Calculator
Example Data Table
| Observation | Value | Observation | Value |
|---|---|---|---|
| 1 | 12.1 | 6 | 12.3 |
| 2 | 11.8 | 7 | 12.4 |
| 3 | 12.5 | 8 | 11.7 |
| 4 | 12.0 | 9 | 12.2 |
| 5 | 11.9 | 10 | 12.1 |
Paste the same values into the calculator to see a working example of the test output.
Formula Used
The Anderson Darling statistic for ordered observations uses cumulative normal probabilities and applies stronger weight to the distribution tails.
A² = -n - (1/n) Σ(2i - 1)[ln(F(xᵢ)) + ln(1 - F(xₙ₊₁₋ᵢ))]
A²* = A² × (1 + 0.75/n + 2.25/n²)
Where:
- n is the sample size.
- xᵢ are the sorted observations.
- F(x) is the normal cumulative distribution function based on the sample mean and sample standard deviation.
- A²* is the small-sample adjusted statistic compared against tabulated critical values.
How to Use This Calculator
- Enter your sample values in the dataset box using commas, spaces, or line breaks.
- Select the significance level that matches your decision threshold.
- Press Submit Test to calculate the Anderson Darling statistic and interpretation.
- Review the result panel above the form for A², adjusted A²*, p-value, and the final decision.
- Use the CSV and PDF buttons to export the test summary for reporting or audit documentation.
Article
Interpreting Normality Output
The Anderson Darling test helps determine whether a sample is consistent with a normal distribution. This calculator reports the adjusted statistic, approximate p-value, sample size, mean, and standard deviation in one decision panel. That combined view supports audit-ready interpretation because users can connect the formal hypothesis result with the underlying descriptive profile of the dataset instead of relying on a single number alone.
Why Analysts Use This Test
This method is valued because it gives greater emphasis to the tails of the distribution. In practical settings, extreme values often influence warranty claims, turnaround delays, laboratory failures, pricing errors, and financial risk more than central observations do. A tail-sensitive procedure can therefore reveal distribution issues that materially affect downstream inference, capability analysis, or model assumptions.
Understanding the Main Decision
The adjusted A²* statistic increases as the sample deviates more strongly from normality. The calculator compares A²* with the selected critical value. When the statistic is larger than that threshold, the null hypothesis of normality is rejected. When it is smaller, the evidence is insufficient to reject normality at the chosen significance level.
Value of the Supporting Table and Graph
The sorted probability table shows each ordered observation, its z-score, and the cumulative probability used in the calculation. The Plotly graph extends that evidence visually by comparing sample positions with expected normal probabilities. Close alignment supports normality, while visible curvature, separation, or tail distortion may indicate skewness, heavy tails, or mixtures.
Effect of Sample Size on Findings
Sample size changes how sensitive the test becomes. Small datasets may miss meaningful departures, while large datasets can flag minor imperfections with limited practical importance. Strong practice is to pair the test with process knowledge, basic descriptive summaries, and visual inspection. Exporting CSV and PDF outputs also improves documentation quality for regulated, operational, or client-facing work. It also supports faster peer review and approval.
Recommended Use in Reporting Workflows
Begin by entering clean numeric observations and selecting the significance level. Review the result panel above the form, then inspect the probability table and graph for visual confirmation. If the data appear non-normal, consider transformation, robust modeling, or nonparametric methods. This structured sequence improves statistical discipline and makes reporting decisions easier to justify.
FAQs
1. What does the Anderson Darling test measure?
It measures how well a sample matches a target distribution, here the normal distribution, with extra sensitivity to differences in the tails.
2. Why is the adjusted A²* statistic used?
The adjustment improves performance for smaller samples, making the reported statistic more appropriate for comparison with published critical values.
3. Can I use this tool with very small datasets?
You can, but the calculator requires at least five observations. Small samples reduce test power and can limit the reliability of any normality conclusion.
4. Does failing to reject normality prove the data are normal?
No. It only means the sample does not provide enough evidence to reject normality at the selected significance level.
5. When should I review the graph as well as the statistic?
Always. The graph can reveal skewness, tail problems, or clusters that help explain why the numerical result looks strong or weak.
6. What should I do if the data are not normal?
Consider transforming the data, using robust estimators, or choosing nonparametric methods that do not rely on normality assumptions.