n n
x x x x
x
1
3 2 1
1 ....
 
(ii) Short cut method 
Arithmetic mean 
n
d
A x
?
? ? ) ( , 
where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the 
individual item, 
?d = sum of deviations and n = number of items. 
(2) Simple arithmetic mean in continuous series (Grouped data) 
(i) Direct method : If the terms of the given series be 
n
x x x ...., , ,
2 1
 and the corresponding 
frequencies be 
n
f f f .... , ,
2 1
,  then the arithmetic mean x is given by,  
?
?
?
?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
n
n n
f
x f
f f f
x f x f x f
x
1
1
2 1
2 2 1 1
....
....
. 
(ii) Short cut method : Arithmetic mean 
f
A x f
A x
?
? ?
? ?
) (
) ( 
Where A = assumed mean, f = frequency and x – A = deviation of each item from the 
assumed mean. 
Measures of Central Tendency  45 
(3) Properties of arithmetic mean 
(i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If 
i i
f x / , i = 1, 2, …, n is the frequency distribution, then 
0 ) (
1
? ?
?
?
x x f
i
n
i
i
, x being the mean of the distribution. 
(ii) The sum of the squares of the deviations of a set of values is minimum when taken 
about mean. 
(iii) Mean of the composite series : If ) ....., , 2 , 1 ( , k i x
i
? are the means of k-component series 
of sizes ) ...., , 2 , 1 ( , k i n
i
? respectively, then the mean x of the composite series obtained on 
combining the component series is given by the formula 
? ?
? ?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
k
k k
n x n
n n n
x n x n x n
x
1 1 2 1
2 2 1 1
....
....
. 
 2.1.3 Geometric Mean. 
If 
n
x x x x ......, , , ,
3 2 1
 are n values of a variate x, none of them being zero, then geometric mean 
(G.M.) is given by 
n
n
x x x x
/ 1
3 2 1
) ...... . . ( G.M. ? ? ) log ..... log (log
1
) G.M. log(
2 1 n
x x x
n
? ? ? ? . 
In case of frequency distribution, G.M. of n values 
n
x x x ..... , ,
2 1
 of a variate x occurring with 
frequency 
n
f f f ....., , ,
2 1
 is given by 
N f
n
f f
n
x x x
/ 1
2 1
) ..... . ( G.M.
2 1
? , where 
n
f f f N ? ? ? ? .....
2 1
. 
 2.1.4 Harmonic Mean. 
The harmonic mean of n items 
n
x x x ......, , ,
2 1
 is defined as 
n
x x x
n
1
.....
1 1
H.M.
2 1
? ? ?
? . 
If the frequency distribution is 
n
f f f f ......, , , ,
3 2 1
 respectively, then 
?
?
?
?
?
?
?
?
? ? ?
? ? ? ?
?
n
n
n
x
f
x
f
x
f
f f f f
.....
.....
H.M.
2
2
1
1
3 2 1
 
 Note : ? A.M. gives more weightage to larger values whereas G.M. and H.M. give more 
weightage to smaller values. 
 
Example: 1 If the mean of the distribution is 2.6, then the value of y is [Kurukshetra CEE 2001] 
Variate x 1 2 3 4 5 
Frequency f of 
x 
4 5 y 1 2 
(a) 24 (b) 13 (c) 8 (d) 3 
Solution: (c) We know that, Mean
?
?
?
?
?
n
i
i
n
i
i i
f
x f
1
1
 
 i.e. 
2 1 5 4
2 5 1 4 3 5 2 4 1
6 . 2
? ? ? ?
? ? ? ? ? ? ? ? ?
?
y
y
 or y y 3 28 6 . 2 2 . 31 ? ? ? or 2 . 3 4 . 0 ? y ? 8 ? y 
Example: 2 In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average 
marks of the complete class are 72, then what are the average marks of the girls [AIEEE 2002] 
Page 3


 
 
 
 
44 Measures of Central Tendency  
 
 
 
 
 
 2.1.1 Introduction. 
An average or a central value of a statistical series in the value of the variable which 
describes the characteristics of the entire distribution. 
The following are the five measures of central tendency. 
(1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) 
Mode 
 2.1.2 Arithmetic Mean. 
Arithmetic mean is the most important among the mathematical mean. 
According to Horace Secrist, 
“The arithmetic mean is the amount secured by dividing the sum of values of the items in a 
series by their number.” 
(1) Simple arithmetic mean in individual series (Ungrouped data) 
(i) Direct method : If the series in this case be 
n
x x x x ......, , , ,
3 2 1
 then the arithmetic mean x 
is given by 
of terms Number 
series of the Sum
? x ,  i.e., 
?
?
?
? ? ? ?
?
n
i
i
n
x
n n
x x x x
x
1
3 2 1
1 ....
 
