D'Agostino-Pearson Test

A comprehensive normality test that combines separate tests for skewness and kurtosis, providing detailed diagnostics about specific departures from normality.

Test Overview

What it measures:

Tests for normality by separately evaluating skewness and kurtosis, then combining these into an omnibus test. This approach provides specific information about why data might not be normal.

Advantages:

• • Provides separate skewness and kurtosis tests
• • Identifies specific normality violations
• • Good statistical power
• • Well-established critical values

Limitations:

• • Requires moderate to large sample sizes
• • More complex than single-statistic tests
• • May be overly sensitive with large samples
• • Limited availability in some software

Omnibus K² Statistic & Formulas

The test standardizes sample skewness (√b1) and sample kurtosis (b2) into approximately normal Z-statistics, then sums their squares into a single chi-square distributed omnibus statistic.

Sample skewness and kurtosis

= k-th sample central moment

= sample standard deviation

D'Agostino-Pearson omnibus statistic

Z(√b1)

= transformed, standardized skewness statistic

Z(b2)

= transformed, standardized kurtosis statistic

K² follows a chi-square distribution with 2 degrees of freedom under the null hypothesis of normality.

Key Assumptions

• • Observations are independent and identically distributed
• • Sample size is at least 20, and ideally 50+, since the Z-transforms of skewness and kurtosis rely on large-sample approximations
• • Data are measured on a continuous scale
• • The null hypothesis is that the population is normally distributed

Reading the D'Agostino-Pearson Combined Plot

Example Result & Decision Rule

Example: 30 cycle-time measurements yield √b1=0.85 (moderate right skew) and b2=4.1 (slightly heavy tails). Standardizing gives Z(√b1)=2.10 and Z(b2)=1.35, so:

Worked example

Comparing K2=6.24 to the chi-square distribution with 2 degrees of freedom gives a p-value of approximately 0.044.

Fail to Reject Normality

• • p-value ≥ 0.05
• • Both Z(√b₁) and Z(b₂) within roughly ±2
• • Treat data as approximately normal for downstream tests

Reject Normality

• • p-value < 0.05 (as in the worked example above)
• • Inspect which component (skewness vs. kurtosis) drives the signal
• • Consider transformations or nonparametric methods

sixsigmacenter.com Implementation Guide

Using sixsigmacenter.com Platform:

Module 3: Understand Your Data Shape (Distribution & Normality) at sixsigmacenter.com completely automates the D'Agostino-Pearson test calculations, providing comprehensive skewness and kurtosis analysis with detailed diagnostic information.

🚀 Automated Analysis Process

• • Comprehensive Testing: Platform automatically performs separate skewness and kurtosis tests plus combined analysis
• • Error-Free Processing: Eliminates complex moment calculations and standard error computations
• • Diagnostic Results: Provides detailed component analysis to identify specific normality departures
• • Statistical Rigor: Ensures proper test implementation with accurate critical values

📊 Publication-Ready Results

• • Component Analysis: Separate results for skewness and kurtosis components
• • Combined Assessment: Overall normality test with chi-square distribution
• • Diagnostic Guidance: Clear identification of which components drive non-normality
• • Publication-Ready: Reports with detailed statistical interpretation

Example Data Layout for D'Agostino-Pearson Test

Single column of continuous data for comprehensive skewness and kurtosis-based normality testing. Sample Size Recommendation: Minimum 20-30 observations for reliable D'Agostino-Pearson test results - adequate sample size ensures stable estimates of third and fourth moments for meaningful skewness and kurtosis assessment.

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.