RS Aggarwal Solutions: Mean, Median and Mode of Ungrouped Data- 2 | Mathematics (Maths) Class 9 PDF Download

 Page 1


      
             
Q u e s t i o n : 4 6
Find the median of
i
2, 10, 9, 9, 5, 2, 3, 7, 11
ii
15, 6, 16, 8, 22, 21, 9, 18, 25
iii
20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22
iv
7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
      2, 2, 3, 5, 7, 9, 9, 10, 11
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 7
ii
Arranging the numbers in ascending order, we get:
6, 8, 9, 15, 16, 18, 21, 22, 25
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 16
iii
Arranging the numbers in ascending order, we get:
    6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25
Here, n is 11, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
11+1
2
th observation             = Value of the 6th observation             = 16
iv
Arranging the numbers in ascending order, we get:
   0, 1, 2, 2, 3, 4, 4, 5, 5, 7, 8, 9, 10
Here, n is 13, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
13+1
2
th observation             = Value of the 7th observation             = 4
Q u e s t i o n : 4 7
Find the median of
i
17, 19, 32, 10, 22, 21, 9, 35
ii
72, 63, 29, 51, 35, 60, 55, 91, 85, 82
iii
10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
9, 10, 17, 19, 21, 22, 32, 35
Here, n is 8, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
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( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
Page 2


      
             
Q u e s t i o n : 4 6
Find the median of
i
2, 10, 9, 9, 5, 2, 3, 7, 11
ii
15, 6, 16, 8, 22, 21, 9, 18, 25
iii
20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22
iv
7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
      2, 2, 3, 5, 7, 9, 9, 10, 11
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 7
ii
Arranging the numbers in ascending order, we get:
6, 8, 9, 15, 16, 18, 21, 22, 25
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 16
iii
Arranging the numbers in ascending order, we get:
    6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25
Here, n is 11, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
11+1
2
th observation             = Value of the 6th observation             = 16
iv
Arranging the numbers in ascending order, we get:
   0, 1, 2, 2, 3, 4, 4, 5, 5, 7, 8, 9, 10
Here, n is 13, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
13+1
2
th observation             = Value of the 7th observation             = 4
Q u e s t i o n : 4 7
Find the median of
i
17, 19, 32, 10, 22, 21, 9, 35
ii
72, 63, 29, 51, 35, 60, 55, 91, 85, 82
iii
10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
9, 10, 17, 19, 21, 22, 32, 35
Here, n is 8, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
Median = Mean of 
8
2
th & 
8
2
+1 th observations              = Mean of the 4th & 5th observations              =
1
2
(19 +21)              = 20
ii
Arranging the numbers in ascending order, we get:
   29, 35, 51, 55, 60, 63, 72, 82, 85, 91
Here, n is 10, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
10
2
th & 
10
2
+1 th observations              = Mean of the 5th & 6th observations              =
1
2
(60 +63)              = 61. 5
iii
Arranging the numbers in ascending order, we get:
    3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81
Here, n is 12, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
12
2
th & 
12
2
+1 th observations              = Mean of the 6th & 7th observations              =
1
2
(15 +17)              = 16
Q u e s t i o n : 4 8
The marks of 15 students in an examination are:
25, 19, 17, 24, 23, 29, 31, 40, 19, 20, 22, 26, 17, 35, 21.
Find the median score.
S o l u t i o n :
Arranging the marks of 15 students in ascending order, we get:
17, 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 35, 40
Here, n is 15, which is an odd number.
We know:
 Median = Value of 
n+1
2
th observation
Thus, we have:
Median score = Value of 
15+1
2
 th observation                        = Value of the 8th observation                        = 23
Q u e s t i o n : 4 9
The heights incm
of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.
Find the median height.
S o l u t i o n :
Arranging the given data in ascending order:
144, 145, 147, 148, 149, 150, 152, 155, 160
Number of terms = 9 odd
? Median =
n+1
2
th
 term                  =
9+1
2
th
 term                  = 5
th
 term                  = 149
Hence, the median height is 149.
Q u e s t i o n : 5 0
The weights inkg
of 8 children are:
13.4, 10.6, 12.7, 17.2, 14.3, 15, 16.5, 9.8.
Find the median weight.
S o l u t i o n :
Arranging the weights inkg
in ascending order, we have:
9.8, 10.6, 12.7, 13.4, 14.3, 15, 16.5, 17.2
Here, n is 8, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median weight = Mean of 
8
2
th & 
8
2
+1 th observations                           = Mean of 4th & 5th observations                           =
1
2
(13. 4 +14. 3)                          = 13. 85               
Hence, the median weight is 13.85 kg.
Q u e s t i o n : 5 1
The ages inyears
of 10 teachers in a school are:
32, 44, 53, 47, 37, 54, 34, 36, 40, 50.
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
Page 3


