Calculate the Upper Quartile (𝑄3) and the Lower Quartile (𝑄1) for the following frequency 
distribution. And hence calculate the quartile deviation and the coefficient of quartile 
deviation. 
Class Interval: 0 – 8 8 – 16 16 – 24 8 – 32 32 – 40
Frequency: 20 12 25 15 8

### Step 1: Determine the Cumulative Frequency

The first step is to compute the cumulative frequency for the given data.

| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|----------------|---------------|--------------------------|
| 0 – 8          | 20            | 20                       |
| 8 – 16         | 12            | 32                       |
| 16 – 24        | 25            | 57                       |
| 24 – 32        | 15            | 72                       |
| 32 – 40        | 8             | 80                       |

### Step 2: Identify Quartiles
The quartiles divide the dataset into four equal parts. Since there are 80 total frequencies, we calculate the positions for \(Q1\) and \(Q3\) as follows:

- \( Q1 \) (Lower Quartile) is at position \( \frac{1}{4} \times 80 = 20 \)
- \( Q3 \) (Upper Quartile) is at position \( \frac{3}{4} \times 80 = 60 \)

### Step 3: Calculate \(Q1\) (Lower Quartile)

\( Q1 \) is at the 20th position, which falls in the class interval 0 – 8 (since the cumulative frequency up to this class is 20). We use the quartile formula:

\[
Q1 = L + \left( \frac{\frac{N}{4} - cf_{\text{prev}}}{f} \right) \times h
\]

Where:
- \( L \) = lower boundary of the class containing \(Q1\) = 0
- \( cf_{\text{prev}} \) = cumulative frequency before the class of \(Q1\) = 0
- \( f \) = frequency of the class containing \(Q1\) = 20
- \( N = 80 \)
- \( h \) = class width = 8

Substitute the values:

\[
Q1 = 0 + \left( \frac{20 - 0}{20} \right) \times 8 = 0 + 8 = 8
\]

So, \( Q1 = 8 \).

### Step 4: Calculate \(Q3\) (Upper Quartile)

\( Q3 \) is at the 60th position, which falls in the class interval 24 – 32 (since the cumulative frequency up to the previous class is 57 and for this class it reaches 72). We use the same quartile formula:

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

Where:
- \( L \) = lower boundary of the class containing \(Q3\) = 24
- \( cf_{\text{prev}} \) = cumulative frequency before the class of \(Q3\) = 57
- \( f \) = frequency of the class containing \(Q3\) = 15
- \( h \) = class width = 8

Substitute the values:

\[
Q3 = 24 + \left( \frac{60 - 57}{15} \right) \times 8 = 24 + \left( \frac{3}{15} \right) \times 8 = 24 + 1.6 = 25.6
\]

So, \( Q3 = 25.6 \).

### Step 5: Calculate Quartile Deviation

The **Quartile Deviation** (QD) is calculated as:

\[
QD = \frac{Q3 - Q1}{2} = \frac{25.6 - 8}{2} = \frac{17.6}{2} = 8.8
\]

### Step 6: Calculate Coefficient of Quartile Deviation

The **Coefficient of Quartile Deviation** (CQD) is calculated as:

\[
CQD = \frac{Q3 - Q1}{Q3 + Q1} = \frac{25.6 - 8}{25.6 + 8} = \frac{17.6}{33.6} \approx 0.524
\]

### Final Results
- \( Q1 = 8 \)
- \( Q3 = 25.6 \)
- Quartile Deviation \( = 8.8 \)
- Coefficient of Quartile Deviation \( \approx 0.524 \)

---

Do you want more details on any step or have any questions? 

Here are 5 related questions to explore further:
1. How is the quartile deviation different from the standard deviation?
2. How would you calculate the median for this frequency distribution?
3. What would the effect of a smaller class width be on quartiles?
4. How is the interquartile range (IQR) related to quartile deviation?
5. How can we interpret the coefficient of quartile deviation in real-world data?

**Tip**: Quartiles are often used to measure the spread of data, making them helpful in understanding variability and identifying outliers.

Calculate the Upper Quartile (𝑄3) and the Lower Quartile (𝑄1) for the following frequency distribution. And hence calculate the quartile deviation and the coefficient of quartile deviation. Class Interval: 0 – 8 8 – 16 16 – 24 24 – 32 32 – 40 Frequency: 20 12 25 15 8

Learn how to calculate the Upper Quartile (Q3), Lower Quartile (Q1), Quartile Deviation, and Coefficient of Quartile Deviation for a frequency distribution with intervals 0-40. This guide walks through each step with detailed explanations and calculations. Suitable for students in Grades 10-12.

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