(ii) Short cut method 
Arithmetic mean 
n
d
A x
?
? ? ) ( , 
where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the 
individual item, 
?d = sum of deviations and n = number of items. 
(2) Simple arithmetic mean in continuous series (Grouped data) 
(i) Direct method : If the terms of the given series be 
n
x x x ...., , ,
2 1
 and the corresponding 
frequencies be 
n
f f f .... , ,
2 1
,  then the arithmetic mean x is given by,  
?
?
?
?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
n
n n
f
x f
f f f
x f x f x f
x
1
1
2 1
2 2 1 1
....
....
. 
(ii) Short cut method : Arithmetic mean 
f
A x f
A x
?
? ?
? ?
) (
) ( 
Where A = assumed mean, f = frequency and x – A = deviation of each item from the 
assumed mean. 
Measures of Central Tendency  45 
(3) Properties of arithmetic mean 
(i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If 
i i
f x / , i = 1, 2, …, n is the frequency distribution, then 
0 ) (
1
? ?
?
?
x x f
i
n
i
i
, x being the mean of the distribution. 
(ii) The sum of the squares of the deviations of a set of values is minimum when taken 
about mean. 
(iii) Mean of the composite series : If ) ....., , 2 , 1 ( , k i x
i
? are the means of k-component series 
of sizes ) ...., , 2 , 1 ( , k i n
i
? respectively, then the mean x of the composite series obtained on 
combining the component series is given by the formula 
? ?
? ?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
k
k k
n x n
n n n
x n x n x n
x
1 1 2 1
2 2 1 1
....
....
. 
 2.1.3 Geometric Mean. 
If 
n
x x x x ......, , , ,
3 2 1
 are n values of a variate x, none of them being zero, then geometric mean 
(G.M.) is given by 
n
n
x x x x
/ 1
3 2 1
) ...... . . ( G.M. ? ? ) log ..... log (log
1
) G.M. log(
2 1 n
x x x
n
? ? ? ? . 
In case of frequency distribution, G.M. of n values 
n
x x x ..... , ,
2 1
 of a variate x occurring with 
frequency 
n
f f f ....., , ,
2 1
 is given by 
N f
n
f f
n
x x x
/ 1
2 1
) ..... . ( G.M.
2 1
? , where 
n
f f f N ? ? ? ? .....
2 1
. 
 2.1.4 Harmonic Mean. 
The harmonic mean of n items 
n
x x x ......, , ,
2 1
 is defined as 
n
x x x
n
1
.....
1 1
H.M.
2 1
? ? ?
? . 
If the frequency distribution is 
n
f f f f ......, , , ,
3 2 1
 respectively, then 
?
?
?
?
?
?
?
?
? ? ?
? ? ? ?
?
n
n
n
x
f
x
f
x
f
f f f f
.....
.....
H.M.
2
2
1
1
3 2 1
 
 Note : ? A.M. gives more weightage to larger values whereas G.M. and H.M. give more 
weightage to smaller values. 
 
