Shapiro Wilk Test For Normality Calculator

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 Test Statistic (W):

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

2. How Does the Test Work?

The test uses the following 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 calculated from the means, variances and covariances of the order statistics
\( x_i \) — Ordered sample values (x₁ ≤ x₂ ≤ ... ≤ xₙ)
\( \bar{x} \) — Sample mean

Explanation: The test compares the ordered sample values with what would be expected if the data were normally distributed.

3. Interpretation of Results

Details: The W statistic ranges from 0 to 1. Values close to 1 indicate normality. The null hypothesis (that the data is normally distributed) is rejected if W is below a critical value (which depends on sample size and significance level).

4. Using the Calculator

Tips: Enter your data as comma-separated values (e.g., 1.2, 3.4, 5.6). The calculator works best with sample sizes between 3 and 5000. For accurate results, use statistical software that implements the complete algorithm with proper coefficients.

5. Frequently Asked Questions (FAQ)

Q1: What sample size is appropriate for the Shapiro-Wilk test?
A: The test works best with sample sizes between 3 and 5000. It's particularly powerful for small samples (n < 50).

Q2: How is W interpreted?
A: W close to 1 suggests normality. Compare your result to critical values for your sample size at your chosen significance level (typically 0.05).

Q3: What are the limitations of this test?
A: The test is sensitive to sample size - with large samples, it may detect trivial departures from normality. It also requires specific coefficients for each sample size.

Q4: When should I use Shapiro-Wilk vs other normality tests?
A: Shapiro-Wilk is generally preferred for small to moderate samples (n < 2000). For larger samples, Kolmogorov-Smirnov or Anderson-Darling tests may be more appropriate.

Q5: Why does my result differ from statistical software?
A: This calculator provides an approximation. Full implementation requires extensive coefficient tables and more complex calculations.