NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 13: Statistics and Probability - Exercise 13.3Question 16
Class 10 - Mathematics - CHAPTER 13: Statistics and Probability - Exercise 13.3
Question. 16
16. Maximum bowling speeds (km/h) of 33 players:
| Speed | 85–100 | 100–115 | 115–130 | 130–145 |
|---|---|---|---|---|
| No. of players | 11 | 9 | 8 | 5 |
Calculate the median speed.
Answer:
≈ 109.17 km/h
Detailed Answer with Explanation:
Step 1: Write the frequency table with cumulative frequencies (CF).
| Speed Interval (km/h) | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 85–100 | 11 | 11 |
| 100–115 | 9 | 20 |
| 115–130 | 8 | 28 |
| 130–145 | 5 | 33 |
Step 2: Find total number of players (N).
\(N = 11 + 9 + 8 + 5 = 33\)
Step 3: Find \(N/2\).
\(N/2 = 33/2 = 16.5\)
Step 4: Identify the median class.
The median class is the class interval whose CF is just greater than 16.5.
Here, CF values are 11, 20, 28, 33. Since 20 is the first CF greater than 16.5, the median class = 100–115 km/h.
Step 5: Write the formula for the median.
\[ ext{Median} = l + left(\dfrac{\dfrac{N}{2} - cf}{f}\right) \times h \]
- \(l = 100\) (lower boundary of median class)
- \(h = 15\) (class width = 115 − 100)
- \(cf = 11\) (cumulative frequency before median class)
- \(f = 9\) (frequency of median class)
- \(N/2 = 16.5\)
Step 6: Substitute the values.
\[ ext{Median} = 100 + \dfrac{16.5 - 11}{9} \times 15 \]
Step 7: Simplify step by step.
\(16.5 - 11 = 5.5\)
\(\dfrac{5.5}{9} = 0.611...\)
\(0.611... \times 15 = 9.166...\)
\(100 + 9.166... = 109.17\)
Final Answer: The median speed is ≈ 109.17 km/h.