Example: 1 If the mean of the distribution is 2.6, then the value of y is [Kurukshetra CEE 2001] 
Variate x 1 2 3 4 5 
Frequency f of 
x 
4 5 y 1 2 
(a) 24 (b) 13 (c) 8 (d) 3 
Solution: (c) We know that, Mean
?
?
?
?
?
n
i
i
n
i
i i
f
x f
1
1
 
 i.e. 
2 1 5 4
2 5 1 4 3 5 2 4 1
6 . 2
? ? ? ?
? ? ? ? ? ? ? ? ?
?
y
y
 or y y 3 28 6 . 2 2 . 31 ? ? ? or 2 . 3 4 . 0 ? y ? 8 ? y 
Example: 2 In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average 
marks of the complete class are 72, then what are the average marks of the girls [AIEEE 2002] 
 
 
 
 
46 Measures of Central Tendency  
(a) 73 (b) 65 (c) 68 (d) 74 
Solution: (b) Let the average marks of the girls students be x, then 
 
100
30 75 70
72
x ? ? ?
? (Number of girls = 100 – 70 = 30) 
   i.e., x ?
?
30
5250 7200
, ? x  = 65. 
Example: 3 If the mean of the set of numbers 
n
x x x x ....., , , ,
3 2 1
 is x , then the mean of the numbers i x
i
2 ? , n i ? ? 1 is 
  [Pb. CET 1988] 
(a) n x 2 ? (b) 1 ? ?n x (c) 2 ? x (d) n x ? 
Solution: (b) We know that 
n
x
x
n
i
i
?
?
?
1
 i.e., x n x
n
i
i
?
?
?1
 
 ? ) 1 (
2
) 1 (
2
) ... 2 1 ( 2
2 ) 2 (
1 1 1
? ? ?
?
?
?
? ? ?
?
?
?
?
? ? ?
? ? ?
n x
n
n n
x n
n
n x n
n
i x
n
i x
n
i
n
i
i
n
i
i
 
Example: 4 The harmonic mean of 4, 8, 16 is [AMU 1995] 
(a) 6.4 (b) 6.7 (c) 6.85 (d) 7.8 
Solution: (c) H.M. of 4, 8, 16 85 . 6
7
48
16
1
8
1
4
1
3
? ?
? ?
? 
Example: 5 The average of n numbers 
n
x x x x ......, , , ,
3 2 1
 is M. If 
n
x is replaced by x ? , then new average is [DCE 2000] 
(a) x x M
n
? ? ? (b) 
n
x x nM
n
? ? ?
 (c) 
n
x M n ? ? ? ) 1 (
 (d) 
n
x x M
n
? ? ?
 
Solution: (b) 
n
x x x x
M
n
......
3 2 1
? ?
? i.e. 
n
x x x x x
x x x x
x x x x x
n
x x nM
x nM
nM
n
n
n n
n
n
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ?
?
?
?
?
1 3 2 1
1 3 2 1
1 3 2 1
.....
.....
.....  
 ? New average
n
x x nM
n
? ? ?
? 
Example: 6 Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were 
wrongly read as 40, 20, 50 respectively. The correct mean is [Kurukshetra CEE 1994] 
(a) 48 (b) 
2
1
82 (c) 50 (d) 80 
Solution: (c) Sum of 100 items 4900 100 49 ? ? ? 
 Sum of items added 210 80 70 60 ? ? ? ? 
 Sum of items replaced 110 50 20 40 ? ? ? ? 
 New sum 5000 110 210 4900 ? ? ? ? 
 ? Correct mean 50
100
5000
? ? 
 2.1.5 Median. 
Median is defined as the value of an item or observation above or below which lies on an 
equal number of observations i.e., the median is the central value of the set of observations 
provided all the observations are arranged in the ascending or descending orders. 
(1) Calculation of median 
(i) Individual series : If the data is raw, arrange in ascending or descending order. Let n 
be the number of observations. 
Page 4


 
 
 
 
44 Measures of Central Tendency  
 
 
 
 
 
 2.1.1 Introduction. 
An average or a central value of a statistical series in the value of the variable which 
describes the characteristics of the entire distribution. 
The following are the five measures of central tendency. 
(1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) 
Mode 
 2.1.2 Arithmetic Mean. 
Arithmetic mean is the most important among the mathematical mean. 
According to Horace Secrist, 
“The arithmetic mean is the amount secured by dividing the sum of values of the items in a 
series by their number.” 
(1) Simple arithmetic mean in individual series (Ungrouped data) 
(i) Direct method : If the series in this case be 
n
x x x x ......, , , ,
3 2 1
 then the arithmetic mean x 
is given by 
of terms Number 
series of the Sum
? x ,  i.e., 
?
?
?
? ? ? ?
?
n
i
i
n
x
n n
x x x x
x
1
3 2 1
1 ....
 
