Central Moments: Key Measures of Statistical Distributions

Central moments are fundamental mathematical tools that characterize probability distributions by measuring deviations from the mean. While you’ve likely encountered variance and skewness in statistical analysis, these familiar concepts belong to a larger mathematical framework—central moments—that provides deeper insights into distribution shape, spread, and behavior. From basic theoretical foundations to practical applications across multiple fields, these numerical descriptors have transformed how we analyze data and understand uncertainty.

What Are Central Moments?

Central moments are statistical measures that quantify the shape and characteristics of probability distributions by examining deviations from the distribution’s mean. Unlike raw moments that calculate expected values of different powers of a random variable, central moments specifically focus on how data clusters and spreads around the central value.

The nth central moment of a random variable X is mathematically defined as:

μₙ = E[(X – μ)ⁿ]

Where:

• μ is the mean of the distribution (first raw moment)
• E represents the expected value operator

This elegant formulation provides remarkably powerful insights into distribution characteristics, with each moment revealing different aspects of the underlying data pattern.

How Central Moments Differ from Raw Moments

To truly understand central moments, we must distinguish them from their raw counterparts:

Order	Raw Moment	Central Moment	Interpretation
1st	E[X] = Mean	E[X-μ] = 0	Mean (always zero for central)
2nd	E[X²]	E[(X-μ)²] = Variance	Spread around mean
3rd	E[X³]	E[(X-μ)³]	Skewness (asymmetry)
4th	E[X⁴]	E[(X-μ)⁴]	Kurtosis (tail behavior)

While raw moments calculate expected values of the powers of X directly, central moments measure the expected values of deviations from the mean raised to various powers. This subtle difference creates measures that more intuitively describe distribution shape and behavior.

The Department of Statistics at Yale University provides comprehensive resources exploring the relationship between different types of moments in probability theory.

The First Four Central Moments Explained

The first four central moments provide particularly valuable insights for statistical analysis:

First Central Moment (μ₁) The first central moment always equals zero by definition: μ₁ = E[X – μ] = E[X] – μ = μ – μ = 0

This mathematical certainty reflects the fact that positive and negative deviations from the mean perfectly balance each other.

Second Central Moment (μ₂): Variance The second central moment, more commonly known as variance, measures the average squared deviation from the mean: μ₂ = E[(X – μ)²] = σ²

This fundamental measure of dispersion indicates how spread out the distribution is from its center. The standard deviation (σ), which equals the square root of variance, provides this measure in the same units as the original data.

Third Central Moment (μ₃): Related to Skewness The third central moment measures distribution asymmetry: μ₃ = E[(X – μ)³]

While the raw third central moment depends on the scale of measurement, statisticians typically standardize it by dividing by σ³, creating the skewness coefficient: γ₁ = μ₃/σ³

This standardized measure reveals:

• Positive values: Right-skewed distribution (longer right tail)
• Zero: Symmetric distribution
• Negative values: Left-skewed distribution (longer left tail)

Fourth Central Moment (μ₄): Related to Kurtosis The fourth central moment relates to the distribution’s tail behavior and peakedness: μ₄ = E[(X – μ)⁴]

Similar to skewness, this is typically standardized as the kurtosis coefficient: γ₂ = μ₄/σ⁴

Often, excess kurtosis is calculated by subtracting 3 (the kurtosis of a normal distribution): γ₂ – 3, which yields:

• Positive values: Leptokurtic (heavier tails than normal)
• Zero: Mesokurtic (normal-like tails)
• Negative values: Platykurtic (lighter tails than normal)

How to Calculate Central Moments

The practical computation of central moments can follow several approaches depending on the available data and distribution information.

For Discrete Distributions

For a discrete random variable with probability mass function P(X = xᵢ), the nth central moment is calculated as:

μₙ = ∑ᵢ (xᵢ – μ)ⁿ · P(X = xᵢ)

This weighted sum accounts for each possible value’s deviation from the mean, raised to the nth power, and weighted by its probability.

For Continuous Distributions

With continuous random variables having a probability density function f(x), the nth central moment becomes:

μₙ = ∫₋∞^∞ (x – μ)ⁿ · f(x) dx

This integral computes the expected value of the nth power of deviations across the entire distribution.

From Sample Data

When working with empirical data rather than theoretical distributions, sample central moments provide estimates of the population parameters:

m̂ₙ = (1/N) · ∑ᵢ₌₁^N (xᵢ – x̄)ⁿ

Where:

• x̄ is the sample mean
• N is the sample size
• xᵢ represents individual observations

For unbiased estimation, particularly for variance, statisticians often adjust the formula with N-1 in the denominator.

The Massachusetts Institute of Technology’s OpenCourseWare provides detailed explanations of estimation techniques for moments from sample data.

Central Moments for Common Distributions

Different probability distributions exhibit characteristic central moment patterns that help identify and distinguish them.

Normal Distribution

The normal distribution (Gaussian), fundamental in statistics and natural sciences, has particularly elegant central moments:

Central Moment	Value for Normal Distribution
μ₁	0 (by definition)
μ₂	σ² (variance parameter)
μ₃	0 (perfectly symmetric)
μ₄	3σ⁴ (excess kurtosis = 0)

These values reflect the distribution’s perfect symmetry (zero skewness) and standard tail behavior (mesokurtic). Higher-order moments follow a specific pattern related to lower-order ones.

Exponential Distribution

The exponential distribution, common in lifetime and waiting time models, has central moments:

Central Moment	Value for Exponential(λ)
μ₁	0
μ₂	1/λ²
μ₃	2/λ³
μ₄	9/λ⁴

The positive third central moment confirms the distribution’s right-skewed nature, while the fourth moment indicates heavier tails than the normal distribution.

Uniform Distribution

The uniform distribution, representing complete randomness within an interval [a,b], has central moments:

Central Moment	Value for Uniform(a,b)
μ₁	0
μ₂	(b-a)²/12
μ₃	0 (symmetric)
μ₄	(b-a)⁴/80 (platykurtic)

The zero third central moment reflects the distribution’s symmetry, while the fourth moment indicates lighter tails than normal (negative excess kurtosis).

Applications of Central Moments

The theoretical elegance of central moments translates into powerful practical applications across numerous fields.

In Statistical Analysis

Statisticians routinely employ central moments to:

• Characterize distribution shape beyond simple measures of center and spread
• Test for normality by examining skewness and kurtosis
• Develop robust statistical methods that account for non-normal data characteristics
• Compare distributions across different datasets or populations

The Jarque-Bera test, which combines sample skewness and kurtosis, exemplifies how central moments contribute to formal statistical inference.

In Finance and Risk Management

Financial analysts leverage central moments to:

• Quantify investment risk beyond simple variance (higher moments capture tail risk)
• Develop portfolio optimization techniques that account for skewness and kurtosis
• Model asset return distributions more accurately than normal assumptions allow
• Price derivatives using distribution characteristics beyond the Black-Scholes framework

In Engineering and Quality Control

Engineers apply central moment concepts to:

• Monitor manufacturing process stability
• Design robust systems that perform well under varied conditions
• Analyze signal processing characteristics
• Develop image recognition algorithms that use moment invariants

These applications demonstrate how abstract mathematical concepts transform into practical tools across diverse disciplines.

Frequently Asked Questions About Central Moments