Normality Test

Why is normality testing important and how do the tests work?

Many statistical methods — including control charts, capability indices (Cpk/Ppk), t-tests, and ANOVA — assume the underlying data follows a normal (Gaussian) distribution. Normality testing verifies this assumption before applying these methods. Violating normality can lead to incorrect confidence intervals, capability estimates, and hypothesis test conclusions.

The Shapiro-Wilk test is the most powerful normality test for small to moderate samples (n < 5000). The Anderson-Darling test is sensitive to deviations in the tails, making it useful for reliability and extreme value applications. The Jarque-Bera test examines skewness and kurtosis, working well for larger samples.

All tests share the same hypothesis structure: H₀ (null) = data is normally distributed; H₁ = data is not normal. If the p-value < α (typically 0.05), reject normality. The QQ-plot provides visual confirmation: points lying on the diagonal line indicate normality; systematic deviations reveal the nature of non-normality.

Formula: Shapiro-Wilk: W = (Σaᵢx(ᵢ))² / Σ(xᵢ - x̄)² Anderson-Darling: A² = -n - Σ[(2i-1)/n](ln F(x(ᵢ)) + ln(1-F(x(n+1-i)))) reported as A*² = A²(1 + 0.75/n + 2.25/n²) Jarque-Bera: JB = (n/6)(S² + K²/4) Reject H₀ if p-value < α

Example Calculation

20 measurements tested: Shapiro-Wilk W = 0.965, p = 0.42; Anderson-Darling A² = 0.31, p = 0.52; Jarque-Bera JB = 1.2, p = 0.55. All p-values > 0.05 — fail to reject normality. The QQ-plot shows points closely following the diagonal. Data can be treated as normally distributed.

When to Use This Calculator

• Before computing Cpk or Ppk — process capability indices assume normal data; non-normal data requires transformation or alternative methods
• Before performing t-tests or ANOVA — these parametric tests assume normally distributed residuals; violations affect confidence interval accuracy
• When setting up SPC control charts — standard control chart limits assume normality; significantly non-normal data needs different chart types
• During incoming inspection when data looks unusual — normality testing helps determine if the distribution has shifted or if a different distribution model is needed
• When preparing data for regression analysis — checking normality of residuals validates the assumptions underlying the regression model

Common Mistakes to Avoid

• Rejecting normality based solely on a p-value with very large samples — with n > 500, tests detect trivially small deviations from normality that have no practical impact; always use the QQ-plot for visual judgment
• Accepting normality without checking with a small sample (n < 20) — tests have low power with small samples and may fail to detect meaningful departures from normality
• Applying normality tests to integer or heavily rounded data — discrete data or data with few unique values can fail normality tests even if the underlying process is normal; ensure measurement resolution is adequate
• Testing raw data when you should be testing residuals — for regression and ANOVA, it is the residuals (not the raw data) that need to be normally distributed

How to Interpret Results

• All tests show p > 0.05 and QQ-plot is linear — data is consistent with normality; proceed with parametric methods and standard capability analysis
• Tests show p < 0.05 but QQ-plot looks approximately linear — with large samples, minor deviations can produce significant p-values; if the QQ-plot is nearly linear, normality is adequate for most practical purposes
• QQ-plot shows S-shaped curve — data has heavier tails than normal (leptokurtic); consider t-distribution or robust methods
• QQ-plot shows systematic curvature (concave or convex) — data is skewed; try log or Box-Cox transformation
• Different tests disagree — Shapiro-Wilk is the most reliable for general use; Anderson-Darling is best when tails matter (reliability); use the QQ-plot as the visual arbiter

Related Standards & References

• ISO 5479 — Statistical interpretation of data: tests for departure from the normal distribution, the international standard covering normality testing methodology
• Shapiro & Wilk (1965), Biometrika — The original paper introducing the W statistic implemented here
• NIST/SEMATECH e-Handbook §1.3.5 — Reference procedures for the Anderson-Darling test and normal probability (Q-Q) plotting
• Jarque & Bera (1987), International Statistical Review — The skewness-kurtosis test for normality used for larger samples
• AIAG SPC Reference Manual — Requires verifying normality before computing capability indices such as Cpk and Ppk

Frequently Asked Questions

What should I do if my data fails the normality test?

Options include: (1) Transform the data using log, Box-Cox, or Johnson transformation to achieve normality. (2) Use non-parametric methods that don't assume normality (e.g., Mann-Whitney instead of t-test). (3) Fit a different distribution (Weibull, lognormal, etc.) that better describes the data. (4) For large samples (n > 30), the Central Limit Theorem often makes normality violations less impactful for mean-based tests.

Why do different normality tests give different results?

Each test is sensitive to different aspects of non-normality. Shapiro-Wilk has the best overall power for detecting various departures. Anderson-Darling emphasizes tail deviations (important for reliability). Jarque-Bera focuses on skewness and kurtosis. With borderline data, one test may reject while others do not. Use the QQ-plot as the visual tiebreaker.