      
             
Q u e s t i o n : 4 6
Find the median of
i
2, 10, 9, 9, 5, 2, 3, 7, 11
ii
15, 6, 16, 8, 22, 21, 9, 18, 25
iii
20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22
iv
7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
      2, 2, 3, 5, 7, 9, 9, 10, 11
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 7
ii
Arranging the numbers in ascending order, we get:
6, 8, 9, 15, 16, 18, 21, 22, 25
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 16
iii
Arranging the numbers in ascending order, we get:
    6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25
Here, n is 11, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
11+1
2
th observation             = Value of the 6th observation             = 16
iv
Arranging the numbers in ascending order, we get:
   0, 1, 2, 2, 3, 4, 4, 5, 5, 7, 8, 9, 10
Here, n is 13, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
13+1
2
th observation             = Value of the 7th observation             = 4
Q u e s t i o n : 4 7
Find the median of
i
17, 19, 32, 10, 22, 21, 9, 35
ii
72, 63, 29, 51, 35, 60, 55, 91, 85, 82
iii
10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
9, 10, 17, 19, 21, 22, 32, 35
Here, n is 8, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
Median = Mean of 
8
2
th & 
8
2
+1 th observations              = Mean of the 4th & 5th observations              =
1
2
(19 +21)              = 20
ii
Arranging the numbers in ascending order, we get:
   29, 35, 51, 55, 60, 63, 72, 82, 85, 91
Here, n is 10, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
10
2
th & 
10
2
+1 th observations              = Mean of the 5th & 6th observations              =
1
2
(60 +63)              = 61. 5
iii
Arranging the numbers in ascending order, we get:
    3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81
Here, n is 12, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
12
2
th & 
12
2
+1 th observations              = Mean of the 6th & 7th observations              =
1
2
(15 +17)              = 16
Q u e s t i o n : 4 8
The marks of 15 students in an examination are:
25, 19, 17, 24, 23, 29, 31, 40, 19, 20, 22, 26, 17, 35, 21.
Find the median score.
S o l u t i o n :
Arranging the marks of 15 students in ascending order, we get:
17, 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 35, 40
Here, n is 15, which is an odd number.
We know:
 Median = Value of 
n+1
2
th observation
Thus, we have:
Median score = Value of 
15+1
2
 th observation                        = Value of the 8th observation                        = 23
Q u e s t i o n : 4 9
The heights incm
of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.
Find the median height.
S o l u t i o n :
Arranging the given data in ascending order:
144, 145, 147, 148, 149, 150, 152, 155, 160
Number of terms = 9 odd
? Median =
n+1
2
th
 term                  =
9+1
2
th
 term                  = 5
th
 term                  = 149
Hence, the median height is 149.
Q u e s t i o n : 5 0
The weights inkg
of 8 children are:
13.4, 10.6, 12.7, 17.2, 14.3, 15, 16.5, 9.8.
Find the median weight.
S o l u t i o n :
Arranging the weights inkg
in ascending order, we have:
9.8, 10.6, 12.7, 13.4, 14.3, 15, 16.5, 17.2
Here, n is 8, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median weight = Mean of 
8
2
th & 
8
2
+1 th observations                           = Mean of 4th & 5th observations                           =
1
2
(13. 4 +14. 3)                          = 13. 85               
Hence, the median weight is 13.85 kg.
Q u e s t i o n : 5 1
The ages inyears
of 10 teachers in a school are:
32, 44, 53, 47, 37, 54, 34, 36, 40, 50.
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
Find the median age.
S o l u t i o n :
Arranging the ages inyears
in ascending order, we have:
32, 34, 36, 37, 40, 44, 47, 50, 53, 54
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median age = Mean of 
10
2
th & 
10
2
+1 th observations                                           = Mean of 5th & 6th observations                                           =
1
2
(40 +44)                                           = 42
Q u e s t i o n : 5 2
If 10, 13, 15, 18, x + 1, x + 3, 30, 32, 35, 41 are ten observation in an ascending order with median 24, find the value of x.
S o l u t i o n :
10, 13, 15, 18, x+1, x+3, 30, 32, 35 and 41 are arranged in ascending order.
Median = 24
We have to find the value of x.
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
? Median = Mean of 
10
2
th & 
10
2
+1 th observations                                  = Mean of 5th & 6th observations                                  =
1
2
(x +1 +x +3)                                  =
1
2
(2x +4)              
Q u e s t i o n : 5 3
The following observations are arranged in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93.
If the median is 65, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93
Number of terms = 10 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 65 = mean of 
10
2
th
 term and 
10
2
+1
th
 term ? 65 = mean of (5)
th
 term and (6)
th
 term ? 65 = mean of [x and x +2] ? 65 =
x+x+2
2
?
Hence, the value of x is 64.
Q u e s t i o n : 5 4
The numbers 50, 42, 35, (2x + 10), (2x – 8), 12, 11, 8 have been written in a descending order. If their median is 25, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
8, 11, 12, (2x – 8), (2x + 10), 35, 42, 50
Number of terms = 8 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 25 = mean of 
8
2
th
 term and 
8
2
+1
th
 term ? 25 = mean of (4)
th
 term and (5)
th
 term ? 25 = mean of [2x -8 and 2x +10] ? 25 =
2x-
Hence, the value of x is 12.
Q u e s t i o n : 5 5
Find the median of the data
46, 41, 77, 58, 35, 64, 87, 92, 33, 55, 90.
In the above data, if 41 and 55 are replaced by 61 and 75 respectively, what will be the new median?
S o l u t i o n :
Arranging the given data in ascending order:
33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 58
Hence, the median of the data is 58.
Now, In the above data, if 41 and 55 are replaced by 61 and 75 respectively.
Then, new data in ascending order is:
33, 35, 46, 58, 61, 64, 75, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 64
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Page 4


