Shapiro-Wilk Test Calculator
Perform the Shapiro-Wilk test , a widely used statistical test for assessing whether a sample comes from a normally distributed population. This calculator computes the W statistic, p-value, and provides a detailed step-by-step breakdown of the calculation, including sorted data, coefficients, and the Q-Q plot.
What Is the Shapiro-Wilk Test?
The Shapiro-Wilk test is a statistical test for the null hypothesis that a sample comes from a normally distributed population. It was developed by Samuel Shapiro and Martin Wilk in 1965 and is one of the most powerful tests for normality.
Null Hypothesis (H₀): The data come from a normal distribution.
Alternative Hypothesis (H₁): The data do not come from a normal distribution.
How the Test Works
- Sort the Data: Arrange all observations in ascending order: x₁ ≤ x₂ ≤ … ≤ xₙ.
- Calculate SS: SS = Σ(xᵢ − x̄)² (the sum of squared deviations from the mean).
- Find Coefficients aᵢ: Use specialized tables (or approximation) to get the coefficients a₁, …, aₘ, where m = n/2 (if n is even) or m = (n−1)/2 (if n is odd).
- Compute b: b = Σ aᵢ · (xₙ−ᵢ₊₁ − xᵢ) for i = 1 to m.
- Calculate W: W = b² / SS. The W statistic ranges from 0 to 1, with values closer to 1 indicating greater normality.
- Determine Significance: Compare W to the critical value Wα. If W ≤ Wα, reject H₀. Alternatively, use the p-value , if p < α, reject H₀.
Key Properties
- Sample Size: The test can be used for sample sizes from 3 to 50 (original test). The expanded version handles up to 5,000 samples.
- Interpretation: W is essentially the square of the correlation coefficient between the ordered data and the expected normal order statistics.
- Sensitivity: The test is very sensitive to ties in the data.
- Significance: If the p-value is less than the chosen significance level α, we reject the null hypothesis of normality.
Why Use This Shapiro-Wilk Test Calculator?
- Complete Analysis: Get W statistic, p-value, critical value, and conclusion.
- Step-by-Step: See the full calculation process with sorted data and coefficients.
- Q-Q Plot: Visualize how well your data aligns with normal quantiles.
- Critical Values: Reference table for key significance levels.
- Free & Private: No registration, no data storage.
❓ Shapiro-Wilk Test Calculator FAQ
What is the Shapiro-Wilk test?
The Shapiro-Wilk test is a statistical test used to assess whether a sample comes from a normally distributed population. It is one of the most powerful tests for normality.
What is the null hypothesis of the Shapiro-Wilk test?
The null hypothesis (H₀) states that the data come from a normally distributed population. The alternative hypothesis (H₁) states that the data do not come from a normal distribution.
What is the W statistic?
The W statistic is the test statistic for the Shapiro-Wilk test. It ranges from 0 to 1, with values closer to 1 indicating stronger evidence of normality. W is the square of the correlation between the ordered data and expected normal order statistics.
How do I interpret the p-value?
If the p-value is less than your significance level (α), you reject the null hypothesis and conclude that the data are not normally distributed. If the p-value is greater than α, you fail to reject the null hypothesis.
What sample size is required for the Shapiro-Wilk test?
The original Shapiro-Wilk test can be used for sample sizes from 3 to 50. An expanded version handles samples up to 5,000.
What are the critical values for the Shapiro-Wilk test?
Critical values (Wα) are used to determine whether to reject the null hypothesis. If W ≤ Wα, reject H₀. Critical values depend on the sample size (n) and the chosen significance level (α).
What does a W value close to 1 mean?
A W value close to 1 indicates that the data closely follow a normal distribution. A low W value (below the critical value) suggests departure from normality.
Is the Shapiro-Wilk test affected by ties?
Yes, the test is very sensitive to ties. If the data has been rounded, ties can affect the test results.
What is a Q-Q plot?
A Q-Q (quantile-quantile) plot compares the quantiles of your data against the quantiles of a normal distribution. If the points fall roughly along a straight line, the data are approximately normal.
What is the difference between the Shapiro-Wilk test and the Kolmogorov-Smirnov test?
Both tests assess normality, but the Shapiro-Wilk test is generally more powerful for detecting departures from normality, especially with smaller sample sizes. The Kolmogorov-Smirnov test is more general and can be used for any distribution.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.