Understanding the Impact on Coefficient of Quartile Deviation When Observations are Doubled

This solution explains how the coefficient of quartile deviation changes when every value in a dataset is multiplied by two. We will look at the definitions of quartile deviation and its coefficient, and then analyze the effect of this transformation.

Defining Quartile Deviation

Quartile Deviation (QD) is a measure used in statistics to describe the spread or variability within the middle 50% of a dataset. It is calculated using the first quartile ($Q_1$) and the third quartile ($Q_3$).

The formula for Quartile Deviation is:

Here, $Q_1$ is the value below which 25% of the data falls, and $Q_3$ is the value below which 75% of the data falls.

Calculating the Coefficient of Quartile Deviation

The Coefficient of Quartile Deviation (CQD) is a relative measure of dispersion. It helps compare the spread of datasets with different scales. It is calculated by dividing the Quartile Deviation by the sum of the upper and lower quartiles.

The formula for the Coefficient of Quartile Deviation is:

Analyzing the Effect of Doubling Observations

Let's consider an original dataset: $x_1, x_2, x_3, ..., x_n$.

Now, imagine each observation in this dataset is doubled. The new dataset becomes: $2x_1, 2x_2, 2x_3, ..., 2x_n$.

When all observations in a dataset are multiplied by a constant value (let's call this constant '$k$'), the quartiles ($Q_1$ and $Q_3$) of the dataset are also multiplied by the same constant '$k$'. In this specific question, $k=2$.

Let the original quartiles be $Q_{1_{\text{original}}}$ and $Q_{3_{\text{original}}}$.

The new quartiles will be:

• $ Q_{1_{\text{new}}} = 2 \times Q_{1_{\text{original}}} $
• $ Q_{3_{\text{new}}} = 2 \times Q_{3_{\text{original}}} $

Now, let's calculate the Coefficient of Quartile Deviation for the new dataset:

$$ \text{CQD}_{\text{new}} = \frac{Q_{3_{\text{new}}} - Q_{1_{\text{new}}}}{Q_{3_{\text{new}}} + Q_{1_{\text{new}}}} $$

Substitute the new quartile values:

$$ \text{CQD}_{\text{new}} = \frac{(2 \times Q_{3_{\text{original}}}) - (2 \times Q_{1_{\text{original}}})}{(2 \times Q_{3_{\text{original}}}) + (2 \times Q_{1_{\text{original}}})} $$

Factor out the '2' from both the numerator and the denominator:

$$ \text{CQD}_{\text{new}} = \frac{2 \times (Q_{3_{\text{original}}} - Q_{1_{\text{original}}})}{2 \times (Q_{3_{\text{original}}} + Q_{1_{\text{original}}})} $$

Cancel out the common factor of 2:

$$ \text{CQD}_{\text{new}} = \frac{Q_{3_{\text{original}}} - Q_{1_{\text{original}}}}{Q_{3_{\text{original}}} + Q_{1_{\text{original}}}} $$

This is the formula for the original Coefficient of Quartile Deviation ($\text{CQD}_{\text{original}}$).

Therefore:

$$ \text{CQD}_{\text{new}} = \text{CQD}_{\text{original}} $$

Conclusion on Coefficient of Quartile Deviation

When each observation in a dataset is doubled, the coefficient of quartile deviation does not change. It remains the same because the scaling factor cancels out in the ratio calculation.