Anderson-Darling A* Calculator
What Is the Anderson-Darling Test?
Theodore Anderson and Donald Darling introduced this test in 1952 as a way to measure how closely a sample's empirical distribution matches a theoretical one. The test statistic A² accumulates squared differences between the empirical cumulative distribution function (ECDF) and the hypothesized CDF, applying weights that increase near the distribution tails. That tail-weighting is the test's main practical advantage over the older Kolmogorov-Smirnov test.
Because the critical values depend on which distribution is being tested and whether parameters are known or estimated from the data, separate tables exist for each distributional family. This table covers the five distributions most commonly tested in practice, all in the "composite null" case where parameters are estimated from the same sample.
Key point: A² is what you calculate; A* is what you look up against the table. A* equals A² multiplied by a distribution-specific correction factor that adjusts for the fact that parameters were estimated rather than known. Without this correction, the test is anticonservative — it would reject H₀ too often.
Anderson-Darling Critical Value Table
Select the tab for your target distribution. Each table shows A* critical values at five significance levels. The modification formula for that distribution appears above the table. Click any cell to load it into the calculator.
Reject H₀ if A* > critical value. All values assume parameters are estimated from the sample (composite null). Sources: D'Agostino & Stephens (1986) for Normal; Stephens (1978) for Exponential; Stephens (1977) for Weibull and Gumbel; Stephens (1979) for Logistic. See NIST e-Handbook §1.3.5.15 for additional background.
How to Use the Anderson-Darling Table
The test follows five steps. Each step below corresponds to a specific calculation or decision in the procedure.
Step 1 — State the Null Hypothesis and Choose α
The null hypothesis H₀ states that the sample comes from a specific distribution (for example, H₀: the data are normally distributed). The alternative H₁ states they do not. Fix α before looking at the data. Most work uses α = 0.05, though reliability engineers often use α = 0.10 when an incorrectly assumed distribution has serious practical consequences.
Step 2 — Estimate Parameters and Sort the Data
Order the sample from smallest to largest: x₁ ≤ x₂ ≤ ... ≤ xₙ. Estimate the distribution parameters from this same sample (e.g., for the normal distribution, compute the sample mean x̄ and standard deviation s). Then evaluate the distribution CDF F at each ordered observation using these estimated parameters.
Step 3 — Compute A²
Apply the Anderson-Darling formula to the ordered CDF values:
A² is always positive. Larger values indicate greater departures from the hypothesized distribution. Statistical software (R, Python scipy, SAS, Minitab) computes A² directly — you only need the manual formula for understanding or hand-calculation practice.
Step 4 — Apply the Modification Factor to Get A*
Each distribution has its own correction formula. Multiply A² by the factor shown below for your distribution:
| Distribution | Modification Formula |
|---|---|
| Normal | A* = A²(1 + 0.75/n + 2.25/n²) |
| Exponential | A* = A²(1 + 0.6/n) |
| Weibull | A* = A²(1 + 0.2/√n) |
| Logistic | A* = A² (no modification needed) |
| Gumbel (EV1) | A* = A²(1 + 0.2/√n) |
Step 5 — Compare A* to the Critical Value and Decide
If A* ≤ A*crit → Fail to reject H₀ → No significant departure detected at α
Note the direction: for Anderson-Darling, rejection occurs when A* is larger than the critical value, not smaller. This contrasts with tests such as Pearson r where rejection occurs when the statistic exceeds a threshold in both directions.
Worked Example: Testing Normality with Anderson-Darling
Scenario: A quality engineer measures the tensile strength (MPa) of n = 40 steel rods and computes A² = 0.638 using statistical software. She wants to know whether the data depart significantly from normality at α = 0.05.
Solution — Step by Step
| Step | Action | Result |
|---|---|---|
| 1 | State hypotheses | H₀: data are Normal; H₁: data are not; α = 0.05 |
| 2 | Record computed A² | A² = 0.638 |
| 3 | Record sample size | n = 40 |
| 4 | Apply modification (Normal) | A* = 0.638 × (1 + 0.75/40 + 2.25/1600) = 0.638 × 1.0190 = 0.650 |
| 5 | Look up A*crit | Normal, α = 0.05 → A*crit = 0.752 |
| 6 | Compare A* vs A*crit | 0.650 < 0.752 → Fail to reject H₀ |
Interpretation
With A*(39) = 0.650 and A*crit(0.05) = 0.752 for the normal distribution, the test finds no statistically significant evidence that the tensile strength data depart from normality. The engineer can proceed with analyses that assume normal errors (t-tests, ANOVA, process capability indices) without significant concern about this assumption at α = 0.05. Had she used α = 0.10 (A*crit = 0.631), the result 0.650 > 0.631 would have triggered rejection — a reminder that the choice of α determines the conclusion in borderline cases.
