Shapiro-Wilk Test

The most powerful normality test for small to moderate sample sizes, providing excellent detection of departures from normality.

Test Overview

What it measures:

The Shapiro-Wilk test is widely considered the most powerful normality test for sample sizes up to 2000. It calculates a test statistic W that represents the ratio of the best linear unbiased estimator of variance to the usual corrected sum of squares estimator of variance. The test achieves its high power by utilizing the exact expected values and covariances of order statistics from a normal distribution.

Key Concept:

The test compares the linear combination of order statistics (which provides the maximum likelihood estimate of variance for normal data) with the standard sample variance. Perfect normality would yield W = 1, while departures from normality result in smaller W values.

Advantages:

• • Highest statistical power for small samples (n ≤ 50)
• • Excellent performance up to n = 2000
• • Well-established and extensively validated
• • Fully automated on the sixsigmacenter.com platform
• • Recommended by most statisticians as first choice
• • Robust critical values and p-value calculations

Limitations:

• • Computationally intensive for large samples
• • Sample size restriction (n ≤ 2000 in most implementations)
• • Sensitive to tied observations
• • Requires complete data (no missing values)
• • Complex algorithm not suitable for manual calculation

W Statistic & Formula Focus

Test Statistic:

Shapiro-Wilk W statistic

= tabulated coefficients derived from covariances of normal order statistics

x(i)

= i-th order statistic (sorted sample value)

= sample mean

W ranges from 0 to 1; values close to 1 indicate close agreement with normality.

Reading the Shapiro-Wilk Order-Statistic Plot

Example Result & Signal Interpretation

For a sample of 24 process measurements, suppose the platform reports W=0.968,p=0.61.

Consistent with Normality

• • p > 0.05: fail to reject normality
• • W close to 1
• • Parametric control charts and Cpk are appropriate

Departure from Normality

• • p ≤ 0.05: reject normality
• • W notably below 1
• • Investigate transformation or non-parametric alternatives

Implementation Guide

Using sixsigmacenter.com Platform:

Module 3: Understand Your Data Shape (Distribution & Normality) at sixsigmacenter.com provides complete Shapiro-Wilk test implementation with the highest computational accuracy and comprehensive result interpretation.

🚀 Automated Analysis Process

• • Instant Calculation: Platform handles complex coefficient calculations and provides immediate results
• • Optimal Accuracy: Uses precise algorithms with exact coefficient tables for different sample sizes
• • Complete Output: Provides W statistic, exact p-value, critical values, and confidence assessment
• • Sample Size Validation: Automatically checks sample size constraints and provides appropriate warnings

📊 Publication-Ready Results

• • Statistical Summary: Complete Shapiro-Wilk test results with detailed interpretation
• • Power Analysis: Information about test power and effect size for your sample
• • Decision Support: Clear recommendations based on W statistic and significance level
• • Publication-Ready: Reports meeting academic and regulatory standards

Example Data Layout for Shapiro-Wilk Test

Single column of continuous data for powerful normality testing. Sample Size Recommendation: Minimum 8-50 observations for optimal power - Shapiro-Wilk is most effective with small to moderate sample sizes (n ≤ 50) and maintains high statistical power.

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.