Exercise
5.4
Miscellaneous
Practice problems
1. Arithmetic mean of 15 observations
was calculated as 85. In doing so an observation was wrongly taken as 73 for 28.
What would be correct mean?
Solution:
Arithmetic mean = Sum of
all observations / Number of observations
85 = Sum of 15 observation / 15
85 × 15 = sum of 15 observations
1275 = sum of 15 observations
Wrong observation = 73
Correct observation = 28
∴ Correct Mean = [Sum – Wrong value + Correct value] / Number of
obervation
= [1275 – 73 + 28] / 15 = [1202 + 28] / 15 = 1230 / 15 = 82
Correct Mean = 82
2. Find the median of 25, 16, 15, 10,
8, 30.
Solution:
Arranging the data
in ascending order: 8, 10, 15, 16, 25, 30
Here n = 6, even
∴ Median = 1/2 {(n / 2)th term + (n/2 + 1)th term
= 1/2 {(6 / 2)th
term + (6/2 + 1)th term
= 1/2 {3th term + 4th term}
= 1/2 {15 + 16} = 1/2 (31)
∴ Median = 15.5
3. Find the mode of 2, 5, 5, 1, 3, 2,
2, 1, 3, 5, 3.
Solution:
Arranging the data in
ascending order: 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5
Here 2, 3 and 5 occurs 3 times each.
Which is the maximum number of times.
∴ Mode is 2, 3 and 5.
4. The marks scored by the students in
social test out of 20 marks are as follows: 12, 10, 8, 18, 14, 16. Find the mean
and median?
Solution:
Arranging the given data in ascending order: 8, 10, 12, 14, 16,
18.
Mean = Sum of all observations / Number of observations
= [ 8 +10 + 12 + 14 + 16 + 18 ] / 6 = 78 / 6
Mean = 13
There are n = 6
observations, which is even
∴ Median = 1/2 {(n / 2)th term + (n/2 + 1)th term
= 1/2 {(6/2)th
term + (6/2 + 1)th term
= 1/2 {3th term + 4th term}
= 1/2 {8+18} = 1/2 (26) =13
∴ Median = 13
5. The number of goals scored by a football
team is given below. Find the mode and median for the data of 2, 3, 2, 4, 6, 1,
3, 2, 4, 1, 6.
Solution:
Arranging the given data in ascending order: 1, 1, 2, 2, 2, 3,
3, 4, 4, 6, 6
Clearly 2 occurs at the maximum of 3 times and so mode = 2
Here number of data of data n = 11, odd.
∴ Median = ([n + 1] / 2)th term
= ([11+ 1] / 2)th
term = (12 / 2)th term
= 6th term
Median = 3
6.
Find the mean and mode of 6, 11, 13, 12, 4, 2.
Solution:
Arranging is ascending order : 2, 4, 6, 11, 12, 13
Mean = Sum of all observations / Number of observations = [ 2 + 4
+ 6 + 11 + 12 + 13] / 6
Mean = 48 / 6 = 8
All observation occurs only once and so there is no mode for
this date.
Challenge
Problems
7.
The average marks of six students is 8. One more student mark is added and the mean
is still 8. Find the student mark that has been added.
Solution:
Average = Sum of all observations / Number of observations
8 = Sum of observation / 6
Sum of observation = 6 × 8
= 48
If one more mark is added then number of observations = 6 + 1 = 7
Let the number be x
Still average = 8
∴ 8 = [48 + x ] / 7
48 + x = 7 × 8
48 + x = 56
48 + x = 56 – 48
x = 8
∴ The number that is added = 8
8.
Calculate the mean, mode and median for the following data: 22, 15, 10, 10, 24,
21.
Solution:
Arranging in ascending order: 10, 10, 15, 21, 22, 24
Mean = Sum of all observations / Number of observations
= [ 10 + 10 + 15 + 21 + 22 + 24 ] / 6
= 102 / 6 = 17
Here n = 6, even
∴ Median = 1/2 {(n / 2)th term + (n/2 + 1)th term
= 1/2 {(6/2)th
term + (6/2 + 1)th term
= 1/2 {3th term + 4th term}
= 1/2 {15 + 21} = 1/2 (36)
∴ Median = 18
Clearly the data 10 occurs maximum number of times and so 10 is
the mode.
∴ Mode = 10
9.
Find the median of the given data: 14, −3, 0, −2, −8, 13, −1, 7.
Solution:
Arranging in ascending order: –8, –3, –2, –1, 0, 7, 13, 14
Here number of data n
= 8, even
∴ Median = 1/2 {(n / 2)th term + ([n/2] + 1)th term
= 1/2 {(8 / 2)th
term + ([8/2] + 1)th term
= 1/2 {4th term + 5th term}
= 1/2 {–1 + 0} = 1/2 (–1) = – 0.5
∴ Median = – 0.5
10.
Find the mean of first 10 prime numbers and first 10 composite numbers.
Solution:
First 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Mean = Sum of all data / number of data
= [ 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 +29 ] / 10
= 129 / 10
Mean = 12.9
Mean of first 10 prime numbers = 12.9
First 10 prime numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
Mean = [ 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 ] / 10
= 112 / 10
= 11.2
Mean of first 10 prime numbers = 11.2
ANSWERS:
Exercise 5.4
1. 82
2. 15.5
3. 2, 3 and 5
4. 13;13
5. 2;3
6. 8; No mode.
Challenge Problems
7. 8
8. 17; 10 ; 18
9. –0.5
10. 12.9 ; 11.2
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