Bowley’s Coefficient of Skewness is a statistical measure that quantifies the degree of asymmetry in a distribution using quartiles. Unlike measures that rely on mean and standard deviation, Bowley's method focuses on the central 50% of the data, making it robust against outliers and extreme values.
Understanding Skewness
Skewness refers to the asymmetry in a distribution. A perfectly symmetrical distribution has a skewness of zero. If the distribution has a longer tail on the right, it's positively skewed; if the tail is on the left, it's negatively skewed.
Bowley's Coefficient of Skewness Formula
Bowley's Coefficient is calculated using the following formula:
SkB = (Q3 + Q1 - 2Q2) / (Q3 - Q1)
- Q1: First quartile (25th percentile)
- Q2: Second quartile or median (50th percentile)
- Q3: Third quartile (75th percentile)
Interpretation of Bowley's Coefficient
- SkB = 0: Symmetrical distribution
- SkB > 0: Positively skewed distribution
- SkB < 0: Negatively skewed distribution
Example
Consider the following dataset representing the number of children in 80 families:
| Number of Children | Number of Families |
|---|---|
| 0 | 12 |
| 1 | 23 |
| 2 | 16 |
| 3 | 9 |
| 4 | 10 |
| 5 | 10 |
Total number of families (N) = 80
To find the quartiles:
- Q1 position = (N + 1) / 4 = (80 + 1) / 4 = 20.25th value
- Q2 position = (N + 1) / 2 = (80 + 1) / 2 = 40.5th value
- Q3 position = 3(N + 1) / 4 = 3(80 + 1) / 4 = 60.75th value
Using cumulative frequencies, we find:
- Q1 ≈ 1
- Q2 ≈ 2
- Q3 ≈ 4
Now, applying the formula:
SkB = (4 + 1 - 2×2) / (4 - 1) = (5 - 4) / 3 = 1 / 3 ≈ 0.333
Interpretation: The distribution is positively skewed.
Merits of Bowley's Coefficient of Skewness
- Robust against outliers and extreme values.
- Useful for open-ended distributions.
- Simple to compute using quartiles.
- Provides a clear indication of the direction of skewness.
Demerits of Bowley's Coefficient of Skewness
- Ignores the tails of the distribution, focusing only on the central 50%.
- Less sensitive to variations in the extremes of the data.
- May not capture the full picture of skewness in certain datasets.
Comparison with Karl Pearson's Coefficient
| Aspect | Bowley's Coefficient | Karl Pearson's Coefficient |
|---|---|---|
| Basis | Quartiles (Q1, Q2, Q3) | Mean, Mode, Standard Deviation |
| Data Sensitivity | Less sensitive to outliers | More sensitive to outliers |
| Applicability | Suitable for open-ended distributions | Requires well-defined mean and mode |
Conclusion
Bowley's Coefficient of Skewness offers a robust measure of asymmetry, especially useful when dealing with datasets that have outliers or are open-ended. By focusing on the central 50% of the data, it provides a clear indication of the direction of skewness without being unduly influenced by extreme values. Understanding this measure is essential for accurate statistical analysis and interpretation.
In the next article, we will explore Kelly’s Coefficient of Skewness and Moments-Based Measures of Skewness, which delve deeper into the shape characteristics of distributions.