The formula to calculate the coefficient of quartile deviation is :

Question

Accepted Answer

The coefficient of quartile deviation is a measure of relative dispersion that is used to understand the spread of data in a dataset. It is particularly useful when dealing with skewed distributions. The formula for calculating the coefficient of quartile deviation is given by:

$$
\text{Coefficient of Quartile Deviation} = \frac{Q_3 - Q_1}{Q_3 + Q_1}
$$

### Explanation for Option 1:

1. **Correct Formula**: Option 1, $\frac{Q_3 - Q_1}{Q_3 + Q_1}$, is the correct formula for the coefficient of quartile deviation. This formula is derived from the quartile deviation (also known as the semi-interquartile range), which is $\frac{Q_3 - Q_1}{2}$. The coefficient of quartile deviation is a normalized measure that expresses the quartile deviation as a proportion of the sum of the third quartile ($Q_3$) and the first quartile ($Q_1$). This normalization allows for comparison across different datasets.

2. **Purpose**: The coefficient of quartile deviation provides a dimensionless number that indicates the extent of variability relative to the central tendency of the dataset. It is particularly useful for comparing the spread of different datasets or distributions.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $\frac{Q_3 + Q_1}{4}$**

This formula does not represent any standard measure of dispersion or relative dispersion. It is simply the average of the first and third quartiles divided by 4, which does not provide meaningful information about the spread of the data.

- **Option 3: $\frac{Q_1 + Q_2}{4}$**

This formula involves $Q_2$, which is the median, and does not relate to the concept of quartile deviation. The quartile deviation focuses on the spread between the first and third quartiles, not the median.

- **Option 4: $\frac{Q_2 + Q_1}{Q_2 - Q_1}$**

This formula is incorrect for the coefficient of quartile deviation as it involves the median ($Q_2$) and does not focus on the interquartile range. The denominator $Q_2 - Q_1$ does not represent any standard measure of dispersion related to quartiles.

In conclusion, Option 1 is the correct formula for the coefficient of quartile deviation, as it accurately captures the relative spread of the data based on the interquartile range. The other options do not align with the standard definition or calculation of the coefficient of quartile deviation.