4.10 Bowley’s Coefficient of Skewness

Bowley’s Coefficient of Skewness is a quartile-based statistical measure used to quantify the asymmetry of a probability distribution. It offers a robust alternative to mean- and standard deviation-based measures like Pearson's skewness, particularly when dealing with datasets that may contain outliers or exhibit significant skewness.

This method leverages the positions of the first, second (median), and third quartiles to assess the distribution's symmetry. It is especially valuable for analyzing ordinal data and skewed distributions where median-based statistics provide a more accurate representation of the data's central tendency.

Formula

The formula for Bowley's Skewness Coefficient (B) is:

B = (Q1 + Q3 - 2 * Q2) / (Q3 - Q1)

Where:

• Q1 = First Quartile (the value below which 25% of the data falls, or the 25th percentile)
• Q2 = Second Quartile (the Median; the value below which 50% of the data falls, or the 50th percentile)
• Q3 = Third Quartile (the value below which 75% of the data falls, or the 75th percentile)

Interpretation of Bowley's Coefficient

The value of Bowley's coefficient provides insight into the shape of the distribution:

• B = 0: Indicates a symmetrical distribution. The data is evenly spread around the median.
• B > 0: Indicates a positively skewed (right-tailed) distribution. The tail on the right side of the distribution is longer or fatter than the left side, suggesting the presence of larger values that pull the distribution toward higher numbers.
• B < 0: Indicates a negatively skewed (left-tailed) distribution. The tail on the left side of the distribution is longer or fatter than the right side, suggesting the presence of smaller values that pull the distribution toward lower numbers.

Example: Step-by-Step Calculation

Let's calculate Bowley's Coefficient of Skewness for the following dataset representing the ages of a group of individuals:

Dataset: 18, 22, 25, 27, 30, 35, 38, 40, 42, 48, 50

Step 1: Sort the Data The dataset is already sorted in ascending order. Number of observations (n) = 11

Step 2: Find the Median (Q2) The median is the middle value. For 11 observations, the median is the (11+1)/2 = 6th value. Q2 = 35

Step 3: Find Q1 (First Quartile) Q1 is the median of the lower half of the data (excluding the median itself if n is odd). Lower half: 18, 22, 25, 27, 30 The median of these 5 values is the (5+1)/2 = 3rd value. Q1 = 25

Step 4: Find Q3 (Third Quartile) Q3 is the median of the upper half of the data (excluding the median itself if n is odd). Upper half: 38, 40, 42, 48, 50 The median of these 5 values is the (5+1)/2 = 3rd value. Q3 = 42

Step 5: Apply the Formula Using the formula: B = (Q1 + Q3 - 2 * Q2) / (Q3 - Q1)

Substitute the calculated values: B = (25 + 42 - 2 * 35) / (42 - 25) B = (67 - 70) / 17 B = -3 / 17 B ≈ -0.176

Conclusion: Since B ≈ -0.176, the dataset exhibits negative skewness. This indicates that the distribution has a longer tail on the left side, meaning there are some relatively smaller values pulling the mean towards the lower end compared to the median.

Why Use Bowley’s Coefficient?

Bowley's Coefficient of Skewness is preferred in several scenarios due to its distinct advantages:

• Robustness to Outliers: It is not affected by extreme values in the dataset, making it reliable for data with outliers.
• Suitability for Ordinal or Skewed Data: It performs well with ordinal data or distributions that are inherently skewed, where mean-based measures might be misleading.
• Simplicity of Interpretation: The interpretation based on quartiles is intuitive and easy to understand.
• Utility in Non-Parametric Analysis: It is a valuable tool in non-parametric statistical analysis, which does not assume a specific distribution for the data.

Bowley's method provides a practical and robust approach for understanding data asymmetry when traditional skewness metrics might be distorted by the influence of extreme observations.

Key Interview Questions

• What is Bowley’s Coefficient of Skewness and how is it calculated?
• How does Bowley’s method differ from Karl Pearson’s skewness coefficients?
• Why is Bowley’s coefficient considered robust?
• What types of data are best suited for Bowley’s skewness calculation?
• What does a negative Bowley’s skewness value indicate about a distribution?
• Can Bowley’s skewness be used for normally distributed data? Explain why or why not.
• Describe the process for calculating Q1, Q2, and Q3 from a given dataset.
• Under what circumstances would you choose Bowley’s skewness over Pearson’s skewness?
• Interpret the meaning of a Bowley’s coefficient of -0.176 in the context of a dataset.
• What are the potential limitations of Bowley’s Coefficient of Skewness?