Shapiro Wilks Calculator Normal Distribution

Shapiro-Wilk Test Formula:

\[ W = \frac{(\sum_{i=1}^n a_i x_i)^2}{\sum_{i=1}^n (x_i - \bar{x})^2} \]

Shapiro-Wilk W Statistic:

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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 Calculator Work?

The calculator uses the Shapiro-Wilk formula:

\[ W = \frac{(\sum_{i=1}^n a_i x_i)^2}{\sum_{i=1}^n (x_i - \bar{x})^2} \]

Where:

\( W \) — Test statistic (0 ≤ W ≤ 1)
\( a_i \) — Coefficients from Shapiro-Wilk tables
\( x_i \) — Ordered sample values (x₁ ≤ x₂ ≤ ... ≤ xₙ)
\( \bar{x} \) — 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 numeric values separated by commas. Minimum 3 data points required. For accurate results, use between 3 and 5000 data points.

5. Frequently Asked Questions (FAQ)

Q1: What's a good W value?
A: Typically W > 0.9 suggests normality, but critical values depend on sample size and significance level.

Q2: How does this compare to other normality tests?
A: Shapiro-Wilk is generally more powerful than Kolmogorov-Smirnov or Anderson-Darling for small samples.

Q3: What sample size works best?
A: Most effective for 3 ≤ n ≤ 50. For n > 5000, other tests may be more appropriate.

Q4: What if my data isn't normal?
A: Consider data transformation or non-parametric statistical tests.

Q5: Are there limitations to this test?
A: The test is sensitive to outliers and requires exact coefficients for precise results.