Understanding Bowley's Coefficient of Skewness

Bowley's coefficient of skewness, also known as the quartile skewness coefficient, is a measure used to assess the asymmetry of a distribution based on its quartiles. It is particularly useful when the data contains extreme values or when dealing with open-ended distributions, as it relies on positional measures rather than moments like the mean and standard deviation.

The formula for Bowley's coefficient of skewness ($S_b$) is given by:

\( S_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \)

Where:

• \(Q_1\) is the first quartile (25th percentile).
• \(Q_2\) is the second quartile (median, 50th percentile).
• \(Q_3\) is the third quartile (75th percentile).

Calculating Skewness when Third Quartile Equals Median

The question provides a specific condition for a data set: the third quartile (\(Q_3\)) is equal to the median (\(Q_2\)). We are asked to find the value of Bowley's coefficient of skewness under this condition.

Given condition: \(Q_3 = Q_2\).

Let's substitute this condition into the formula for Bowley's coefficient of skewness:

\( S_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \)

Replace \(Q_3\) with \(Q_2\) in the numerator:

Numerator \( = Q_2 + Q_1 - 2Q_2 \)

Simplify the numerator:

Numerator \( = Q_1 + Q_2 - 2Q_2 \)

Numerator \( = Q_1 - Q_2 \)

Now, replace \(Q_3\) with \(Q_2\) in the denominator:

Denominator \( = Q_2 - Q_1 \)

So, the formula for Bowley's coefficient of skewness under the given condition becomes:

\( S_b = \frac{Q_1 - Q_2}{Q_2 - Q_1} \)

Interpreting the Result for Skewness

We have the expression \( S_b = \frac{Q_1 - Q_2}{Q_2 - Q_1} \). Let's analyze this expression.

The term \((Q_1 - Q_2)\) is the negative of the term \((Q_2 - Q_1)\). That is, \((Q_1 - Q_2) = -(Q_2 - Q_1)\).

So, we can rewrite the expression as:

\( S_b = \frac{-(Q_2 - Q_1)}{Q_2 - Q_1} \)

Assuming that the denominator \((Q_2 - Q_1)\) is not zero (i.e., \(Q_1 \neq Q_2\)), we can cancel the common term \((Q_2 - Q_1)\) from the numerator and the denominator:

\( S_b = -1 \)

In a typical distribution, the first quartile (\(Q_1\)) is less than or equal to the median (\(Q_2\)). If \(Q_1 < Q_2\), then \((Q_2 - Q_1)\) is positive, and the coefficient is indeed -1.

If \(Q_1 = Q_2\), then since \(Q_3 = Q_2\) is given, it implies \(Q_1 = Q_2 = Q_3\). When all three quartiles are equal, the distribution is symmetric around the median, and the skewness is 0. However, in this specific case, the denominator \((Q_3 - Q_1)\) becomes \((Q_2 - Q_2) = 0\), and the formula for \(S_b\) becomes \(\frac{0}{0}\), which is indeterminate. Bowley's coefficient is most informative when the quartiles are distinct.

Given the multiple-choice options provided, -1 is a specific value that arises directly from the formula when \(Q_3 = Q_2\) and \(Q_1 < Q_2\). This scenario indicates a distribution where the data points in the upper half (above the median) are very close to the median compared to the spread in the lower half, leading to a left-skewed distribution. A negative skewness value, such as -1, confirms this leftward skew.

Therefore, if the third quartile and median are equal (\(Q_3 = Q_2\)) for a data set where \(Q_1 < Q_2\), Bowley's coefficient of skewness is -1.

Conclusion on Bowley's Skewness Calculation

When the third quartile and median of a dataset are equal (\(Q_3 = Q_2\)), substituting this condition into Bowley's coefficient formula \(\left( S_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \right)\) leads to the expression \(\left( S_b = \frac{Q_1 - Q_2}{Q_2 - Q_1} \right)\). Assuming \(Q_1 < Q_2\), this simplifies to \(S_b = -1\).

Revision Table: Key Statistics Concepts

Additional Information: Types of Skewness

Skewness describes the degree of asymmetry of a distribution. Understanding the type of skewness is important for interpreting data.

• Negative Skewness (Left-Skewed): The tail on the left side of the distribution is longer or fatter than the tail on the right side. The mean is typically less than the median. In terms of quartiles, it often means \(Q_3 - Q_2 < Q_2 - Q_1\). Bowley's coefficient is negative.
• Positive Skewness (Right-Skewed): The tail on the right side of the distribution is longer or fatter than the tail on the left side. The mean is typically greater than the median. In terms of quartiles, it often means \(Q_3 - Q_2 > Q_2 - Q_1\). Bowley's coefficient is positive.
• Zero Skewness (Symmetric): The distribution is symmetric, meaning it looks the same on both sides of the center. The mean, median, and mode are typically equal. In terms of quartiles, it often means \(Q_3 - Q_2 = Q_2 - Q_1\). Bowley's coefficient is zero (if the formula is defined).

In the case where \(Q_3 = Q_2\), the distance from the median to the third quartile (\(Q_3 - Q_2\)) is zero. If \(Q_1 < Q_2\), the distance from the first quartile to the median (\(Q_2 - Q_1\)) is positive. This scenario indicates a distribution heavily concentrated towards the upper end of the lower half, implying a significant leftward skew, consistent with a negative skewness coefficient like -1.