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Shapiro-Wilk, Anderson-Darling and Lilliefors Normality Tutor

· Last updated：2026-09-07

Run the three workhorse normality tests (Shapiro-Wilk AS R94, Anderson-Darling, Lilliefors) with a Q-Q plot and a Monte-Carlo power comparison.

Details

What this tool helps you do

Exact ports of the reference implementations: R’s swilk.c for Shapiro-Wilk, nortest’s ad.test and lillie.test for the other two, validated digit-for-digit against scipy. Includes Blom Q-Q plotting and seeded power simulations across five alternative shapes.

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Written by the Elysia Tools team

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Science & Education
Input types
textarea, select, checkbox
Output type
html
Sample coverage
4
API ready
Yes

Overview

Assess dataset normality with three standard goodness-of-fit algorithms: Shapiro-Wilk (AS R94), Anderson-Darling, and Lilliefors. This tool generates diagnostic reports complete with Blom Q-Q plots, exact p-values, sample descriptive statistics, and Monte-Carlo power curve simulations across multiple alternative distribution shapes.

When to use

  • Checking normality assumptions prior to running parametric analyses such as t-tests, ANOVA, or linear regression.
  • Evaluating dataset tails, asymmetry, and potential outliers using quantitative tests alongside a Blom Q-Q plot.
  • Comparing statistical power and detection sensitivity across different non-normal distribution alternatives.

How it works

  1. 1Paste numeric observations separated by commas, spaces, or line breaks into the input field.
  2. 2Select your target significance level (α = 0.01, 0.05, or 0.10) and enable the optional Blom Q-Q plot.
  3. 3Choose an alternative distribution profile (such as exponential, bimodal, uniform, or Student t3) to run Monte-Carlo power simulations.
  4. 4Review the resulting HTML report containing test statistics (W, A², D), exact p-values, descriptive metrics, and decision summaries.

Use cases

  • Validating experimental assay residuals in laboratory and clinical research datasets.
  • Demonstrating normality test mechanics and power trade-offs in statistics and data science courses.
  • Verifying process distribution shapes for statistical quality control in manufacturing workflows.

Examples

1. Penicillin Yield Outlier Analysis

Biostatistician

Background

Evaluating thirty fermentation batch yield values where low-end outliers could indicate process inconsistency.

Problem

Confirm whether extreme negative values violate the normality assumption required for standard batch comparison models.

How to use

Enter the 30 yield values, set the significance level to α = 0.05, and enable both the Q-Q plot and the exponential power curve.

data: 0.0987, 0.0533, 0.0293, 0.0246, 0.0200, 0.0194, 0.0191, 0.0180, 0.0172, 0.0132, 0.0102, 0.0084, 0.0077, 0.0058, 0.0016, 0.0000, -0.0026, -0.0036, -0.0042, -0.0114, -0.0139, -0.0222, -0.0333, -0.0348, -0.0363, -0.0363, -0.0402, -0.0583, -0.1184, -0.1420
alpha: 0.05
showQqPlot: true
powerAlternative: exponential
showPowerCurve: true

Outcome

Shapiro-Wilk (W = 0.8922, p = 0.0054) and Anderson-Darling (A² = 1.2245, p = 0.0029) reject normality, while Lilliefors fails to reject (D = 0.1582, p = 0.0535).

2. Sensor Calibration Baseline Check

Quality Engineer

Background

Testing 24 baseline temperature sensor readings before calibrating automated anomaly detection thresholds.

Problem

Verify that sensor measurement noise follows an ordinary normal distribution without severe skew.

FAQ

Which normality test is best for general use?

Shapiro-Wilk is generally recommended as the most powerful omnibus test, while Anderson-Darling provides superior sensitivity to tail departures.

How are the p-values and test statistics calculated?

Calculations use exact ports of standard R implementations: swilk.c (AS R94) for Shapiro-Wilk, and nortest for Anderson-Darling and Lilliefors, validated against SciPy.

What formula is used to plot the Q-Q line?

The Q-Q plot uses Blom plotting positions to calculate empirical quantiles against theoretical normal quantiles with a robust quartile reference line.

What does the Monte-Carlo power simulation represent?

It simulates empirical rejection rates across various sample sizes to show how effectively each test detects departures from normality for a chosen alternative distribution.

What data formats are accepted?

The tool accepts numeric samples entered as raw values separated by spaces, commas, or standard line breaks.

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How to use

Paste the 24 sensor measurements, set α = 0.05, and select the Q-Q plot option to view the quartile fit.

data: 33.6, 40.8, 41.0, 42.8, 44.3, 44.9, 45.2, 45.6, 46.7, 47.1, 47.4, 48.6, 48.7, 51.7, 51.9, 52.2, 52.5, 52.8, 55.0, 55.2, 55.5, 56.1, 57.5, 62.2
alpha: 0.05
showQqPlot: true
powerAlternative: exponential
showPowerCurve: true

Outcome

All three tests fail to reject normality (Shapiro-Wilk p = 0.9438, Anderson-Darling p = 0.8414, Lilliefors p = 0.5896), confirming normal distribution parameters.