Moments of a Distribution: Mean to Kurtosis Explained

Moments are a family of summary numbers, each capturing a deeper feature of a distribution: the first moment gives the centre (mean), the second gives the spread (variance), the third gives asymmetry (skewness), and the fourth gives tail weight (kurtosis). One framework, four insights.

They matter because moments unify the whole of descriptive statistics — instead of four disconnected formulas, you learn one idea (averaging powers of deviations) and derive centre, spread, skew and kurtosis from it.

Key Concepts

Raw moment (about zero): μ'ᵣ = Σxʳ/n — the r-th power averaged. Central moment (about the mean): μᵣ = Σ(x − x̄)ʳ/n — deviations averaged. The four: μ₁ = 0 always; μ₂ = variance; μ₃ feeds skewness; μ₄ feeds kurtosis. Standardised moments divide by powers of σ to become unit-free: skewness = μ₃/σ³, kurtosis = μ₄/σ⁴.

Formulas

First raw moment: μ'₁ = Σx/n = x̄ (the mean). Second central moment: μ₂ = Σ(x − x̄)²/n = σ². Third central moment: μ₃ = Σ(x − x̄)³/n; skewness γ₁ = μ₃/σ³. Fourth central moment: μ₄ = Σ(x − x̄)⁴/n; kurtosis γ₂ = μ₄/σ⁴. Relation: μ₂ = μ'₂ − (μ'₁)². Central moments carry units of (data unit)ʳ; standardised ones are unit-free.

Worked Example

Data: 2, 4, 6, 8 (n = 4), mean x̄ = 5. Raw moments: μ'₁ = 20/4 = 5; μ'₂ = (4+16+36+64)/4 = 120/4 = 30. Central moments: deviations −3, −1, 1, 3. μ₂ = (9+1+1+9)/4 = 20/4 = 5 = variance (check: 30 − 25 = 5 ✓). μ₃ = (−27 −1 +1 +27)/4 = 0/4 = 0 — the distribution is symmetric, so skewness = 0. μ₄ = (81+1+1+81)/4 = 164/4 = 41. Kurtosis: γ₂ = 41/5² = 41/25 = 1.64 — platykurtic (< 3), light tails, as expected for evenly spaced values with no extremes. One table of powers delivered mean, variance, symmetry and tail weight.

Common Misconceptions

1. "The first central moment is the mean." No — the first central moment μ₁ is always zero; the mean is the first raw moment. 2. "Moments only work for normal data." They are defined for any distribution; normality only gives them neat reference values (skew 0, kurtosis 3). 3. "Higher moments are always more informative." Beyond the fourth, moments become extremely sensitive to outliers and are rarely used in practice.

Real-World Applications

Moments underpin portfolio theory in finance (mean-variance-skewness preferences), signal processing (characterising noise), physics (centre of mass as a first moment, moment of inertia as a second), and statistical software, which computes skewness and kurtosis from central moments internally.

Moments are power-averaged summaries: the first raw moment is the mean, and the second, third and fourth central moments give variance, skewness and kurtosis. Learning this single pattern replaces four isolated formulas with one ladder that climbs from centre to shape.