This calculator helps you assess whether your data follows a normal distribution using three powerful statistical tests: Shapiro-Wilk, Kolmogorov-Smirnov, and Anderson-Darling. It also generates Q-Q plots and histograms to visualize your data distribution. Normality is a crucial assumption for many parametric statistical procedures, including t-tests, ANOVA, and linear regression. Simply input your data, select the column to analyze, choose which tests to run, and get comprehensive results with visual plots to help you make informed decisions about your data.
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Normality Tests
Normality tests help determine whether your sample data comes from a normally distributed population. This is a critical assumption for many statistical procedures, including t-tests, ANOVA, and linear regression. Different normality tests have varying sensitivities and are better suited for different scenarios and sample sizes.
Shapiro-Wilk Test
Best for:
- Small to moderate sample sizes (n between 3 and 5000)
- Considered the most powerful normality test for small samples
How it works:
The test statistic W compares the ordered sample values with the corresponding normal order statistics. Values close to 1 indicate normality.
Key strengths:
- High power for detecting non-normality
- Works well for symmetric distributions
- Reliable for small samples
Kolmogorov-Smirnov Test
Best for:
- Moderate to large sample sizes
- Comparing sample with any theoretical distribution
How it works:
The K-S test compares your empirical distribution function with the cumulative distribution function of the reference distribution (normal). The test statistic D is the maximum vertical distance between these functions.
Key strengths:
- Distribution-free test
- Applicable to continuous distributions
- Good for detecting shifts in distribution
Anderson-Darling Test
Best for:
- All sample sizes
- Detecting deviations in distribution tails
How it works:
A modification of the Kolmogorov-Smirnov test that gives more weight to the tails of the distribution. The test statistic A² measures the integrated squared difference between the empirical and theoretical distribution functions.
Key strengths:
- More sensitive to deviations in tails
- Often more powerful than K-S test
- Good for detecting outliers
Interpreting Normality Test Results
All normality tests use the following hypothesis structure:
- Null Hypothesis (H₀): The data follows a normal distribution.
- Alternative Hypothesis (H₁): The data does not follow a normal distribution.
The significance level (typically 0.05) is your threshold for rejecting the null hypothesis of normality.
- p-value ≥ α: Fail to reject normality
- p-value < α: Reject normality
What to Do If Your Data Isn't Normal
Data Transformations
Apply mathematical transformations to normalize your data:
- Right-skewed data: Log, square root, or reciprocal transformations
- Left-skewed data: Square or cube transformations
- Remember: Transformations affect interpretation
Non-parametric Methods
Use tests that don't assume normality:
- Mann-Whitney U Test (instead of t-test)
- Wilcoxon Signed-Rank Test (paired data)
- Kruskal-Wallis Test (instead of ANOVA)
- Spearman's Correlation (instead of Pearson's)
Robust Methods
Use procedures less sensitive to non-normality:
- Bootstrapping (resampling techniques)
- Permutation tests
- Trimmed means and robust variance estimators
- Generalized linear models
Sample Size Considerations
The performance of normality tests varies significantly with sample size:
- Very small samples (n < 10): Low power to detect non-normality. Shapiro-Wilk test is preferred, but even it struggles with very small samples.
- Small samples (10 ≤ n < 30): Shapiro-Wilk test offers the best power.
- Medium samples (30 ≤ n < 300): Any of the tests work well. Anderson-Darling is particularly good at detecting tail deviations.
- Large samples (n ≥ 300): Normality tests become overly sensitive. Minor, practically insignificant deviations can lead to rejection of normality.
Recommended Approach by Sample Size
| Sample Size | Approach |
|---|---|
| n < 30 | Use Shapiro-Wilk test + Q-Q plots |
| 30 ≤ n < 100 | Use any test, with visual confirmation |
| 100 ≤ n < 300 | Prioritize visual methods over test p-values |
| n ≥ 300 | Rely on Central Limit Theorem or use visual methods |