Calculation of Skewness and Kurtosis using Pearson’s Beta and Gamma Coefficients
Subtitle: Data Science and A.I. Lecture Series
Author: Bindeshwar Singh Kushwaha
Institute: PostNetwork Academy
Date: December 4, 2024
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Pearson’s Beta and Gamma Coefficients
Karl Pearson defined the following coefficients based on the first four central moments:
- β1: β1 = μ32 / μ23 (Skewness)
- γ1: γ1 = √β1 = μ3 / (μ2)3/2 (Directionality of Skewness)
Pearson’s Beta and Gamma Coefficients (Continued)
- β2: β2 = μ4 / μ22 (Kurtosis)
- γ2: γ2 = β2 – 3 (Standardized Kurtosis)
Frequency Distribution Table
Problem: For the given data, calculate the first four moments about the mean and find β1, β2, γ1, and γ2.
| Marks (x) | Frequency (f) |
|---|---|
| 5 | 4 |
| 10 | 10 |
| 15 | 20 |
| 20 | 36 |
| 25 | 16 |
| 30 | 12 |
| 35 | 2 |
Calculations of Moments
- First Moment: μ1 = -0.30
- Second Moment: μ2 = 44.50
- Third Moment: μ3 = -52.50
- Fourth Moment: μ4 = 5462.50
Skewness and Kurtosis
- β1 (Skewness): 0.001785
- γ1 (Standardized Skewness): -0.0422 (Negative Skewness)
- β2 (Kurtosis): 2.7499
- γ2 (Standardized Kurtosis): -0.2501 (Platykurtic)
PDF Presentation
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