Q1. [4] § 13.4 Median of Grouped Data
Activities like running or cycling reduce stress and the risk of mental disorders like depression. Running helps build endurance. Children develop stronger bones and muscles and are less prone to gain weight. The physical education teacher of a school has decided to conduct an inter school running tournament in his school premises. The time taken by a group of students to run 100 m, was noted as follows :
Time (in seconds): 0–20, 20–40, 40–60, 60–80, 80–100
Number of students: 8, 10, 13, 6, 3
Based on the above, answer the following questions :
- (i) What is the median class of the above given data? [1]
- (ii) Find the mean time taken by the students to finish the race, OR find the mode of the above given data. [2]
- (iii) How many students took time less than 60 seconds? [1]
Previously asked in CBSE board exam
2024 30/5/1 Q38
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding stimulus
Model Answer
(i) Median Class:
Total students = 8 + 10 + 13 + 6 + 3 = 40; N/2 = 20
Cumulative frequencies: 8, 18, 31, 37, 40
Since 20th observation falls in the class 40–60, the median class is 40–60.
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(ii) Mean:
| Class | $x_i$ (mid-value) | $f_i$ | $f_i x_i$ |
|-------|-------------------|--------|------------|
| 0–20 | 10 | 8 | 80 |
| 20–40 | 30 | 10 | 300 |
| 40–60 | 50 | 13 | 650 |
| 60–80 | 70 | 6 | 420 |
| 80–100 | 90 | 3 | 270 |
| Total | | 40 | 1720 |
$$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1720}{40} = \textbf{43 seconds}$$
OR Mode:
Modal class = 40–60 (highest frequency = 13)
$$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h = 40 + \frac{13-10}{26-10-6} \times 20 = 40 + \frac{3}{10} \times 20 = \textbf{46 seconds}$$
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(iii) Students who took less than 60 seconds = 8 + 10 + 13 = 31 students
Source: Statistics, Data Analysis
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Explanation
- (i) Always find cumulative frequency first; whichever class contains the N/2-th value is the median class.
- (ii) For mean, use midpoints. For mode, identify modal class (highest frequency) then apply the formula carefully — examiners check correct substitution of $f_0$, $f_1$, $f_2$.
- (iii) "Less than 60 seconds" means add frequencies of 0–20, 20–40, and 40–60 only.
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