Jarque-Bera Test

Combines skewness and kurtosis measures for comprehensive normality assessment.

Test Overview

What it measures:

The Jarque-Bera test is a comprehensive goodness-of-fit test that determines whether sample data has the skewness and kurtosis matching a normal distribution. It combines measures of asymmetry (skewness) and tail heaviness (kurtosis) into a single test statistic, making it particularly effective for detecting departures from normality in large samples. Named after Carlos Jarque and Anil Bera, this test is widely used in econometrics and financial analysis.

Key Concept:

The test evaluates whether the third moment (skewness) and fourth moment (kurtosis) of the sample data match those expected from a normal distribution. Normal distributions have zero skewness and kurtosis of 3, providing clear benchmarks for comparison.

Advantages:

• • Tests both skewness and kurtosis simultaneously
• • Excellent performance with large samples (n ≥ 50)
• • Available on the sixsigmacenter.com platform
• • Based on well-understood statistical moments
• • Provides insight into type of non-normality
• • Simple chi-square distribution for critical values

Limitations:

• • Poor performance with small samples (n < 50)
• • Only tests third and fourth moments
• • May miss other forms of non-normality
• • Asymptotic test (requires large sample theory)
• • Can be overly sensitive with very large samples

Statistical Foundation

Test Statistic (JB):

The Jarque-Bera statistic combines standardized skewness and kurtosis measures into a chi-square distributed test statistic under the null hypothesis of normality.

Jarque-Bera test statistic

= sample size

= sample skewness

= sample kurtosis

Where S is sample skewness, K is sample kurtosis, and n is sample size. Under normality, JB follows a χ²(2) distribution.

Components:

Skewness (S): Measures asymmetry

• Normal distribution: S = 0

• Right-skewed: S > 0

• Left-skewed: S < 0

Kurtosis (K): Measures tail heaviness

• Normal distribution: K = 3

• Heavy tails: K > 3

• Light tails: K < 3

Distribution Properties:

• JB ~ χ²(2) under null hypothesis

• Degrees of freedom: 2

• Critical value (α=0.05): 5.99

• Critical value (α=0.01): 9.21

• Critical value (α=0.10): 4.61

Mathematical Formulation:

Sample skewness

= individual observation

= sample mean

= sample standard deviation

Sample kurtosis

Reading the Jarque-Bera Skewness and Kurtosis Plot

Interpretation Guidelines

Decision Framework:

Normal Distribution (p > 0.05):

• • Skewness and kurtosis are within normal limits
• • Parametric statistical tests are appropriate
• • JB < 5.99 (α = 0.05)
• • Distribution appears symmetric with normal tail behavior
• • Proceed with confidence intervals and hypothesis tests

Non-Normal (p ≤ 0.05):

• • Significant departure from normality detected
• • Consider non-parametric alternatives
• • JB ≥ 5.99 (α = 0.05)
• • Investigate specific departures (skewness vs kurtosis)
• • Consider data transformation or robust methods

Component Analysis:

Skewness Component Dominates:

When S² >> (1/4)(K-3)² in the JB formula:

• • Primary issue is asymmetry
• • Consider log or Box-Cox transformation
• • Look for outliers in one tail

Kurtosis Component Dominates:

When (1/4)(K-3)² >> S² in the JB formula:

• • Primary issue is tail behavior
• • Examine for outliers or heavy tails
• • Consider robust statistical methods

Sample Size Guidelines:

Small (n < 30)

Not recommended - use Shapiro-Wilk instead

Medium (30-50)

Acceptable but interpret cautiously

Large (n ≥ 50)

Optimal performance and reliability

sixsigmacenter.com Implementation Guide

Assumptions & Limitations

Key Assumptions:

Independent Observations

Data points must be independent - no correlation between observations

Large Sample Theory

Test relies on asymptotic properties - requires sufficient sample size (n ≥ 30)

Continuous Data

Designed for continuous variables - not appropriate for discrete or ordinal data

When to Use Jarque-Bera:

✅ Recommended When:

• • Sample size ≥ 50
• • Need diagnostic information about normality departures
• • Financial or economic data analysis
• • Regression model diagnostics
• • Computational simplicity is important

❌ Avoid When:

• • Sample size < 30
• • Data has known ties or discreteness
• • Suspected non-linear relationships
• • Time series with autocorrelation
• • Need maximum power for small departures

Limitations in Practice:

• Moment-based only: Only examines third and fourth moments, may miss other distributional issues

• Large sample sensitivity: With very large samples (n > 1000), may detect trivial departures

• No directional information: Doesn't indicate which specific transformation might help

• Assumption violations: Performance degrades with correlated or non-independent data

Example Data Layout for Jarque-Bera Test

Single column of continuous data for skewness and kurtosis-based normality testing. Sample Size Recommendation: Minimum 20-30 observations for reliable skewness and kurtosis estimates, optimal with 50+ observations.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.