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 two estimates of the variance of the data.

2. How Does the Calculator Work?

The calculator uses the Shapiro-Wilk formula:

Where:

• \( W \) — Test statistic (0 ≤ W ≤ 1)
• \( a_i \) — Coefficients from Shapiro-Wilk tables
• \( x_i \) — Ordered sample values
• \( \bar{x} \) — Sample mean
• \( n \) — Sample size

Explanation: The numerator is based on the expected normal order statistics, while the denominator is the sample variance. Values of W close to 1 suggest normality.

3. Importance of Normality Testing

Details: Many statistical tests assume normally distributed data. The Shapiro-Wilk test is particularly powerful for detecting departures from normality across a wide range of alternatives.

4. Using the Calculator

Tips: Enter numeric values separated by commas. The test works best with sample sizes between 3 and 5000. For accurate results, use statistical software that includes exact coefficients.

5. Frequently Asked Questions (FAQ)

Q1: What does the W value mean?
A: W ranges from 0 to 1, with values closer to 1 indicating better fit to normality. Small p-values (typically <0.05) suggest non-normal data.

Q2: How many data points are needed?
A: The test works with as few as 3 observations, but is more reliable with 20+ points. Maximum recommended is 5000.

Q3: What are the limitations?
A: The test is sensitive to sample size - with large samples, it may detect trivial departures from normality.

Q4: How does it compare to other normality tests?
A: Shapiro-Wilk is generally more powerful than Kolmogorov-Smirnov or Anderson-Darling for most alternatives.

Q5: Should I always test for normality?
A: It depends on your analysis. Some tests are robust to non-normality, while others require strict normality assumptions.