Determine quartile deviation and co efficient of quartile deviation for the following distribution

Weight (kg) number of boys
30-34. 5
35-39. 4
40-44. 26
45-49. 10
50-54. 8

To determine the **Quartile Deviation** and the **Coefficient of Quartile Deviation** for the given distribution, we need to first find the **lower quartile (Q1)** and the **upper quartile (Q3)**. Let's break this process into steps.

### Step 1: Prepare the Cumulative Frequency Table
We first need the cumulative frequencies (CF) to identify the quartiles.

| Weight (kg) | Frequency (f) | Cumulative Frequency (CF) |
|-------------|---------------|---------------------------|
| 30-34       | 5             | 5                         |
| 35-39       | 4             | 9                         |
| 40-44       | 26            | 35                        |
| 45-49       | 10            | 45                        |
| 50-54       | 8             | 53                        |

### Step 2: Total Number of Boys
The total number of boys is the sum of the frequencies:

\[
N = 5 + 4 + 26 + 10 + 8 = 53
\]

### Step 3: Calculate Quartiles (Q1 and Q3)

1. **Position of Q1** (25th percentile):
   \[
   Q1 = \frac{N+1}{4} = \frac{53+1}{4} = \frac{54}{4} = 13.5
   \]

2. **Position of Q3** (75th percentile):
   \[
   Q3 = \frac{3(N+1)}{4} = \frac{3 \times 54}{4} = \frac{162}{4} = 40.5
   \]

### Step 4: Identify the Class Intervals for Q1 and Q3

- For **Q1** (13.5th value), the cumulative frequency just before 13.5 is 9, and the next class is 40-44 with cumulative frequency 35. So, the 13.5th value lies in the 40-44 class.
  
- For **Q3** (40.5th value), the cumulative frequency just before 40.5 is 35, and the next class is 45-49 with cumulative frequency 45. So, the 40.5th value lies in the 45-49 class.

### Step 5: Calculate Q1 and Q3

To calculate the exact values of Q1 and Q3, we'll use the following formula for quartiles:

\[
Q1 = L + \left( \frac{\frac{N}{4} - CF_{\text{previous}}}{f} \right) \times h
\]
\[
Q3 = L + \left( \frac{\frac{3N}{4} - CF_{\text{previous}}}{f} \right) \times h
\]

Where:
- \(L\) = Lower boundary of the class containing the quartile
- \(CF_{\text{previous}}\) = Cumulative frequency of the class before the quartile class
- \(f\) = Frequency of the quartile class
- \(h\) = Class width (in this case, \(h = 5\))

#### For Q1 (in class 40-44):
\[
L = 39.5, \quad CF_{\text{previous}} = 9, \quad f = 26, \quad h = 5
\]
\[
Q1 = 39.5 + \left( \frac{13.5 - 9}{26} \right) \times 5 = 39.5 + \left( \frac{4.5}{26} \right) \times 5
\]
\[
Q1 = 39.5 + 0.865 = 40.365
\]

#### For Q3 (in class 45-49):
\[
L = 44.5, \quad CF_{\text{previous}} = 35, \quad f = 10, \quad h = 5
\]
\[
Q3 = 44.5 + \left( \frac{40.5 - 35}{10} \right) \times 5 = 44.5 + \left( \frac{5.5}{10} \right) \times 5
\]
\[
Q3 = 44.5 + 2.75 = 47.25
\]

### Step 6: Calculate the Quartile Deviation (QD)

\[
QD = \frac{Q3 - Q1}{2} = \frac{47.25 - 40.365}{2} = \frac{6.885}{2} = 3.4425
\]

### Step 7: Calculate the Coefficient of Quartile Deviation (CQD)

\[
CQD = \frac{Q3 - Q1}{Q3 + Q1} = \frac{47.25 - 40.365}{47.25 + 40.365} = \frac{6.885}{87.615} \approx 0.0786
\]

### Final Answer:
- **Quartile Deviation** = 3.4425
- **Coefficient of Quartile Deviation** = 0.0786

---

Would you like further clarification on any of the steps, or should we explore more questions? Here are some related questions:

1. How do you interpret the coefficient of quartile deviation?
2. Can you explain the relationship between quartiles and percentiles?
3. What are other methods to measure dispersion in a dataset?
4. What is the significance of choosing quartiles over standard deviation?
5. How do you calculate deciles and percentiles for a frequency distribution?

**Tip:** Quartile deviation is useful when the dataset contains extreme values, as it is less affected by outliers than the standard deviation.

Determine quartile deviation and coefficient of quartile deviation for the following distribution: Weight (kg) number of boys 30-34: 5, 35-39: 4, 40-44: 26, 45-49: 10, 50-54: 8.

Learn how to calculate the quartile deviation and coefficient of quartile deviation for a weight distribution of boys. The solution explains step-by-step how to determine Q1 and Q3 using cumulative frequency tables and includes key formulas. Suitable for students in Grades 10-12.

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