1. Moments

Moments are measures that describe the shape and spread of a distribution relative to its mean. They are the expected values of powers of deviations from a central value.

Types of Moments

1. Raw Moments (about the origin) $$\mu'_r = \frac{\sum x^r}{N}$$

• $r$ = order of the moment (1, 2, 3…)
• N = number of observations

2. Central Moments (about the mean) $$\mu_r = \frac{\sum (x - \bar{x})^r}{N}$$

• r = order
• $\mu_1 = 0$ (by definition)
• $\mu_2 = \text{Variance}$
• $\mu_3$ and $\mu_4$ are used for skewness and kurtosis

2. Skewness

Skewness measures the asymmetry of a probability distribution around its mean.

• Symmetrical distribution → Skewness = 0
• Positive skew (Right-skewed) → Tail on right side, Skewness > 0
• Negative skew (Left-skewed) → Tail on left side, Skewness < 0

Formula (using central moments)

• $\mu_3$ = third central moment
• $\mu_2$ = variance

Interpretation

Example: Income distributions are usually positively skewed (few high incomes).

3. Kurtosis

Kurtosis measures the peakedness or flatness of a distribution compared to the normal distribution.

• High kurtosis (Leptokurtic) → Sharp peak, heavy tails
• Low kurtosis (Platykurtic) → Flat peak, light tails
• Normal distribution (Mesokurtic) → Kurtosis ≈ 3

Formula (using central moments)

• $\mu_4$ = fourth central moment
• $\mu_2$ = variance

Interpretation

Summary Table

Key Points to Remember

• Central moments are preferred over raw moments for shape analysis.
• Skewness tells direction of asymmetry.
• Kurtosis tells height/flatness of the distribution peak.
• Both skewness and kurtosis are dimensionless (unit-free).