(ii) Short cut method 
Arithmetic mean 
n
d
A x
?
? ? ) ( , 
where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the 
individual item, 
?d = sum of deviations and n = number of items. 
(2) Simple arithmetic mean in continuous series (Grouped data) 
(i) Direct method : If the terms of the given series be 
n
x x x ...., , ,
2 1
 and the corresponding 
frequencies be 
n
f f f .... , ,
2 1
,  then the arithmetic mean x is given by,  
?
?
?
?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
n
n n
f
x f
f f f
x f x f x f
x
1
1
2 1
2 2 1 1
....
....
. 
(ii) Short cut method : Arithmetic mean 
f
A x f
A x
?
? ?
? ?
) (
) ( 
Where A = assumed mean, f = frequency and x – A = deviation of each item from the 
assumed mean. 
Measures of Central Tendency  45 
(3) Properties of arithmetic mean 
(i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If 
i i
f x / , i = 1, 2, …, n is the frequency distribution, then 
0 ) (
1
? ?
?
?
x x f
i
n
i
i
, x being the mean of the distribution. 
(ii) The sum of the squares of the deviations of a set of values is minimum when taken 
about mean. 
(iii) Mean of the composite series : If ) ....., , 2 , 1 ( , k i x
i
? are the means of k-component series 
of sizes ) ...., , 2 , 1 ( , k i n
i
? respectively, then the mean x of the composite series obtained on 
combining the component series is given by the formula 
? ?
? ?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
k
k k
n x n
n n n
x n x n x n
x
1 1 2 1
2 2 1 1
....
....
. 
 2.1.3 Geometric Mean. 
If 
n
x x x x ......, , , ,
3 2 1
 are n values of a variate x, none of them being zero, then geometric mean 
(G.M.) is given by 
n
n
x x x x
/ 1
3 2 1
) ...... . . ( G.M. ? ? ) log ..... log (log
1
) G.M. log(
2 1 n
x x x
n
? ? ? ? . 
In case of frequency distribution, G.M. of n values 
n
x x x ..... , ,
2 1
 of a variate x occurring with 
frequency 
n
f f f ....., , ,
2 1
 is given by 
N f
n
f f
n
x x x
/ 1
2 1
) ..... . ( G.M.
2 1
? , where 
n
f f f N ? ? ? ? .....
2 1
. 
 2.1.4 Harmonic Mean. 
The harmonic mean of n items 
n
x x x ......, , ,
2 1
 is defined as 
n
x x x
n
1
.....
1 1
H.M.
2 1
? ? ?
? . 
If the frequency distribution is 
n
f f f f ......, , , ,
3 2 1
 respectively, then 
?
?
?
?
?
?
?
?
? ? ?
? ? ? ?
?
n
n
n
x
f
x
f
x
f
f f f f
.....
.....
H.M.
2
2
1
1
3 2 1
 
 Note : ? A.M. gives more weightage to larger values whereas G.M. and H.M. give more 
weightage to smaller values. 
 
Example: 1 If the mean of the distribution is 2.6, then the value of y is [Kurukshetra CEE 2001] 
Variate x 1 2 3 4 5 
Frequency f of 
x 
4 5 y 1 2 
(a) 24 (b) 13 (c) 8 (d) 3 
Solution: (c) We know that, Mean
?
?
?
?
?
n
i
i
n
i
i i
f
x f
1
1
 
 i.e. 
2 1 5 4
2 5 1 4 3 5 2 4 1
6 . 2
? ? ? ?
? ? ? ? ? ? ? ? ?
?
y
y
 or y y 3 28 6 . 2 2 . 31 ? ? ? or 2 . 3 4 . 0 ? y ? 8 ? y 
Example: 2 In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average 
marks of the complete class are 72, then what are the average marks of the girls [AIEEE 2002] 
 
 
 
 
46 Measures of Central Tendency  
(a) 73 (b) 65 (c) 68 (d) 74 
Solution: (b) Let the average marks of the girls students be x, then 
 
