Moments and Pearson’s Coefficient Simplified | Data Science & A.I. Lecture Series

Introduction

Welcome to the Data Science and A.I. Lecture Series presented by PostNetwork Academy. In this session, we’ll focus on key statistical concepts: moments about the mean, skewness, and kurtosis. These concepts are essential in understanding data distribution characteristics and play a significant role in data science, artificial intelligence, and statistical analysis.

In this lecture, we aim to simplify the calculation of central moments and Pearson’s coefficients, making it accessible to learners of all levels.

The Problem

Consider the following question:

The first four moments of a distribution about the value 5 are given as 1, 10, 20, and 25. Find the central moments (\(μ_2, μ_3, μ_4\)), the skewness (\(β_1\)), and the kurtosis (\(β_2\)).

Step-by-Step Solution

Step 1: Calculate the Second Moment about the Mean

The second moment about the mean is given by:

Formula:

μ₂ = μ’₂ – (μ’₁)²

Solution:

μ₂ = 10 – (1)² = 9

Step 2: Calculate the Third Moment about the Mean

Formula:

μ₃ = μ’₃ – 3μ’₂μ’₁ + 2(μ’₁)³

Solution:

μ₃ = 20 – 3 × 10 × 1 + 2(1)³ = -8

Step 3: Calculate the Fourth Moment about the Mean

Formula:

μ₄ = μ’₄ – 4μ’₃μ’₁ + 6μ’₂(μ’₁)² – 3(μ’₁)⁴

Solution:

μ₄ = 25 – 4 × 20 × 1 + 6 × 10 × 1 – 3(1)⁴ = 2

Skewness and Kurtosis

Step 4: Calculate Skewness (β₁)

Formula:

β₁ = (μ₃)² / (μ₂)³

Solution:

β₁ = (-8)² / (9)³ = 64 / 729 ≈ 0.0877

Step 5: Calculate Kurtosis (β₂)

Formula:

β₂ = μ₄ / (μ₂)²

Solution:

β₂ = 2 / 81 ≈ 0.0247