1. What is the Shapiro-Wilk Test?

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

2. How Does the Calculator Work?

The calculator uses the Shapiro-Wilk formula:

Where:

• \( W \) — Test statistic (0 ≤ W ≤ 1)
• \( x_i \) — Ordered sample values (x₁ ≤ x₂ ≤ ... ≤ xₙ)
• \( a_i \) — Coefficients generated from the means of ordered statistics
• \( \bar{x} \) — Sample mean

Interpretation: Values of W close to 1 indicate normality. Small values suggest non-normality.

3. Importance of Normality Testing

Details: Many statistical tests (t-tests, ANOVA, etc.) assume normally distributed data. The Shapiro-Wilk test helps verify this assumption before applying parametric tests.

4. Using the Calculator

Tips: Enter your numerical data points separated by commas. The test works best with sample sizes between 3 and 5000 observations.

5. Frequently Asked Questions (FAQ)

Q1: What sample size is appropriate?
A: The test works for 3-5000 samples but is most powerful for n ≤ 50. For larger samples, consider other tests like Kolmogorov-Smirnov.

Q2: What's a good W value?
A: Typically W > 0.9 suggests normality, but critical values depend on sample size and significance level (usually 0.05).

Q3: How does this compare to other normality tests?
A: Shapiro-Wilk is generally more powerful than Kolmogorov-Smirnov or Anderson-Darling for small samples.

Q4: What if my data isn't normal?
A: Consider data transformations (log, square root) or non-parametric statistical tests.

Q5: Are there limitations?
A: The test is sensitive to outliers. It may detect non-normality in large samples even for trivial deviations.