100
30 75 70
72
x ? ? ?
? (Number of girls = 100 – 70 = 30) 
   i.e., x ?
?
30
5250 7200
, ? x  = 65. 
Example: 3 If the mean of the set of numbers 
n
x x x x ....., , , ,
3 2 1
 is x , then the mean of the numbers i x
i
2 ? , n i ? ? 1 is 
  [Pb. CET 1988] 
(a) n x 2 ? (b) 1 ? ?n x (c) 2 ? x (d) n x ? 
Solution: (b) We know that 
n
x
x
n
i
i
?
?
?
1
 i.e., x n x
n
i
i
?
?
?1
 
 ? ) 1 (
2
) 1 (
2
) ... 2 1 ( 2
2 ) 2 (
1 1 1
? ? ?
?
?
?
? ? ?
?
?
?
?
? ? ?
? ? ?
n x
n
n n
x n
n
n x n
n
i x
n
i x
n
i
n
i
i
n
i
i
 
Example: 4 The harmonic mean of 4, 8, 16 is [AMU 1995] 
(a) 6.4 (b) 6.7 (c) 6.85 (d) 7.8 
Solution: (c) H.M. of 4, 8, 16 85 . 6
7
48
16
1
8
1
4
1
3
? ?
? ?
? 
Example: 5 The average of n numbers 
n
x x x x ......, , , ,
3 2 1
 is M. If 
n
x is replaced by x ? , then new average is [DCE 2000] 
(a) x x M
n
? ? ? (b) 
n
x x nM
n
? ? ?
 (c) 
n
x M n ? ? ? ) 1 (
 (d) 
n
x x M
n
? ? ?
 
Solution: (b) 
n
x x x x
M
n
......
3 2 1
? ?
? i.e. 
n
x x x x x
x x x x
x x x x x
n
x x nM
x nM
nM
n
n
n n
n
n
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ?
?
?
?
?
1 3 2 1
1 3 2 1
1 3 2 1
.....
.....
.....  
 ? New average
n
x x nM
n
? ? ?
? 
Example: 6 Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were 
wrongly read as 40, 20, 50 respectively. The correct mean is [Kurukshetra CEE 1994] 
(a) 48 (b) 
2
1
82 (c) 50 (d) 80 
Solution: (c) Sum of 100 items 4900 100 49 ? ? ? 
 Sum of items added 210 80 70 60 ? ? ? ? 
 Sum of items replaced 110 50 20 40 ? ? ? ? 
 New sum 5000 110 210 4900 ? ? ? ? 
 ? Correct mean 50
100
5000
? ? 
 2.1.5 Median. 
Median is defined as the value of an item or observation above or below which lies on an 
equal number of observations i.e., the median is the central value of the set of observations 
provided all the observations are arranged in the ascending or descending orders. 
(1) Calculation of median 
(i) Individual series : If the data is raw, arrange in ascending or descending order. Let n 
be the number of observations. 
  
 
 
 
Measures of Central Tendency  47 
If n is odd,   Median = value of 
th
n
?
?
?
?
?
? ?
2
1
 item. 
If n is even, Median = 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
item 1
2
of  value item 
2
of  value
2
1
th th
n n
 
(ii) Discrete series : In this case, we first find the cumulative frequencies of the variables 
arranged in ascending or descending order and the median is given by  
Median = 
th
n
?
?
?
?
?
? ?
2
1
observation, where n is the cumulative frequency. 
(iii) For grouped or continuous distributions : In this case, following formula can be 
used 
(a) For series in ascending order,  Median = i
f
C
N
l ?
?
?
?
?
?
?
?
?
2
 
Where l =  Lower limit of the median class 
f =  Frequency of the median class 
N = The sum of all frequencies 
 i = The width of the median class 
C = The cumulative frequency of the class preceding to median class. 
(b) For series in descending order 
Median = i
f
C
N
u ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
,  where u = upper limit of the median class 
          
