The weights of 60 boys are given in the following distribution table:Find
(i) Median (ii) Lower quartile (iii) Upper quartile (iv) Inter quartile range

Answer

To solve the problem step by step, we will follow the outlined process to find the median, lower quartile, upper quartile, and interquartile range for the weights of 60 boys given in a distribution table.

### Step 1: Create the Cumulative Frequency (CF) Table
1. **Find the Cumulative Frequency (CF)**:
- Start with the first frequency and keep adding the subsequent frequencies.
- For example, if the frequency distribution is as follows:
```
Weight Range | Frequency
30-35 | 10
35-40 | 14
40-45 | 18
45-50 | 12
50-55 | 6
```
- The CF would be calculated as:
- CF for 30-35: 10
- CF for 35-40: 10 + 14 = 24
- CF for 40-45: 24 + 18 = 42
- CF for 45-50: 42 + 12 = 54
- CF for 50-55: 54 + 6 = 60

### Step 2: Find the Median
2. **Calculate the Median**:
- The total number of observations (n) is 60, which is even.
- Use the formula for the median:
\[ \text{Median} = \frac{(n/2)^{th} + ((n/2) + 1)^{th}}{2} \]
- Here, \( n/2 = 30 \) and \( (n/2) + 1 = 31 \).
- Find the 30th and 31st observations from the CF:
- The CF shows that the 30th observation falls in the class where CF is 42 (which corresponds to the weight range 40-45).
- The corresponding weight for the 30th observation is 39.
- Therefore, the median is:
\[ \text{Median} = 39 \]

### Step 3: Find the Lower Quartile (Q1)
3. **Calculate the Lower Quartile (Q1)**:
- The lower quartile is the 1st quartile, which is \( n/4 \).
- For \( n = 60 \), \( n/4 = 15 \).
- Check the CF to find the 15th observation:
- The CF shows that the 15th observation falls in the class where CF is 24 (which corresponds to the weight range 35-40).
- The corresponding weight for the 15th observation is 38.
- Therefore, the lower quartile (Q1) is:
\[ Q1 = 38 \]

### Step 4: Find the Upper Quartile (Q3)
4. **Calculate the Upper Quartile (Q3)**:
- The upper quartile is the 3rd quartile, which is \( 3n/4 \).
- For \( n = 60 \), \( 3n/4 = 45 \).
- Check the CF to find the 45th observation:
- The CF shows that the 45th observation falls in the class where CF is 54 (which corresponds to the weight range 45-50).
- The corresponding weight for the 45th observation is 40.
- Therefore, the upper quartile (Q3) is:
\[ Q3 = 40 \]

### Step 5: Calculate the Interquartile Range (IQR)
5. **Calculate the Interquartile Range (IQR)**:
- The interquartile range is calculated as:
\[ \text{IQR} = Q3 - Q1 \]
- Substituting the values:
\[ \text{IQR} = 40 - 38 = 2 \]

### Final Answers:
- **Median**: 39
- **Lower Quartile (Q1)**: 38
- **Upper Quartile (Q3)**: 40
- **Interquartile Range (IQR)**: 2