1. What is the Shapiro-Wilk Test?

The Shapiro-Wilk test is a statistical test of normality. It evaluates whether a given sample of data comes from a normally distributed population. The test statistic W compares the ordered sample values with the corresponding order statistics from a normal distribution.

2. How Does the Calculator Work?

The calculator uses the Shapiro-Wilk formula:

Where:

• \( W \) — Test statistic (0 ≤ W ≤ 1)
• \( a_i \) — Coefficients based on expected normal order statistics
• \( x_i \) — Ordered sample values
• \( \bar{x} \) — Sample mean

Explanation: The numerator captures how well the data matches a normal distribution's expected order statistics, while the denominator is the sample variance.

3. Interpretation of Results

Details: W values close to 1 indicate normality. Small values suggest non-normality. The exact critical values depend on sample size and significance level.

4. Using the Calculator

Tips: Enter at least 3 and up to 5000 numeric values, separated by commas. The calculator will sort the values and compute W.

5. Frequently Asked Questions (FAQ)

Q1: What sample sizes work best?
A: The test works best for sample sizes between 3 and 5000. For n > 50, approximations are used.

Q2: How to interpret the W value?
A: Values closer to 1 suggest normality. Compare with critical values for your sample size at your chosen α level.

Q3: What are the limitations?
A: The test is sensitive to sample size - large samples may reject normality for trivial deviations.

Q4: Alternatives to Shapiro-Wilk?
A: For large samples, Kolmogorov-Smirnov or Anderson-Darling tests may be used.

Q5: What if my data fails the test?
A: Consider data transformations or non-parametric statistical methods.