?
?
?
n
i
i
f N
1
 
As median divides a distribution into two equal parts, similarly the quartiles, quantiles, 
deciles and percentiles divide the distribution respectively into 4, 5, 10 and 100 equal parts. The 
j
th
 quartile is given by 3 , 2 , 1 ;
4
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? j i
f
C
N
j
l Q
j
. 
1
Q is the lower quartile, 
2
Q is the median and 
3
Q is called the upper quartile. 
(2) Lower quartile 
(i) Discrete series : item 
4
1
of  size
th
1
?
?
?
?
?
? ?
?
n
Q 
(ii) Continuous series : i
f
C
N
l Q ?
?
?
?
?
?
?
?
? ?
4
1
 
(3) Upper quartile 
Page 5


 
 
 
 
44 Measures of Central Tendency  
 
 
 
 
 
 2.1.1 Introduction. 
An average or a central value of a statistical series in the value of the variable which 
describes the characteristics of the entire distribution. 
The following are the five measures of central tendency. 
(1) Arithmetic mean (2) Geometric mean (3) Harmonic mean (4) Median (5) 
Mode 
 2.1.2 Arithmetic Mean. 
Arithmetic mean is the most important among the mathematical mean. 
According to Horace Secrist, 
“The arithmetic mean is the amount secured by dividing the sum of values of the items in a 
series by their number.” 
(1) Simple arithmetic mean in individual series (Ungrouped data) 
(i) Direct method : If the series in this case be 
n
x x x x ......, , , ,
3 2 1
 then the arithmetic mean x 
is given by 
of terms Number 
series of the Sum
? x ,  i.e., 
?
?
?
? ? ? ?
?
n
i
i
n
x
n n
x x x x
x
1
3 2 1
1 ....
 
(ii) Short cut method 
Arithmetic mean 
n
d
A x
?
? ? ) ( , 
where, A = assumed mean, d = deviation from assumed mean = x – A, where x is the 
individual item, 
?d = sum of deviations and n = number of items. 
(2) Simple arithmetic mean in continuous series (Grouped data) 
(i) Direct method : If the terms of the given series be 
n
x x x ...., , ,
2 1
 and the corresponding 
frequencies be 
n
f f f .... , ,
2 1
,  then the arithmetic mean x is given by,  
?
?
?
?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
n
n n
f
x f
f f f
x f x f x f
x
1
1
2 1
2 2 1 1
....
....
. 
(ii) Short cut method : Arithmetic mean 
f
A x f
A x
?
? ?
? ?
) (
) ( 
Where A = assumed mean, f = frequency and x – A = deviation of each item from the 
assumed mean. 
Measures of Central Tendency  45 
(3) Properties of arithmetic mean 
(i) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If 
i i
f x / , i = 1, 2, …, n is the frequency distribution, then 
0 ) (
1
? ?
?
?
x x f
i
n
i
i
, x being the mean of the distribution. 
(ii) The sum of the squares of the deviations of a set of values is minimum when taken 
about mean. 
(iii) Mean of the composite series : If ) ....., , 2 , 1 ( , k i x
i
? are the means of k-component series 
of sizes ) ...., , 2 , 1 ( , k i n
i
? respectively, then the mean x of the composite series obtained on 
combining the component series is given by the formula 
? ?
? ?
?
? ? ?
? ? ?
?
n
i
i
n
i
i i
k
k k
n x n
n n n
x n x n x n
x
1 1 2 1
2 2 1 1
....
....
. 
 2.1.3 Geometric Mean. 
If 
n
x x x x ......, , , ,
3 2 1
 are n values of a variate x, none of them being zero, then geometric mean 
(G.M.) is given by 
n
n
x x x x
/ 1
3 2 1
) ...... . . ( G.M. ? ? ) log ..... log (log
1
) G.M. log(
2 1 n
x x x
n
? ? ? ? . 
In case of frequency distribution, G.M. of n values 
n
x x x ..... , ,
2 1
 of a variate x occurring with 
frequency 
n
f f f ....., , ,
2 1
 is given by 
N f
n
f f
n
x x x
/ 1
2 1
) ..... . ( G.M.
2 1
? , where 
n
f f f N ? ? ? ? .....
2 1
. 
 2.1.4 Harmonic Mean. 
The harmonic mean of n items 
n
x x x ......, , ,
2 1
 is defined as 
n
x x x
n
1
.....
1 1
H.M.
2 1
? ? ?
? . 
If the frequency distribution is 
n
f f f f ......, , , ,
3 2 1
 respectively, then 
?
?
?
?
?
?
?
?
? ? ?
? ? ? ?
?
n
n
n
x
f
x
f
x
f
f f f f
.....
.....
H.M.
2
2
1
1
3 2 1
 
 Note : ? A.M. gives more weightage to larger values whereas G.M. and H.M. give more 
weightage to smaller values. 
 
