Shapiro Wilk Test Calculator

Shapiro-Wilk Normality Test

Sample Dataset Enter raw numeric data separated by commas, spaces, or newlines ($n \ge 3$)

Please enter at least 3 valid numeric values.

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Parametric Assumption Verification

What Is the Shapiro-Wilk Test?

The Shapiro-Wilk Test is one of the most powerful and widely utilized statistical tests for assessing whether a continuous dataset originates from a normally distributed population.

The test computes the ratio of the best linear unbiased estimator of the population variance to the sample variance. A test statistic $W$ close to 1 indicates normality, whereas small $W$ values indicate deviation from a normal curve.

When to Use It

Pre-Test Assumption Checking: Verifying the normality assumption before running independent T-Tests, Paired T-Tests, or One-Way ANOVA.
Linear Regression Residuals: Testing whether regression model residuals follow a normal distribution $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$.
Quality Control & Process Capability: Assessing whether manufacturing dimensions or process metrics conform to standard Gaussian tolerances.
Selecting Non-Parametric Alternatives: Directing analysis toward the Mann-Whitney U, Wilcoxon Signed-Rank, or Kruskal-Wallis tests when $p \le \alpha$.

The Mathematical Formulas

$$W = \frac{b^2}{SS} = \frac{\left( \sum_{i=1}^n a_i x_{(i)} \right)^2}{\sum_{i=1}^n (x_i - \bar{x})^2}$$ $$\mathbf{a} = \frac{\mathbf{m}^T \mathbf{V}^{-1}}{\sqrt{\mathbf{m}^T \mathbf{V}^{-1} \mathbf{V}^{-1} \mathbf{m}}}$$ $$y = \ln(1 - W), \quad Z = \frac{y - \mu_y}{\sigma_y}, \quad p = 1 - \Phi(Z)$$

Where $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$ represent the sorted sample order statistics, $\mathbf{a}$ are the normalized Shapiro-Wilk weights derived from expected normal order statistics $\mathbf{m}$, and $Z$ is Royston's standardized normalizing transformation.

Worked Example

Scenario: 8 lab specimens yield values $[14.2, 15.1, 14.8, 15.4, 14.9, 15.2, 14.6, 15.0]$ at $\alpha = 0.05$.

Step 1: Calculate Mean & Sum of Squares:
$\bar{x} = 14.90, \quad SS = \sum (x_i - 14.90)^2 = 0.9400$.

Step 2: Sort Order Statistics & Apply Weights:
$x_{(i)} = [14.2, 14.6, 14.8, 14.9, 15.0, 15.1, 15.2, 15.4]$
$b = \sum a_i x_{(i)} = 0.9634 \implies b^2 = (0.9634)^2 = 0.9281$.

Step 3: Compute W-Statistic & P-Value:
$W = \frac{0.9281}{0.9400} = 0.9874 \implies p = 0.9904$.

Conclusion: Since $p = 0.9904 > 0.05$, fail to reject $H_0$. The sample data conforms to a normal distribution.

Computation Complete

Shapiro-Wilk Test Results

Computed W-Statistic & P-Value

W = 0.9874, p = 0.9904

Data Follows a Normal Distribution (Fail to Reject H₀ at α = 0.05)

W-Statistic 0.9874

P-Value 0.9904

Sample Size (n) 8

Sample Mean (x̄) 14.9000

Sample Std Dev (s) 0.3664

Skewness -0.4851

Shapiro Wilk Test Calculator

Shapiro-Wilk Normality Test

Shapiro-Wilk Test Results

Step-by-Step Calculation

Interpretation

Normal Quantile-Quantile (Q-Q) Plot

Perfect for students, researchers, and professionals looking to build real statistical skills.