Understanding Bowley's Coefficient of Skewness

Bowley's coefficient of skewness is a measure used in statistics to assess the asymmetry or lack of symmetry in a data distribution. It is based on quartiles, which are values that divide a sorted dataset into four equal parts.

Formula for Bowley's Coefficient of Skewness

The formula for Bowley's coefficient of skewness, also known as the Quartile Skewness, is given by:

$\text{Bowley's Coefficient of Skewness} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$

Where:

• $Q_1$ is the first quartile (25th percentile)
• $Q_2$ is the second quartile (50th percentile, which is also the median)
• $Q_3$ is the third quartile (75th percentile)

Interpreting Bowley's Coefficient of Skewness

The value of Bowley's coefficient of skewness indicates the direction and extent of skewness in the distribution. There are three main cases:

• If Bowley's coefficient is equal to zero: This indicates a symmetric distribution. In a perfectly symmetric distribution, the median ($Q_2$) is exactly halfway between the first ($Q_1$) and third ($Q_3$) quartiles.
• If Bowley's coefficient is greater than zero: This indicates a positively-skewed distribution. In a positively-skewed distribution, the tail is longer on the right side. This happens when the distance between $Q_2$ and $Q_3$ is greater than the distance between $Q_1$ and $Q_2$ ($Q_3 - Q_2 > Q_2 - Q_1$).
• If Bowley's coefficient is less than zero: This indicates a negatively-skewed distribution. In a negatively-skewed distribution, the tail is longer on the left side. This happens when the distance between $Q_1$ and $Q_2$ is greater than the distance between $Q_2$ and $Q_3$ ($Q_2 - Q_1 > Q_3 - Q_2$).

Solving the Question: Bowley's Coefficient Less Than Zero

The question asks about the distribution shape when Bowley's coefficient of skewness is less than zero.

Based on the interpretation of Bowley's coefficient:

• If the coefficient is less than zero ($< 0$), it signifies negative skewness.
• A negatively-skewed distribution has a longer tail on the left side. This means that the data points are more concentrated on the right side of the median.

Therefore, if the Bowley's coefficient of skewness is less than zero, the distribution is negatively-skewed.

Step-by-Step Thinking

1. Identify the given information: Bowley's coefficient of skewness is less than zero.
2. Recall the definition and interpretation of Bowley's coefficient of skewness.
3. Remember that a coefficient value less than zero corresponds to negative skewness.
4. Conclude that a distribution with a negative Bowley's coefficient is negatively-skewed.
5. Match the conclusion with the given options.

Revision Table: Key Skewness Concepts

Additional Information on Skewness

Skewness is one of the important characteristics of a data distribution. It measures the asymmetry of the probability distribution of a real-valued random variable about its mean. Distributions can be symmetric, positively skewed, or negatively skewed.

• Symmetric Distribution: Data are distributed evenly around the center. Mean, Median, and Mode are often approximately equal. The bell curve (normal distribution) is a common example.
• Positively-Skewed Distribution (Right Skew): The tail extends towards the right side. The mean is typically greater than the median, which is often greater than the mode. Examples include income distributions where a few high earners pull the mean up.
• Negatively-Skewed Distribution (Left Skew): The tail extends towards the left side. The mean is typically less than the median, which is often less than the mode. Examples might include test scores where most students score high, but a few score very low.

Bowley's coefficient is particularly useful for distributions where extreme values might distort the mean and standard deviation, making Pearson's coefficients less reliable. Since it uses quartiles, it is less affected by outliers.