Question

Quartile coefficient of skewness is

• A. \( \frac{Q_3 + Q_2 - 2Q_1}{Q_3 - Q_1} \)
• B. \( \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \)
• C. \( \frac{Q_3 + Q_2 - 2Q_1}{Q_1 - Q_3} \)
• D. \( \frac{Q_3 + Q_1 - 2Q_2}{Q_2 - Q_1} \)

Hint:

The numerator \( Q_3 + Q_1 - 2Q_2 \) can be rewritten as \( (Q_3 - Q_2) - (Q_2 - Q_1) \). If \( Q_3 - Q_2>Q_2 - Q_1 \), the skewness is positive. If \( Q_3 - Q_2<Q_2 - Q_1 \), the skewness is negative. If \( Q_3 - Q_2 = Q_2 - Q_1 \), the skewness is zero (symmetric).

Answer 1

The Correct Option is B

Step 1: Understand the concept of skewness.
Skewness is a measure of the asymmetry of a probability distribution or a dataset. A distribution can be positively skewed (tail extends to the right), negatively skewed (tail extends to the left), or symmetric (no skew). Step 2: Recall the definition of quartiles.
Quartiles divide a dataset into four equal parts.
The first quartile (\( Q_1 \)) is the value below which 25\% of the data falls.
The second quartile (\( Q_2 \)) is the median, the value below which 50\% of the data falls.
The third quartile (\( Q_3 \)) is the value below which 75\% of the data falls.
Step 3: Understand Bowley's coefficient of skewness.
Bowley's coefficient of skewness uses the quartiles to measure the asymmetry of the distribution. It compares the relative positions of the quartiles. Step 4: Recall the formula for Bowley's coefficient of skewness.
The formula is given by:
$$ \text{Quartile Coefficient of Skewness} = \frac{(Q_3 - Q_2) - (Q_2 - Q_1)}{Q_3 - Q_1} $$
This formula compares the difference between the third quartile and the median with the difference between the median and the first quartile, relative to the interquartile range (\( Q_3 - Q_1 \)). Step 5: Simplify the formula.
$$ \text{Quartile Coefficient of Skewness} = \frac{Q_3 - Q_2 - Q_2 + Q_1}{Q_3 - Q_1} $$
$$ \text{Quartile Coefficient of Skewness} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} $$
This matches option (2).