      
             
Q u e s t i o n : 4 6
Find the median of
i
2, 10, 9, 9, 5, 2, 3, 7, 11
ii
15, 6, 16, 8, 22, 21, 9, 18, 25
iii
20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22
iv
7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
      2, 2, 3, 5, 7, 9, 9, 10, 11
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 7
ii
Arranging the numbers in ascending order, we get:
6, 8, 9, 15, 16, 18, 21, 22, 25
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 16
iii
Arranging the numbers in ascending order, we get:
    6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25
Here, n is 11, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
11+1
2
th observation             = Value of the 6th observation             = 16
iv
Arranging the numbers in ascending order, we get:
   0, 1, 2, 2, 3, 4, 4, 5, 5, 7, 8, 9, 10
Here, n is 13, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
13+1
2
th observation             = Value of the 7th observation             = 4
Q u e s t i o n : 4 7
Find the median of
i
17, 19, 32, 10, 22, 21, 9, 35
ii
72, 63, 29, 51, 35, 60, 55, 91, 85, 82
iii
10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
9, 10, 17, 19, 21, 22, 32, 35
Here, n is 8, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
Median = Mean of 
8
2
th & 
8
2
+1 th observations              = Mean of the 4th & 5th observations              =
1
2
(19 +21)              = 20
ii
Arranging the numbers in ascending order, we get:
   29, 35, 51, 55, 60, 63, 72, 82, 85, 91
Here, n is 10, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
10
2
th & 
10
2
+1 th observations              = Mean of the 5th & 6th observations              =
1
2
(60 +63)              = 61. 5
iii
Arranging the numbers in ascending order, we get:
    3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81
Here, n is 12, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
12
2
th & 
12
2
+1 th observations              = Mean of the 6th & 7th observations              =
1
2
(15 +17)              = 16
Q u e s t i o n : 4 8
The marks of 15 students in an examination are:
25, 19, 17, 24, 23, 29, 31, 40, 19, 20, 22, 26, 17, 35, 21.
Find the median score.
S o l u t i o n :
Arranging the marks of 15 students in ascending order, we get:
17, 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 35, 40
Here, n is 15, which is an odd number.
We know:
 Median = Value of 
n+1
2
th observation
Thus, we have:
Median score = Value of 
15+1
2
 th observation                        = Value of the 8th observation                        = 23
Q u e s t i o n : 4 9
The heights incm
of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.
Find the median height.
S o l u t i o n :
Arranging the given data in ascending order:
144, 145, 147, 148, 149, 150, 152, 155, 160
Number of terms = 9 odd
? Median =
n+1
2
th
 term                  =
9+1
2
th
 term                  = 5
th
 term                  = 149
Hence, the median height is 149.
Q u e s t i o n : 5 0
The weights inkg
of 8 children are:
13.4, 10.6, 12.7, 17.2, 14.3, 15, 16.5, 9.8.
Find the median weight.
S o l u t i o n :
Arranging the weights inkg
in ascending order, we have:
9.8, 10.6, 12.7, 13.4, 14.3, 15, 16.5, 17.2
Here, n is 8, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median weight = Mean of 
8
2
th & 
8
2
+1 th observations                           = Mean of 4th & 5th observations                           =
1
2
(13. 4 +14. 3)                          = 13. 85               
Hence, the median weight is 13.85 kg.
Q u e s t i o n : 5 1
The ages inyears
of 10 teachers in a school are:
32, 44, 53, 47, 37, 54, 34, 36, 40, 50.
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
Find the median age.
S o l u t i o n :
Arranging the ages inyears
in ascending order, we have:
32, 34, 36, 37, 40, 44, 47, 50, 53, 54
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median age = Mean of 
10
2
th & 
10
2
+1 th observations                                           = Mean of 5th & 6th observations                                           =
1
2
(40 +44)                                           = 42
Q u e s t i o n : 5 2
If 10, 13, 15, 18, x + 1, x + 3, 30, 32, 35, 41 are ten observation in an ascending order with median 24, find the value of x.
S o l u t i o n :
10, 13, 15, 18, x+1, x+3, 30, 32, 35 and 41 are arranged in ascending order.
Median = 24
We have to find the value of x.
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
? Median = Mean of 
10
2
th & 
10
2
+1 th observations                                  = Mean of 5th & 6th observations                                  =
1
2
(x +1 +x +3)                                  =
1
2
(2x +4)              
Q u e s t i o n : 5 3
The following observations are arranged in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93.
If the median is 65, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93
Number of terms = 10 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 65 = mean of 
10
2
th
 term and 
10
2
+1
th
 term ? 65 = mean of (5)
th
 term and (6)
th
 term ? 65 = mean of [x and x +2] ? 65 =
x+x+2
2
?
Hence, the value of x is 64.
Q u e s t i o n : 5 4
The numbers 50, 42, 35, (2x + 10), (2x – 8), 12, 11, 8 have been written in a descending order. If their median is 25, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
8, 11, 12, (2x – 8), (2x + 10), 35, 42, 50
Number of terms = 8 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 25 = mean of 
8
2
th
 term and 
8
2
+1
th
 term ? 25 = mean of (4)
th
 term and (5)
th
 term ? 25 = mean of [2x -8 and 2x +10] ? 25 =
2x-
Hence, the value of x is 12.
Q u e s t i o n : 5 5
Find the median of the data
46, 41, 77, 58, 35, 64, 87, 92, 33, 55, 90.
In the above data, if 41 and 55 are replaced by 61 and 75 respectively, what will be the new median?
S o l u t i o n :
Arranging the given data in ascending order:
33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 58
Hence, the median of the data is 58.
Now, In the above data, if 41 and 55 are replaced by 61 and 75 respectively.
Then, new data in ascending order is:
33, 35, 46, 58, 61, 64, 75, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 64
( ) ( )
( ) ( )
( ) ( )
( ) ( )
[ ( ) ( ) ] [ ( ) ( ) ]
[ ]
[ ( ) ( ) ] [ ( ) ( ) ]
[ ]
( ) ( )
( ) ( )
Hence, the new median of the data is 64.
      
           
        
           
      
   
         
        
        
        
      
   
                
                
         
                
      
   
          
           
   
        
           
      
    
                                  
  
 
  
  
              
      
      
           
     
      
      
   
       
                                      
      
          
     
    
      
      
        
      
   
Page 5


      
             
