Kelly’s Coefficient of Skewness is an important statistical measure used to understand the asymmetry of a distribution. It is an extension of Bowley’s method but uses deciles instead of quartiles, allowing it to consider a broader portion of the data.
What is Skewness?
Before diving into Kelly’s measure, recall that Skewness indicates how symmetric or asymmetric a dataset is around its central value. A perfectly symmetric distribution has zero skewness. If the right tail is longer, the distribution is positively skewed. If the left tail is longer, it's negatively skewed.
Why Kelly’s Method?
Bowley’s method uses the middle 50% of data through quartiles (Q1, Q2, Q3). However, Kelly felt that considering a wider middle range would give a more representative measure of skewness. Hence, his method uses the middle 80% of data using deciles.
Formula for Kelly’s Coefficient of Skewness
The formula for Kelly’s Skewness is:
SkK = (D9 + D1 - 2D5) / (D9 - D1)
- D1: First decile (10th percentile)
- D5: Fifth decile or median (50th percentile)
- D9: Ninth decile (90th percentile)
This formula parallels Bowley’s, but instead of using quartiles, it uses deciles to extend the analysis to a broader part of the distribution.
Interpretation
- SkK = 0: Perfectly symmetric distribution
- SkK > 0: Positively skewed distribution
- SkK < 0: Negatively skewed distribution
Example
Let us take a grouped frequency distribution of exam scores of 100 students:
| Marks Range | Frequency |
|---|---|
| 0–10 | 5 |
| 10–20 | 10 |
| 20–30 | 15 |
| 30–40 | 20 |
| 40–50 | 25 |
| 50–60 | 15 |
| 60–70 | 10 |
Total N = 100
- D1 = value at 10th position
- D5 = median = value at 50th position
- D9 = value at 90th position
From cumulative frequencies, find the class intervals containing 10th, 50th, and 90th values. Then use the formula for deciles:
Dk = L + [(kN/10 – F) / f] × h
- L: lower boundary of the decile class
- F: cumulative frequency before decile class
- f: frequency of decile class
- h: width of class interval
Calculate D1, D5, and D9 using the above formula and substitute into the main formula to get SkK.
Merits of Kelly’s Coefficient
- Uses 80% of the central data, more representative than quartile-based skewness.
- Less affected by extreme values compared to mean-based measures.
- Can be applied even when data is open-ended, provided deciles can be calculated.
- More accurate than Bowley’s when the middle 50% is not sufficient to reflect the shape.
Demerits of Kelly’s Coefficient
- More complex to compute as it involves deciles, which are not as intuitive as quartiles.
- Requires accurate cumulative frequency tables and interpolation.
- Still ignores extreme tails beyond the 10th and 90th percentiles.
Bowley vs Kelly
| Feature | Bowley’s Skewness | Kelly’s Skewness |
|---|---|---|
| Based on | Quartiles (Q1, Q2, Q3) | Deciles (D1, D5, D9) |
| Data Range Used | Middle 50% | Middle 80% |
| Coverage | Limited coverage | Wider coverage |
| Computation | Simple | Relatively complex |
| Sensitivity to shape | Lower | Higher |
When to Use Kelly’s Coefficient?
- When you want a broader view of skewness in the distribution.
- When quartile-based measures are not sensitive enough.
- When dealing with moderately large datasets where deciles can be calculated meaningfully.
Conclusion
Kelly’s Coefficient of Skewness is a valuable tool when a wider measure of asymmetry is needed. By using deciles, it offers a more refined look at the distribution's central tendency and skewness. Though it involves more computation, its interpretive power justifies the effort, especially in academic and analytical contexts. For UGC NET aspirants, understanding all three major measures Karl Pearson’s, Bowley’s, and Kelly’s ensures thorough preparation for both theory and application-based questions.
Next in this series, we will explore Skewness Based on Moments, which uses the mathematical moments of a distribution to determine asymmetry with precision.