Example: 1 If the mean of the distribution is 2.6, then the value of y is [Kurukshetra CEE 2001] 
Variate x 1 2 3 4 5 
Frequency f of 
x 
4 5 y 1 2 
(a) 24 (b) 13 (c) 8 (d) 3 
Solution: (c) We know that, Mean
?
?
?
?
?
n
i
i
n
i
i i
f
x f
1
1
 
 i.e. 
2 1 5 4
2 5 1 4 3 5 2 4 1
6 . 2
? ? ? ?
? ? ? ? ? ? ? ? ?
?
y
y
 or y y 3 28 6 . 2 2 . 31 ? ? ? or 2 . 3 4 . 0 ? y ? 8 ? y 
Example: 2 In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average 
marks of the complete class are 72, then what are the average marks of the girls [AIEEE 2002] 
 
 
 
 
46 Measures of Central Tendency  
(a) 73 (b) 65 (c) 68 (d) 74 
Solution: (b) Let the average marks of the girls students be x, then 
 
100
30 75 70
72
x ? ? ?
? (Number of girls = 100 – 70 = 30) 
   i.e., x ?
?
30
5250 7200
, ? x  = 65. 
Example: 3 If the mean of the set of numbers 
n
x x x x ....., , , ,
3 2 1
 is x , then the mean of the numbers i x
i
2 ? , n i ? ? 1 is 
  [Pb. CET 1988] 
(a) n x 2 ? (b) 1 ? ?n x (c) 2 ? x (d) n x ? 
Solution: (b) We know that 
n
x
x
n
i
i
?
?
?
1
 i.e., x n x
n
i
i
?
?
?1
 
 ? ) 1 (
2
) 1 (
2
) ... 2 1 ( 2
2 ) 2 (
1 1 1
? ? ?
?
?
?
? ? ?
?
?
?
?
? ? ?
? ? ?
n x
n
n n
x n
n
n x n
n
i x
n
i x
n
i
n
i
i
n
i
i
 
Example: 4 The harmonic mean of 4, 8, 16 is [AMU 1995] 
(a) 6.4 (b) 6.7 (c) 6.85 (d) 7.8 
Solution: (c) H.M. of 4, 8, 16 85 . 6
7
48
16
1
8
1
4
1
3
? ?
? ?
? 
Example: 5 The average of n numbers 
n
x x x x ......, , , ,
3 2 1
 is M. If 
n
x is replaced by x ? , then new average is [DCE 2000] 
(a) x x M
n
? ? ? (b) 
n
x x nM
n
? ? ?
 (c) 
n
x M n ? ? ? ) 1 (
 (d) 
n
x x M
n
? ? ?
 
Solution: (b) 
n
x x x x
M
n
......
3 2 1
? ?
? i.e. 
n
x x x x x
x x x x
x x x x x
n
x x nM
x nM
nM
n
n
n n
n
n
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ?
?
?
?
?
1 3 2 1
1 3 2 1
1 3 2 1
.....
.....
.....  
 ? New average
n
x x nM
n
? ? ?
? 
Example: 6 Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were 
wrongly read as 40, 20, 50 respectively. The correct mean is [Kurukshetra CEE 1994] 
(a) 48 (b) 
2
1
82 (c) 50 (d) 80 
Solution: (c) Sum of 100 items 4900 100 49 ? ? ? 
 Sum of items added 210 80 70 60 ? ? ? ? 
 Sum of items replaced 110 50 20 40 ? ? ? ? 
 New sum 5000 110 210 4900 ? ? ? ? 
 ? Correct mean 50
100
5000
? ? 
 2.1.5 Median. 
Median is defined as the value of an item or observation above or below which lies on an 
equal number of observations i.e., the median is the central value of the set of observations 
provided all the observations are arranged in the ascending or descending orders. 
(1) Calculation of median 
(i) Individual series : If the data is raw, arrange in ascending or descending order. Let n 
be the number of observations. 
  
