1. What is the Shapiro-Wilk Test?

The Shapiro-Wilk test is a statistical test of normality. It determines whether a given sample of data comes from a normally distributed population. The test statistic W quantifies how well the data fits 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 sample size
• \( x_i \) — Ordered sample values
• \( \bar{x} \) — Sample mean
• \( n \) — Sample size

Explanation: The numerator is a weighted sum of the ordered data, squared. The denominator is the sum of squared deviations from the mean. W values close to 1 indicate normality.

3. Importance of Normality Testing

Details: Many statistical tests assume normally distributed data. The Shapiro-Wilk test helps verify this assumption before applying parametric tests.

4. Using the Calculator

Tips: Enter your sample data points separated by commas or spaces. The calculator works best with sample sizes between 3 and 5000. Results are more reliable with larger samples.

5. Frequently Asked Questions (FAQ)

Q1: What does the W statistic mean?
A: W ranges from 0 to 1, with values closer to 1 indicating better fit to normality. Small W values suggest non-normal data.

Q2: What sample size is appropriate?
A: The test works for 3-5000 samples, but is most reliable for 20-50 samples. Very large samples may detect trivial deviations from normality.

Q3: How to interpret the p-value?
A: A p-value < 0.05 typically indicates non-normal data. However, this calculator provides W only; exact p-values require coefficient tables.

Q4: What are alternatives to Shapiro-Wilk?
A: Kolmogorov-Smirnov, Anderson-Darling, and D'Agostino-Pearson tests are other normality tests.

Q5: When is normality testing most important?
A: Crucial for small samples in parametric tests (t-tests, ANOVA). Less critical for large samples due to Central Limit Theorem.