1. What is the Shapiro-Wilk Test?

The Shapiro-Wilk test is a statistical test of normality that determines whether a given sample of data comes from a normally distributed population. It's particularly effective for small to medium sample sizes (3 ≤ n ≤ 5000).

2. How Does the Test Work?

The test calculates a W statistic that compares the ordered sample values with the corresponding expected values from a normal distribution:

Where:

• \( W \) — Test statistic (0 ≤ W ≤ 1)
• \( x_{(i)} \) — i-th ordered sample value
• \( a_i \) — Coefficients generated from the means, variances and covariances of the order statistics
• \( \bar{x} \) — Sample mean

Explanation: Higher values of W (closer to 1) indicate the sample is likely from a normal distribution.

3. Interpreting the Results

Guidelines:

• W > 0.95 (for large samples) may indicate normality
• Compare against critical values from Shapiro-Wilk tables
• Small p-values (typically < 0.05) suggest non-normality

4. Using the Calculator

Tips: Enter your data as comma-separated values. The calculator works best with sample sizes between 3-5000. For accurate results, use statistical software for formal testing.

5. Frequently Asked Questions (FAQ)

Q1: What sample size works best with Shapiro-Wilk?
A: The test is most reliable for sample sizes between 3 and 5000. For larger samples, other tests like Kolmogorov-Smirnov may be more appropriate.

Q2: What's the difference between W and p-value?
A: W is the test statistic (0-1), while p-value indicates significance. Typically p < 0.05 rejects normality.

Q3: When should I test for normality?
A: Before using parametric tests (t-tests, ANOVA, etc.) that assume normally distributed data.

Q4: Are there alternatives to Shapiro-Wilk?
A: Yes, including Kolmogorov-Smirnov, Anderson-Darling, and D'Agostino's K-squared tests.

Q5: Can I use this for very small samples (n < 5)?
A: The test works but has limited power with very small samples. Graphical methods (Q-Q plots) may be more informative.