Skewness and Kurtosis

Overview

Skewness and kurtosis are measures of the shape of a distribution, describing its asymmetry and tail behavior.

Skewness

Definition

Skewness measures the asymmetry of a distribution.

$$\text{Skewness} = \frac{\sum \left(\frac{x_i - \bar{x}}{s}\right)^3}{n}$$

Or using the moment coefficient:

$$\gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3}$$

Types of Skewness

Type	Value	Description	Tail
Symmetric	≈ 0	Mean ≈ Median	Equal
Right (positive)	> 0	Mean > Median	Right tail longer
Left (negative)	< 0	Mean < Median	Left tail longer

Visual Guide

Left-skewed        Symmetric         Right-skewed
(negative)         (γ₁ ≈ 0)          (positive)
                       
    ╱╲                ╱╲                ╱╲
   ╱  ╲              ╱  ╲              ╱  ╲
  ╱    ╲            ╱    ╲            ╱    ╲
╱      ╲───       ╱      ╲          ───╱    ╲
         tail           tail        tail
Mean < Median     Mean ≈ Median     Mean > Median

Pearson's Coefficient of Skewness

$$\text{Skewness} = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}}$$

Rule of Thumb

Skewness	Interpretation
$\lvert \gamma_1 \rvert < 0.5$	Approximately symmetric
$0.5 \leq \lvert \gamma_1 \rvert < 1$	Moderately skewed
$\lvert \gamma_1 \rvert \geq 1$	Highly skewed

Kurtosis

Definition

Kurtosis measures the "tailedness" of a distribution—how heavy or light the tails are.

$$\text{Kurtosis} = \frac{\sum \left(\frac{x_i - \bar{x}}{s}\right)^4}{n}$$

Excess Kurtosis

The normal distribution has kurtosis = 3. We often use excess kurtosis:

$$\text{Excess Kurtosis} = \text{Kurtosis} - 3$$

Types of Kurtosis

Type	Excess Kurtosis	Description
Mesokurtic	≈ 0	Normal-like tails
Leptokurtic	> 0	Heavy tails, sharp peak
Platykurtic	< 0	Light tails, flat peak

Visual Guide

Leptokurtic     Mesokurtic      Platykurtic
(heavy tails)   (normal)        (light tails)
                    
     ╱╲             ╱╲             ╱──╲
    ╱  ╲           ╱  ╲           ╱    ╲
   │    │         ╱    ╲         │      │
  ╱      ╲       ╱      ╲        │      │
─╱        ╲─   ╱        ╲      ╱        ╲

Examples of Common Distributions

Distribution	Skewness	Excess Kurtosis
Normal	0	0
Exponential	2	6
Uniform	0	-1.2
Student's t (df=5)	0	6
Chi-square (df=4)	1.41	3

Practical Applications

Skewness

• Income data: Usually right-skewed (few high earners)
• Age at death: Usually left-skewed
• Housing prices: Typically right-skewed
• Exam scores: May be left-skewed (easy test) or right-skewed (hard test)

Kurtosis

• Financial returns: Often leptokurtic (fat tails = more extreme events)
• Quality control: Detect unusual variance patterns
• Risk assessment: Higher kurtosis = more extreme outcomes

Effect on Statistical Analysis

Issue	Skewness Effect	Kurtosis Effect
Mean vs Median	Mean pulled toward tail	Less affected
Standard deviation	May not represent typical spread	Underestimates tail risk
Normal-based tests	May be invalid	May be invalid
Outliers	More in direction of skew	More with high kurtosis

Transformations for Skewed Data

Skewness	Common Transformations
Right-skewed	$\log(x)$, $\sqrt{x}$, $1/x$
Left-skewed	$x^2$, $e^x$