 
 
 
Measures of Central Tendency  47 
If n is odd,   Median = value of 
th
n
?
?
?
?
?
? ?
2
1
 item. 
If n is even, Median = 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
item 1
2
of  value item 
2
of  value
2
1
th th
n n
 
(ii) Discrete series : In this case, we first find the cumulative frequencies of the variables 
arranged in ascending or descending order and the median is given by  
Median = 
th
n
?
?
?
?
?
? ?
2
1
observation, where n is the cumulative frequency. 
(iii) For grouped or continuous distributions : In this case, following formula can be 
used 
(a) For series in ascending order,  Median = i
f
C
N
l ?
?
?
?
?
?
?
?
?
2
 
Where l =  Lower limit of the median class 
f =  Frequency of the median class 
N = The sum of all frequencies 
 i = The width of the median class 
C = The cumulative frequency of the class preceding to median class. 
(b) For series in descending order 
Median = i
f
C
N
u ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
,  where u = upper limit of the median class 
          
?
?
?
n
i
i
f N
1
 
As median divides a distribution into two equal parts, similarly the quartiles, quantiles, 
deciles and percentiles divide the distribution respectively into 4, 5, 10 and 100 equal parts. The 
j
th
 quartile is given by 3 , 2 , 1 ;
4
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? j i
f
C
N
j
l Q
j
. 
1
Q is the lower quartile, 
2
Q is the median and 
3
Q is called the upper quartile. 
(2) Lower quartile 
(i) Discrete series : item 
4
1
of  size
th
1
?
?
?
?
?
? ?
?
n
Q 
(ii) Continuous series : i
f
C
N
l Q ?
?
?
?
?
?
?
?
? ?
4
1
 
(3) Upper quartile 
 
 
 
 
48 Measures of Central Tendency  
(i) Discrete series : item 
4
) 1 ( 3
of  size
th
3
?
?
?
?
?
? ?
?
n
Q 
(ii) Continuous series : i
f
C
N
l Q ?
?
?
?
?
?
?
?
? ?
4
3
3
 
(4) Decile : Decile divide total frequencies N into ten equal parts. 
i
f
C
j N
l D
j
?
?
?
? ?
10
 [j = 1, 2, 3, 4, 5, 6, 7, 8, 9] 
If j = 5, then i
f
C
N
l D ?
?
? ?
2
5
. Hence 
5
D is also known as median. 
(5) Percentile : Percentile divide total frequencies N into hundred equal parts 
   i
f
C
k N
l P
k
?
?
?
? ?
100
 
where k = 1, 2, 3, 4, 5,.......,99. 
 
Example: 7 The following data gives the distribution of height of students 
Height (in cm) 160 150 152 161 156 154 155 
Number of 
students 
12 8 4 4 3 3 7 
The median of the distribution is      [AMU 1994] 
(a) 154 (b) 155 (c) 160 (d) 161 
Solution: (b) Arranging the data in ascending order of magnitude, we obtain 
 
Height (in cm) 150 152 154 155 156 160 161 
Number of 
students 
8 4 3 7 3 12 4 
Cumulative 
frequency  
8 12 15 22 25 37 41 
 Here, total number of items is 41, i.e. an odd number. Hence, the median is 
2
1 41 ?
th i.e. 21
st
 item. 
 From cumulative frequency table, we find that median i.e. 21
st
 item is 155. 
 (All items from 16 to 22
nd
 are equal, each = 155) 
Example: 8 The median of a set of 9 distinct observation is 20.5. If each of the largest 4 observation of the set is 
increased by 2, then the median of the new set [AIEEE 2003] 
(a) Is increased by 2   (b) Is decreased by 2  
(c) Is two times the original median (d) Remains the same as that of the original set 
Solution: (d) n = 9, then median term  term 5
2
1 9
th
th
? ?
?
?
?
?
? ?
? . Since last four observation are increased by 2. 
 ? The median is 5
th
 observation which is remaining unchanged. 
 ? There will be no change in median. 
Example: 9 Compute the median from the following table 
Marks 
obtained 
No. of students