Anderson–Darling test

Test a sample’s agreement with a specified continuous distribution by integrating squared empirical-CDF deviations with extra weight in the tails.

Version

v2 · 2026-09-19 · History

Domain-specific #

3290

Domain group

Formal Sciences

Origin domain

Experimental Design & Statistics

Subdomain

Goodness of Fit Testing → Experimental Design & Statistics

Core Idea

The Anderson–Darling statistic is a Cramér–von Mises-type goodness-of-fit measure weighted by [F(x)(1−F(x))]^{-1}, emphasizing tail departures. Data are transformed through the null CDF, ordered, and compared with uniform order-statistic expectations; logarithmic tail terms accumulate discrepancies into A². The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of statistics. It is the Anderson–Darling tail-weighted statistic and distribution-specific calibration. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if Kolmogorov–Smirnov critical values are reused, fitted parameters are ignored, discrete data use continuous tables without adjustment, or a p-value is treated as fit magnitude.

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The Edge-Watching Fit Check

Say you guess what shape a pile of numbers should make, like most in the middle and a few at the ends. The Anderson–Darling test is a way to check how badly the numbers disagree with your guess. It looks extra hard at the far ends, where the rare numbers are. If it finds little disagreement, that does not prove your guess was right, only that it did not catch a problem.

Does-It-Fit Tail Test

Scientists often assume their data follow a certain pattern, like the bell-shaped curve. The Anderson–Darling test checks that assumption by comparing the real data to the pattern and adding up how much they disagree. Unlike some similar checks, it pays extra attention to the extremes, the very small and very large values, where mismatches are easy to miss. To decide whether the disagreement is big, it uses its own special tables made for the pattern being tested. If the test finds no clear problem, that does not prove the data follow the pattern.

Tail-Weighted Goodness-of-Fit Test

The Anderson–Darling test is a goodness-of-fit test: it measures how far a sample's distribution is from a proposed (null) distribution. It first converts each data point using the null distribution's cumulative distribution function (CDF), sorts them, and compares them with what sorted values would look like if the null were true. Its statistic, A^2, gives extra weight to the tails, so disagreement at the extremes counts more than in similar tests like Kolmogorov–Smirnov. Its critical values depend on which distribution is tested and on whether its parameters were estimated from the data, so borrowing another test's tables gives wrong answers. A small p-value signals poor fit, but a p-value is not a measure of how big the misfit is, and a non-significant result does not prove the fit is good.

 

The Anderson–Darling statistic is a Cramér–von Mises-type measure of the distance between the empirical CDF and a hypothesized CDF F, weighted by [F(x)(1−F(x))]^{-1} to emphasize the tails. Computationally, the data are transformed through the null CDF (yielding values that should be uniform if the null is correct), sorted, and compared with the expected behavior of uniform order statistics; logarithmic terms that are large near 0 and 1 accumulate the discrepancies into A². Its null distribution is specific to the hypothesized family and to whether parameters are known or estimated, so Kolmogorov–Smirnov critical values cannot be substituted, and estimated parameters require adjusted calibration. Discrete data, ties, and censoring need their own adjustments. Sound use also means inspecting effect size and plots, not treating the p-value as a magnitude of misfit, and not concluding the null is true from nonsignificance. Applications include testing distributional assumptions, detecting tail deviations, and k-sample comparisons.

Scope of Application

Anderson–Darling test belongs to statistics and is useful where the analyst can specify an ordered sample, a fully specified or fitted continuous null cumulative distribution, and the transformed empirical distribution, then evaluate the declared tail-weighted empirical-versus-null CDF statistic and matching null calibration determine the test. The scope is broad within that domain but bounded by the need for the declared tail-weighted empirical-versus-null CDF statistic and matching null calibration determine the test. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the declared tail-weighted empirical-versus-null CDF statistic and matching null calibration determine the test the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Anderson–Darling test can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Anderson–Darling test. Anderson–Darling test compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: an ordered sample, a fully specified or fitted continuous null cumulative distribution, and the transformed empirical distribution. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express the declared tail-weighted empirical-versus-null CDF statistic and matching null calibration determine the test independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of statistics because they reuse an ordered sample, a fully specified or fitted continuous null cumulative distribution, and the transformed empirical distribution, Data are transformed through the null CDF, ordered, and compared with uniform order-statistic expectations; logarithmic tail terms accumulate discrepancies into A²., and state whether parameters are known or estimated, use the correct finite-sample calibration, handle ties and censoring, inspect effect size and plots, and avoid accepting the null from nonsignificance. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.

Relationships to Other Abstractions

Current abstraction Anderson–Darling test Domain-specific

Parents (1) — outgoing typed relations

• Anderson–Darling test is a kind of Hypothesis Testing (Null vs. Alternative) Prime

  The procedure literally compares sample evidence with a null distribution through a rejection rule.

Hierarchy paths (4) — routes to 4 parentless roots

• Anderson–Darling test → is a kind of Hypothesis Testing (Null vs. Alternative) → is a kind of Statistical Inference → is a kind of Inductive Reasoning

Show alternative paths (3)

Neighborhood in Abstraction Space

Anderson–Darling test sits in a crowded region of the domain-specific corpus (36^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Descriptive Statistics & Correlation Measures (40 abstractions)

Nearest neighbors

• Z-test — 0.89
• Portmanteau test — 0.88
• Logrank test — 0.88
• Exact test — 0.87
• Normality test — 0.87

Computed from structural-signature embeddings · 2026-10-25