How the Same A² Produces Different Decisions Across α Levels
Using the worked example above (A* = 0.650, Normal distribution), the table below shows how the conclusion changes depending on the chosen significance level.
| Significance Level α | Critical Value A*crit | A* = 0.650 vs A*crit | Decision |
|---|---|---|---|
| 0.15 | 0.561 | 0.650 > 0.561 | Reject H₀ |
| 0.10 | 0.631 | 0.650 > 0.631 | Reject H₀ |
| 0.05 | 0.752 | 0.650 < 0.752 | Fail to Reject |
| 0.025 | 0.873 | 0.650 < 0.873 | Fail to Reject |
| 0.01 | 1.035 | 0.650 < 1.035 | Fail to Reject |
Normal distribution, n = 40, A² = 0.638, A* = 0.650. The borderline nature of this result (failing at α = 0.10, passing at α = 0.05) is exactly why the significance level should be fixed before the test is run.
Anderson-Darling vs Other Normality Tests
Several tests target the same question of distributional fit. The choice between them depends on sample size, what kind of departure matters most, and whether the test applies to distributions other than the normal.
Anderson-Darling vs Shapiro-Wilk
The Shapiro-Wilk test generally has more power than Anderson-Darling for detecting non-normality when n < 50, making it the first choice for small samples. Anderson-Darling gains an advantage at larger n and — crucially — extends to distributions other than the normal. If your goal is normality testing alone and n ≤ 50, Shapiro-Wilk is typically the better choice. For n > 50 or for testing Weibull, exponential, or logistic fits, Anderson-Darling is standard.
Anderson-Darling vs Kolmogorov-Smirnov
The Kolmogorov-Smirnov (KS) test uses the maximum vertical distance between the ECDF and the theoretical CDF. Anderson-Darling weights the squared distance across the whole distribution, giving additional emphasis to the tails. In practice, Anderson-Darling detects tail departures more reliably than KS, which is why it has largely replaced the KS test for composite null problems in applied work.
Anderson-Darling vs Chi-Square Goodness-of-Fit
The chi-square goodness-of-fit test groups data into bins and compares observed versus expected counts. It applies to discrete and continuous data alike but loses information through binning and requires n > 5 per cell. Anderson-Darling uses the full ordered data without binning, making it more efficient for continuous data when sample sizes are moderate (n ≥ 8).
| Test | Best for Small n | Tail Sensitivity | Non-Normal Distributions | Software Support |
|---|---|---|---|---|
| Anderson-Darling | Moderate | High | Yes (5 families) | R, Python, Minitab, SAS |
| Shapiro-Wilk | High | Moderate | Normal only | R, SPSS, SAS |
| Kolmogorov-Smirnov | Low | Low | Yes (many) | R, Python, SPSS |
| Chi-Square | Requires large n | Low | Yes (any) | All packages |
Applied Examples Across Research Fields
The Anderson-Darling test appears across manufacturing, health research, finance, and reliability engineering. Each application below shows a different distribution family.
Manufacturing: Checking Dimensional Measurements (Normal)
A machinist measuring bore diameters across 50 parts computes A² = 0.421. With n = 50, the modification gives A* = 0.421(1 + 0.75/50 + 2.25/2500) = 0.421 × 1.0152 = 0.428. Since 0.428 < 0.752 (critical at α = 0.05), the measurements are consistent with normality. Process capability indices such as Cpk can be reported without concern about the normality assumption at this α level.
Reliability Engineering: Component Lifetime Modeling (Weibull)
A reliability engineer fits a Weibull distribution to n = 30 component failure times and obtains A² = 0.743. Applying the Weibull correction: A* = 0.743(1 + 0.2/√30) = 0.743 × 1.0365 = 0.770. Comparing to A*crit = 0.877 (α = 0.05), the engineer fails to reject the Weibull assumption and proceeds with Weibull-based reliability predictions.
Environmental Science: Rainfall Extremes (Gumbel)
Annual maximum rainfall data at a weather station (n = 25 years) are tested against a Gumbel extreme value distribution. The computed A² = 0.496. With the Gumbel modification: A* = 0.496(1 + 0.2/√25) = 0.496 × 1.040 = 0.516. Since 0.516 < 0.757 (critical at α = 0.05), the Gumbel fit is not rejected, and the distribution can be used for flood frequency estimation.
Healthcare: Time Between Patient Arrivals (Exponential)
An operations researcher at a hospital emergency department records n = 60 inter-arrival times and computes A² = 1.024. With the exponential correction: A* = 1.024(1 + 0.6/60) = 1.024 × 1.010 = 1.034. Comparing to A*crit = 1.341 (α = 0.05), the exponential fit is retained. The Poisson arrival assumption underlying queueing models for this ED is supported at α = 0.05.
Anderson-Darling Decision Rule Summary
The decision rule is the same regardless of distribution or α level. Only the critical value changes.
