Karl Pearson’s Coefficient of Skewness is a well-known and widely-used method to measure the skewness (asymmetry) of a distribution. This coefficient helps us understand whether a distribution is symmetrical, positively skewed (tail on the right), or negatively skewed (tail on the left).
Understanding Skewness
Before diving into Karl Pearson's formula, let's recall that skewness refers to the degree of asymmetry observed in a frequency distribution. If a distribution is perfectly symmetrical, the mean, median, and mode all coincide, and the skewness is zero.
- Positive Skewness: Mean > Median > Mode
- Negative Skewness: Mean < Median < Mode
Formula of Karl Pearson’s Coefficient of Skewness
There are two forms of the formula depending on the availability of the mode:
1. When Mode is Known:
Skp = (Mean - Mode) / Standard Deviation
2. When Mode is Not Clearly Defined:
Skp = 3(Mean - Median) / Standard Deviation
Where,
- Mean: The average value of the dataset.
- Mode: The value that appears most frequently.
- Median: The middle value when data is arranged in ascending order.
- Standard Deviation (SD): Measures the dispersion of data from the mean.
Range of Karl Pearson’s Coefficient
The value of Skp generally lies between -3 and +3. However, in practice, most values lie between -1 and +1 in real-life data.
Interpretation
- If Skp = 0 → Symmetrical distribution
- If Skp > 0 → Positively skewed distribution (tail on the right)
- If Skp < 0 → Negatively skewed distribution (tail on the left)
Example 1: (With Mode)
Suppose the following statistics are given for a dataset:
- Mean = 60
- Mode = 55
- Standard Deviation = 10
Solution:
Skp = (Mean - Mode) / SD = (60 - 55) / 10 = 5 / 10 = +0.5
Interpretation: The distribution is positively skewed.
Example 2: (Without Mode)
Mean = 45, Median = 42, Standard Deviation = 6
Skp = 3(Mean - Median) / SD = 3(45 - 42) / 6 = 9 / 6 = +1.5
Interpretation: Highly positively skewed distribution.
Merits of Karl Pearson’s Coefficient of Skewness
- It gives a numerical value indicating the direction and degree of skewness.
- Useful for comparing different datasets.
- Easy to calculate when mean, mode (or median), and SD are available.
- Helps in statistical modeling and decision-making.
Demerits / Limitations
- Not suitable if mode is ill-defined or absent.
- Assumes the data follows a continuous distribution.
- Heavily influenced by extreme values (outliers).
- Less informative when used alone without visual aids like histograms.
Comparison Table
| Measure | Formula | When to Use |
|---|---|---|
| Karl Pearson’s Skp (Mode Known) | (Mean - Mode) / SD | When Mode is reliable and well-defined |
| Karl Pearson’s Skp (Mode Unknown) | 3(Mean - Median) / SD | When Mode is unreliable or not available |
Real-Life Applications
- Economics: Understanding income distributions
- Business: Analyzing customer behavior and purchase patterns
- Education: Interpreting examination results
- Finance: Analyzing stock return distributions
Conclusion
Karl Pearson’s Coefficient of Skewness is a foundational tool in statistical analysis that offers a direct and interpretable way to assess the symmetry of a distribution. Understanding its formula, application, and interpretation equips students and researchers to draw meaningful conclusions from data. It sets the stage for learning more advanced concepts like Bowley’s Coefficient of Skewness and Kelly's Coefficient of Skewness, which we will cover in subsequent articles.