Question: The marks obtained by 40 students in an examination are as follows: 91 81 78 67 59 84 60 72 47 60 33 36 60 57 63 42 18 42 36 27 66 54 27 42 63 27 18 9 33 24 39 45 24 39 49 33 45 12 14 72 (a) Using the class intervals 1-10, 11-20, 21-30, ... construct a cumulative frequency table for the distribution. (b) Draw the cumulative frequency curve. (c) Use your curve to find (i) the median mark (ii) the probability that a student selected at random scored a mark less than 32.5.
1. **Problem Statement:**
We have marks of 40 students and need to:
- (a) Construct a cumulative frequency table using class intervals 1-10, 11-20, 21-30, etc.
- (b) Draw the cumulative frequency curve.
- (c) Use the curve to find the median mark and the probability a student scored less than 32.5.
2. **Step (a): Constructing the cumulative frequency table**
- Class intervals: 1-10, 11-20, 21-30, 31-40, 41-50, 51-60, 61-70, 71-80, 81-90, 91-100
- Count frequencies for each interval:
- 1-10: marks = 9, 12, 14 (12 and 14 are >10, so only 9) → frequency = 1
- 11-20: 18, 18, 12, 14 → frequency = 4 (including 12,14)
- 21-30: 27, 27, 27, 24, 24 → frequency = 5
- 31-40: 33, 36, 36, 33, 39, 39, 33 → frequency = 7
- 41-50: 42, 42, 42, 45, 45, 47, 49 → frequency = 7
- 51-60: 54, 57, 59, 60, 60, 60 → frequency = 6
- 61-70: 63, 63, 66, 67 → frequency = 4
- 71-80: 72, 72, 78 → frequency = 3
- 81-90: 81, 84 → frequency = 2
- 91-100: 91 → frequency = 1
- Calculate cumulative frequency (CF):
- CF for 1-10 = 1
- CF for 11-20 = 1 + 4 = 5
- CF for 21-30 = 5 + 5 = 10
- CF for 31-40 = 10 + 7 = 17
- CF for 41-50 = 17 + 7 = 24
- CF for 51-60 = 24 + 6 = 30
- CF for 61-70 = 30 + 4 = 34
- CF for 71-80 = 34 + 3 = 37
- CF for 81-90 = 37 + 2 = 39
- CF for 91-100 = 39 + 1 = 40
| Class Interval | Frequency | Cumulative Frequency |
|---------------|-----------|----------------------|
| 1 - 10 | 1 | 1 |
| 11 - 20 | 4 | 5 |
| 21 - 30 | 5 | 10 |
| 31 - 40 | 7 | 17 |
| 41 - 50 | 7 | 24 |
| 51 - 60 | 6 | 30 |
| 61 - 70 | 4 | 34 |
| 71 - 80 | 3 | 37 |
| 81 - 90 | 2 | 39 |
| 91 - 100 | 1 | 40 |
3. **Step (b): Drawing the cumulative frequency curve**
- Plot cumulative frequency (y-axis) against upper class boundaries (x-axis):
- Points: (10,1), (20,5), (30,10), (40,17), (50,24), (60,30), (70,34), (80,37), (90,39), (100,40)
- Connect points smoothly to form the ogive (cumulative frequency curve).
4. **Step (c): Using the curve**
(i) **Median mark:**
- Median position = $\frac{N}{2} = \frac{40}{2} = 20$th value
- Find class interval where CF ≥ 20: CF at 50 is 24, so median class is 41-50
- Use median formula:
$$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$
where
$L = 40.5$ (lower boundary of median class),
$N = 40$ (total frequency),
$F = 17$ (CF before median class),
$f = 7$ (frequency of median class),
$h = 10$ (class width)
Calculate:
$$\text{Median} = 40.5 + \left(\frac{20 - 17}{7}\right) \times 10 = 40.5 + \frac{3}{7} \times 10 = 40.5 + 4.29 = 44.79$$
(ii) **Probability that a student scored less than 32.5:**
- Find CF at 30 = 10
- Total students = 40
- Probability = $\frac{\text{Number of students scoring less than 32.5}}{\text{Total students}} = \frac{10}{40} = 0.25$
**Final answers:**
- Median mark = 44.79
- Probability (score < 32.5) = 0.25