Q u e s t i o n : 4 6
Find the median of
i
2, 10, 9, 9, 5, 2, 3, 7, 11
ii
15, 6, 16, 8, 22, 21, 9, 18, 25
iii
20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22
iv
7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
      2, 2, 3, 5, 7, 9, 9, 10, 11
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 7
ii
Arranging the numbers in ascending order, we get:
6, 8, 9, 15, 16, 18, 21, 22, 25
Here, n is 9, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
9+1
2
th observation             = Value of the 5th observation             = 16
iii
Arranging the numbers in ascending order, we get:
    6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25
Here, n is 11, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
11+1
2
th observation             = Value of the 6th observation             = 16
iv
Arranging the numbers in ascending order, we get:
   0, 1, 2, 2, 3, 4, 4, 5, 5, 7, 8, 9, 10
Here, n is 13, which is an odd number.
If n is an odd number, we have:
Median = Value of 
n+1
2
th observation
Now,
Median = Value of 
13+1
2
th observation             = Value of the 7th observation             = 4
Q u e s t i o n : 4 7
Find the median of
i
17, 19, 32, 10, 22, 21, 9, 35
ii
72, 63, 29, 51, 35, 60, 55, 91, 85, 82
iii
10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27
S o l u t i o n :
i
Arranging the numbers in ascending order, we get:
9, 10, 17, 19, 21, 22, 32, 35
Here, n is 8, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
Median = Mean of 
8
2
th & 
8
2
+1 th observations              = Mean of the 4th & 5th observations              =
1
2
(19 +21)              = 20
ii
Arranging the numbers in ascending order, we get:
   29, 35, 51, 55, 60, 63, 72, 82, 85, 91
Here, n is 10, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
10
2
th & 
10
2
+1 th observations              = Mean of the 5th & 6th observations              =
1
2
(60 +63)              = 61. 5
iii
Arranging the numbers in ascending order, we get:
    3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81
Here, n is 12, which is an even number.
If n is an even number, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Now,
Median = Mean of 
12
2
th & 
12
2
+1 th observations              = Mean of the 6th & 7th observations              =
1
2
(15 +17)              = 16
Q u e s t i o n : 4 8
The marks of 15 students in an examination are:
25, 19, 17, 24, 23, 29, 31, 40, 19, 20, 22, 26, 17, 35, 21.
Find the median score.
S o l u t i o n :
Arranging the marks of 15 students in ascending order, we get:
17, 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 35, 40
Here, n is 15, which is an odd number.
We know:
 Median = Value of 
n+1
2
th observation
Thus, we have:
Median score = Value of 
15+1
2
 th observation                        = Value of the 8th observation                        = 23
Q u e s t i o n : 4 9
The heights incm
of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.
Find the median height.
S o l u t i o n :
Arranging the given data in ascending order:
144, 145, 147, 148, 149, 150, 152, 155, 160
Number of terms = 9 odd
? Median =
n+1
2
th
 term                  =
9+1
2
th
 term                  = 5
th
 term                  = 149
Hence, the median height is 149.
Q u e s t i o n : 5 0
The weights inkg
of 8 children are:
13.4, 10.6, 12.7, 17.2, 14.3, 15, 16.5, 9.8.
Find the median weight.
S o l u t i o n :
Arranging the weights inkg
