Moment-Based Measures of Skewness

Moment-based skewness is one of the most precise and mathematically sound methods of measuring asymmetry in a distribution. Unlike positional methods like Karl Pearson’s, Bowley’s, or Kelly’s, this method relies on Statistical Moments which consider each value of the dataset in relation to the mean, weighted by the frequency or probability.

Understanding the Concept of Moments

Moments are quantitative measures used to describe the shape of a distribution. Just like we describe a person by their height and weight, a distribution can be described by its moments.

There are four moments commonly used:

• First moment about the mean (μ₁): Always zero. It indicates the mean position.
• Second moment about the mean (μ₂): Indicates the variance (i.e., dispersion).
• Third moment about the mean (μ₃): Indicates skewness.
• Fourth moment about the mean (μ₄): Indicates kurtosis (peakedness).

Formula for Moment-Based Skewness

The skewness coefficient based on moments is defined as:

β₁ = μ₃² / μ₂³     or     γ₁ = μ₃ / σ³

Where:

• μ₂: Second central moment = Variance = Σ(f(x − ȳ)²) / N
• μ₃: Third central moment = Σ(f(x − ȳ)³) / N
• σ: Standard deviation = √μ₂
• γ₁: Skewness coefficient in actual value
• β₁: Square of γ₁, also called the non-dimensional form

Interpretation of Skewness

• γ₁ = 0: Symmetrical distribution
• γ₁ > 0: Positively skewed (right-tailed)
• γ₁ < 0: Negatively skewed (left-tailed)

Example:

Consider a dataset: 4, 6, 8, 10, 12

Step 1: Calculate mean (ȳ)

ȳ = (4 + 6 + 8 + 10 + 12) / 5 = 40 / 5 = 8

Step 2: Calculate deviations from mean and compute central moments:

x	x − ȳ	(x − ȳ)²	(x − ȳ)³
4	-4	16	-64
6	-2	4	-8
8	0	0	0
10	2	4	8
12	4	16	64

μ₂ = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8

μ₃ = (−64 − 8 + 0 + 8 + 64) / 5 = 0 / 5 = 0

γ₁ = μ₃ / σ³ = 0 / (√8)³ = 0

Interpretation: The distribution is symmetrical.

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Merits of Moment-Based Skewness

• Most scientific and comprehensive method.
• Takes into account all observations and their deviations from the mean.
• Can be applied to any type of data: raw, discrete, or continuous.
• More suitable for theoretical and research analysis.

Demerits of Moment-Based Skewness

• Computation is lengthy, especially with large datasets.
• Values can be difficult to interpret intuitively without mathematical background.
• More prone to distortion due to extreme values, as all values are involved in calculations.

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Comparison of Skewness Methods

Method	Data Considered	Best For	Ease of Use	Sensitivity
Karl Pearson’s	Mean, Median, Mode	Symmetrical distributions	Simple	Moderate
Bowley’s	Q1, Q2, Q3	Open-ended classes	Very easy	Low
Kelly’s	D1, D5, D9	Broad central data	Moderate	High
Moment-Based	All values (mean-based)	Theoretical, precise analysis	Complex	Very high

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Application in Real Life

• Financial analysts use moment-based skewness to detect risk asymmetry in return distributions.
• Social scientists use it to understand population characteristics when modeling data.
• Researchers in economics and quality control use it for hypothesis testing and variability assessments.

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Key Points for UGC NET Preparation

• Remember that μ₁ = 0 always (first moment about the mean).
• β₁ is unitless; γ₁ gives direction of skewness.
• Moment-based skewness is more comprehensive but harder to compute manually.
• For MCQs, you may be asked the formula directly or to interpret skewness value.

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Conclusion

Moment-based measures of skewness offer a statistically rigorous way to quantify the asymmetry of a distribution. Though more complex than other methods, they provide a complete picture by involving every data point’s deviation from the mean. Mastering this technique equips students with a deeper analytical skill set, especially useful in advanced research, financial modeling, and econometrics.

This completes the detailed exploration of various skewness measures. Up next, we’ll move toward understanding Kurtosis, the measure of how peaked or flat a distribution is compared to a normal curve.