Effective ICSE Class 10 Maths Solutions S Chand Chapter 18 Arithmetic Mean, Median, Mode and Quartiles Ex 18(d) can help bridge the gap between theory and application.
S Chand Class 10 ICSE Maths Solutions Chapter 18 Arithmetic Mean, Median, Mode and Quartiles Ex 18(d)
Question 1.
Find the mode of the following data :
(i) 8, 5, 6, 8, 8, 4, 6, 10, 8, 2;
(ii) 1, 2, 3, 3, 3, 5, 6, 8, 8, 8, 9;
(iii) 3, 5, 6, 6, 5, 3, 5, 3, 6, 5, 3, 5, 7, 6, 5, 7, 5;
(iv) 3, 4, 7, 11, 4, 3, 4, 5, 6, 4, 1, 4, 2, 4, 4.
Solution:
(i)
| x | f |
| 2 | 1 |
| 4 | 1 |
| 5 | 1 |
| 6 | 2 |
| 8 | 4 |
| 10 | 1 |
| Total | 10 |
∵ Frequency of 8 is 4 which is the greatest
∴ Mode = 4
(ii)
| x | f |
| 1 | 1 |
| 2 | 1 |
| 3 | 3 |
| 5 | 1 |
| 6 | 1 |
| 8 | 3 |
| 9 | 1 |
| Total | 11 |
∵ Frequency of 3 and 8 is 3 in each case
∴ Mode is 3 and 8 both
(iii)
| x | f |
| 3 | 4 |
| 5 | 7 |
| 6 | 4 |
| 7 | 2 |
| Total | 17 |
∵ Frequency of 5 is the greatest
∴ Mode = 5
(i)
| x | f |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 7 |
| 5 | 1 |
| 6 | 1 |
| 7 | 1 |
| 11 | 1 |
| Total | 15 |
∵ The frequency of 4 is the greatest
∴ Mode = 4
Question 2.
Find the median and mode for the set of numbers : 2, 2, 3, 5, 5, 5, 6, 8, 9
Solution:
Arranging in ascending order 2, 2, 3, 5, 5, 5, 6, 8, 9
Here n = 9 which is odd
∴ Median = \(\frac{n+1}{2}\)th term = \(\frac{9+1}{2}\) = 5th term
Which is 5
∵ The frequency of 5 is the greatest
∴ Mode = 5
Question 3.
A boy scored the following marks in various class tests during a term, each test being marked out of 20.
15, 17, 16, 7, 10, 12, 14, 16, 19, 12, 16
(i) What are his modal marks ?
(ii) What are his median marks ?
(iii) What are his mean marks ?
Solution:
Arranging in ascending order,
7, 10, 12, 12, 14, 15, 16, 16, 16, 17, 19
(i) ∵ Frequency of 16 is the greatest
∴ Mode = 16
(ii) Here n= 11 which is odd
∴ Median = \(\frac{n+1}{2}\)th term = \(\frac{11+1}{2}\) = 6th term
Which is 15
∴ Median = 15 marks
(iii) Mean = \(\frac{\sum x_i}{n}\)
= \(\frac{7+10+12+12+14+15+16+16+16+17+19}{11}\)
= \(\frac{154}{11}\) = 14 Marks
Question 4.
Find the mean, median and mode of the following marks obtained by 16 students in a class test marked out of 10 marks :
0, 0, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7, 8
Solution:
(iii) Mode
∵ 5 has greatest frequency
∴ Mode = 5 marks
Question 5.
Find the mode from the following distributions :
| Marks | 10 | 12 | 15 | 20 | 25 | 35 | 45 | 50 | 60 |
| No. of students | 4 | 6 | 10 | 14 | 20 | 19 | 10 | 6 | 3 |
Solution:
Here in this distribution, frequency of 25 is greatest
∴ Mode = 25 marks
Question 6.
At a shooting competition the scores of a competitor were as given below:
| Score | 0 | 1 | 2 | 3 | 4 | 5 |
| No. of shots | 0 | 3 | 6 | 4 | 7 | 5 |
(i) What was his modal score ?
(ii) What was his median score ?
(iii) What was his total score ?
(iv) What was his mean score ?
Solution:
| Score (x) | No. of shots (f) | c.f. | f × x |
| 0 | 0 | 0 | 0 |
| 1 | 3 | 3 | 3 |
| 2 | 6 | 9 | 12 |
| 3 | 4 | 13 | 12 |
| 4 | 7 | 20 | 28 |
| 5 | 5 | 25 | 25 |
| Total | 25 | 80 |
(i) Modal score is 4
∵ It has the maximum frequency i.e. 7
∴ Mode = 4
(ii) Here n = 25
∴ Median = \(\frac{n+1}{2} \text { th }\) term = \(\frac{25+1}{2} \text { th }\) = 13 th term
Which is 3
∴ Median = 3
(iii) Total score = 80
(iv) Mean = \(\frac{\sum f x}{\sum f}\) = \(\frac{80}{25}\) = \(\frac{16}{5}\) = 3.2
Question 7.
For what value of x, the mode of the following data is 17 ?
15, 16, 17, 13, 17, 16, 14, , 17, 16, 15, 15
Solution:
∵ Mode = 17
∴ The data has 17 at the most
The given data except x can be veritex as
| Number | Frequency |
| 13 | 1 |
| 14 | 1 |
| 15 | 3 |
| 16 | 3 |
| 17 | 3 |
We see that frequency of 15, 26 and 17 is 3
∵ Mode = 17
∴ Frequency of 17 must be greater than of 15 or 16
∴ x = 17
Question 8.
Find the value of k for which the mode of the following is 7 ?
3, 5, 5, 7, 3, 6, 7, 9, 6, 7, 3, 5, 7, 3, k
Solution:
Mode = 7
The given data except k, can be represented as given,
| Number | Frequency |
| 3 | 4 |
| 5 | 3 |
| 6 | 2 |
| 7 | 4 |
| 9 | 1 |
We see that frequency of 3 and 7 is 4 each
But mode = 7
∴ 7 has the greatest frequency or more than frequency of 3
∴ k = 7
Find the mode of the distributions given in problems 9-13 by drawing a histogram.
Question 9.
A box contains nails of different lengths, measured to the nearest half centimetres; the frequency distribution is as follows :
| Length (class mark) | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
| No. of nails (frequency) | 10 | 35 | 50 | 45 | 35 | 25 |
| Cumulative frequency | 10 | 45 | 95 | 140 | 175 | 200 |
(i) State the upper boundary of the last class;
(ii) State the class size
(iii) State the modal class
(iv) Determine the class which contains the median of the distribution.
Solution:
We are given class marks such as 2.5, 3, 3.5, 4 etc.
So, forming the classes accordingly :
2.25-2.75, 2.75-3.25, 3.25-3.75 etc.
Now
| Class | Frequency |
| 2.25-2.75 | 10 |
| 2.75-3.25 | 35 |
| 3.25-3.75 | 50 |
| 3.75-4.25 | 45 |
| 4.25-4.75 | 35 |
| 4.75-5.25 | 25 |
(i) Upper boundry of the last class is 5.25
(ii) Class size is 0.5
(iii) Modal class is 3.25-3.75 as it has the greatest frequency which is 50
(iv) Here n = 200
∴ Median = \(\frac { n }{ 2 }\)th = \(\frac { 200 }{ 2 }\) = 100th term
∴ Median lies between 96-140, which contains the class 3.75-4.25