in ascending order, we have:
9.8, 10.6, 12.7, 13.4, 14.3, 15, 16.5, 17.2
Here, n is 8, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median weight = Mean of 
8
2
th & 
8
2
+1 th observations                           = Mean of 4th & 5th observations                           =
1
2
(13. 4 +14. 3)                          = 13. 85               
Hence, the median weight is 13.85 kg.
Q u e s t i o n : 5 1
The ages inyears
of 10 teachers in a school are:
32, 44, 53, 47, 37, 54, 34, 36, 40, 50.
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
Find the median age.
S o l u t i o n :
Arranging the ages inyears
in ascending order, we have:
32, 34, 36, 37, 40, 44, 47, 50, 53, 54
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
Median age = Mean of 
10
2
th & 
10
2
+1 th observations                                           = Mean of 5th & 6th observations                                           =
1
2
(40 +44)                                           = 42
Q u e s t i o n : 5 2
If 10, 13, 15, 18, x + 1, x + 3, 30, 32, 35, 41 are ten observation in an ascending order with median 24, find the value of x.
S o l u t i o n :
10, 13, 15, 18, x+1, x+3, 30, 32, 35 and 41 are arranged in ascending order.
Median = 24
We have to find the value of x.
Here, n is 10, which is an even number.
Thus, we have:
Median = Mean of 
n
2
th & 
n
2
+1 th observations
? Median = Mean of 
10
2
th & 
10
2
+1 th observations                                  = Mean of 5th & 6th observations                                  =
1
2
(x +1 +x +3)                                  =
1
2
(2x +4)              
Q u e s t i o n : 5 3
The following observations are arranged in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93.
If the median is 65, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
26, 29, 42, 53, x, x + 2, 70, 75, 82, 93
Number of terms = 10 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 65 = mean of 
10
2
th
 term and 
10
2
+1
th
 term ? 65 = mean of (5)
th
 term and (6)
th
 term ? 65 = mean of [x and x +2] ? 65 =
x+x+2
2
?
Hence, the value of x is 64.
Q u e s t i o n : 5 4
The numbers 50, 42, 35, (2x + 10), (2x – 8), 12, 11, 8 have been written in a descending order. If their median is 25, find the value of x.
S o l u t i o n :
Arranging the given data in ascending order:
8, 11, 12, (2x – 8), (2x + 10), 35, 42, 50
Number of terms = 8 even
? Median = mean of 
n
2
th
 term and 
n
2
+1
th
 term ? 25 = mean of 
8
2
th
 term and 
8
2
+1
th
 term ? 25 = mean of (4)
th
 term and (5)
th
 term ? 25 = mean of [2x -8 and 2x +10] ? 25 =
2x-
Hence, the value of x is 12.
Q u e s t i o n : 5 5
Find the median of the data
46, 41, 77, 58, 35, 64, 87, 92, 33, 55, 90.
In the above data, if 41 and 55 are replaced by 61 and 75 respectively, what will be the new median?
S o l u t i o n :
Arranging the given data in ascending order:
33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 58
Hence, the median of the data is 58.
Now, In the above data, if 41 and 55 are replaced by 61 and 75 respectively.
Then, new data in ascending order is:
33, 35, 46, 58, 61, 64, 75, 77, 87, 90, 92
Number of terms = 11 odd
? Median =
n+1
2
th
 term                  =
11+1
2
th
 term                  = 6
th
 term                  = 64
( ) ( )
( ) ( )
( ) ( )
( ) ( )
[ ( ) ( ) ] [ ( ) ( ) ]
[ ]
[ ( ) ( ) ] [ ( ) ( ) ]
[ ]
( ) ( )
( ) ( )
Hence, the new median of the data is 64.
      