If A* > A*crit → Reject H₀ → Data depart significantly from the distribution at α
If A* ≤ A*crit → Fail to reject H₀ → No significant departure detected at α
Software (R, SciPy, Minitab, SAS) typically reports a p-value directly, so you can skip the table lookup. The p-value and the table approach give the same decision: reject H₀ when p < α. The table is still useful for manual calculation, exam work, and for checking software output against a published reference. For R users: the ad.test() function in the nortest package and scipy.stats.anderson() in Python both implement this test with p-value output.
Symbol and Concept Glossary
Every symbol appearing in the Anderson-Darling procedure is defined below, along with the most common source of confusion for each.
| Symbol | Name | Definition | Common Error |
|---|---|---|---|
| A² | Anderson-Darling statistic | Computed from data; always positive; larger = more departure | Comparing A² directly to critical values without modification |
| A* | Modified A² statistic | A² after applying the distribution-specific correction for estimated parameters | Using the wrong modification formula for the distribution |
| n | Sample size | Number of observations; appears in the modification factor | Omitting the modification entirely for small n |
| α | Significance level | Probability of rejecting H₀ when it is true; selects table column | Setting α after seeing the test result |
| F(x) | Hypothesized CDF | CDF of the distribution being tested, evaluated with estimated parameters | Using population parameters instead of sample estimates |
| H₀ | Null hypothesis | Sample comes from the specified distribution | Treating failure to reject as proof of H₀ |
| p-value | Observed significance | Probability of A* ≥ observed value if H₀ is true; reject when p < α | Confusing with the probability H₀ is true |
Conditions for Valid Anderson-Darling Results
The test and its critical values rest on a few requirements. Meeting them does not guarantee a correct conclusion, but violating them can make the test misleading.
Independence
Observations must be independent of each other. Time-series data with autocorrelation, or clustered observations from the same group, violate this. The test was derived under the assumption of iid data. See study design for how independence is achieved through random sampling.
Continuous Data
The Anderson-Darling test assumes a continuous distribution. It can be applied to discrete data as an approximation, but the critical values are derived for continuous distributions and tie-heavy discrete data will produce unreliable results.
Sample Size (n ≥ 8)
The modification factors produce reliable results for n ≥ 8. Below that, the asymptotic critical values may not be accurate. For very small samples (n < 8), exact critical values or permutation-based methods should be consulted.
Correct Distribution Family
Use the critical values and modification factor that correspond to the distribution being tested. Applying normal critical values when testing an exponential fit will produce wrong decisions. Check the assumptions guide for distribution selection guidance.
nortest package with the ad.test() function. The output includes the test statistic and p-value, allowing you to determine whether the sample significantly departs from the specified distribution.
Sources & Further Reading
The critical values in this table come from the following primary and secondary sources. Each citation includes the specific table or section most relevant to the Anderson-Darling procedure.
D'Agostino, R. B., & Stephens, M. A. (Eds.). (1986). Goodness-of-Fit Techniques. Marcel Dekker. — The canonical textbook reference for Anderson-Darling critical values and modification factors across distributional families. Table 4.2 contains the normal distribution critical values used here.
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730–737. — Original derivation of modification factors and critical values for the normal distribution. doi:10.2307/2286009
Stephens, M. A. (1977). Goodness of Fit for the Extreme Value Distribution. Biometrika, 64(3), 583–588. — Source for Weibull and Gumbel (extreme value) critical values and modification factors used in this table. doi:10.2307/2345336
NIST/SEMATECH e-Handbook of Statistical Methods (2023). Section 1.3.5.15: Anderson-Darling Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference with formulas, critical values, and worked examples for the Anderson-Darling test.
Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Annals of Mathematical Statistics, 23(2), 193–212. — Original paper introducing the Anderson-Darling test statistic and its asymptotic distribution under a fully-specified (known-parameter) null hypothesis. doi:10.1214/aoms/1177729437
Related Statistical Tables & Resources
Understanding What the Anderson-Darling Test Measures
Why Tail Weighting Matters
The Kolmogorov-Smirnov test gives equal weight to every point of the distribution. Anderson-Darling weights observations near the tails more heavily. This matters in practice because many important assumptions — such as the normality required for t-tests and ANOVA — are most sensitive to departures in the tails, where extreme observations live. A symmetric distribution may pass a center-sensitive test while failing the tail-sensitive Anderson-Darling.
The Role of the Modification Factor
Each distribution's modification factor corrects the raw A² for the fact that estimated parameters pull the fitted distribution toward the data, reducing the apparent distance. Without correction, A² would be biased downward, causing the test to reject H₀ less often than the nominal α. The larger the sample, the smaller the correction (factors approach 1 as n → ∞), which is why the adjustment mostly matters for n < 50. Check the sampling distributions section for related ideas.
Practical vs Statistical Significance
Whether a departure from a distribution is statistically significant depends partly on sample size. With n = 1,000, a tiny tail irregularity that has no practical consequence for any subsequent analysis will produce a highly significant A*. With n = 15, a fairly substantial departure may go undetected. This means you should combine the Anderson-Darling result with a graphical check — a Q-Q plot or probability plot — before deciding whether a distribution assumption is acceptable for the intended application.