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 $\approx 0.524$

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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.