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 one of the most powerful tests for normality, especially for small sample sizes.

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, variances and covariances of the order statistics of a sample of size n from a normal distribution
• \( \bar{x} \) — Sample mean

Interpretation: Values of W close to 1 indicate normality. Small values indicate 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 data points separated by commas. The test works best with sample sizes between 3 and 5000. For very large samples, even small deviations from normality may be significant.

5. Frequently Asked Questions (FAQ)

Q1: What sample size works best for Shapiro-Wilk?
A: The test is most reliable for small to moderate sample sizes (3-2000). For very large samples, consider Q-Q plots instead.

Q2: What's a good W value?
A: Generally, W > 0.9 suggests normality, but the exact critical value depends on sample size and significance level.

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 fails the normality test?
A: Consider data transformations or non-parametric tests that don't assume normality.

Q5: Can I use this for very small samples?
A: Yes, Shapiro-Wilk works with samples as small as 3, but with very small n, power to detect non-normality is limited.