           
        
           
      
   
         
        
        
        
      
   
                
                
         
                
      
   
          
           
   
        
           
      
    
                                  
  
 
  
  
              
      
      
           
     
      
      
   
       
                                      
      
          
     
    
      
      
        
      
   
Q u e s t i o n : 1
Find the mean of:
i
the first eight natural numbers
ii
the first ten odd numbers
iii
the first seven multiples of 5
iv
all the factors of 20
v
all prime numbers between 50 and 80.
S o l u t i o n :
We know:
Mean =
Sum of observations
Number of observations
i
The first eight natural numbers are 1, 2, 3, 4, 5, 6, 7 and 8.
Mean of these numbers:
1+2+3+4+5+6+7+8
8
=
36
8
= 4. 5 
ii
The first ten odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19.
Mean of these numbers:
1+3+5+7+9+11+13+15+17+19
10
=
100
10
= 10
iii
The first seven multiples of 5 are 5, 10, 15, 20, 25, 30 and 35.
Mean of these numbers:
 
5+10+15+20+25+30+35
7
=
140
7
= 20
iv
The factors of 20 are 1, 2, 4, 5, 10 and 20.
Mean of these numbers:
1+2+4+5+10+20
6
= 
42
6
= 7
v
The prime numbers between 50 and 80 are 53, 59, 61, 67, 71, 73 and 79.
Mean of these numbers:
 
53+59+61+67+71+73+79
7
=
463
7
= 66. 14
Q u e s t i o n : 2
The number of children in 10 families of a locality are
2, 4, 3, 4, 2, 0, 3, 5, 1, 6.
Find the mean number of children per family.
S o l u t i o n :
Numbers of children in 10 families = 2, 4, 3, 4, 2, 0, 3, 5, 1 and 6.
Thus, we have:
Mean =
Sum of observations
Number of observations
 
= 
2+4+3+4+2+0+3+5+1+6
10
=
30
10
= 3
Q u e s t i o n : 3
The following are the number of books issued in a school library during a week:
105, 216, 322, 167, 273, 405 and 346.
Find the average number of books issued per day.
S o l u t i o n :
Numbers of books issued in the school library: 105, 216, 322, 167, 273, 405 and 346
Thus, we have:
Mean =
Sum of observations
Number of observations
 
=
105+216+322+167+273+405+346
7
= 
346
7
= 262