D'Agostino's K-squared test¶

An omnibus sample-normality test that transforms sample skewness and kurtosis into approximately standard-normal components and sums their squares, testing an i.i.d. Gaussian null specifically against skewness and tail/peakedness departures.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8837
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomains: Normality Testing, Hypothesis Testing → Experimental Design & Statistics

Core Idea¶

D'Agostino's K-squared test is an omnibus normality test that transforms sample skewness and kurtosis into approximately standard-normal components and sums their squares for comparison with a reference distribution under an i.i.d. Gaussian null. Under the declared i.i.d. Under the declared i.i.d.

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Is It a Bell Hill?

Imagine measuring lots of kids' heights and stacking a block for each one to make a hill. A usual "bell" hill is the same on both sides and has edges that are not too thick or too thin. D'Agostino's K-squared test checks just those two things: is the hill lopsided, and are its edges wrong? If either is off, it says the hill isn't a bell, but a hill can pass these two checks and still be a different shape.

Lopsided-or-Tails Check

Many things in statistics assume numbers follow a normal distribution, the bell curve. D'Agostino's K-squared test checks whether data look like they came from one. It measures two features: skewness, which is how lopsided the data are, and kurtosis, which is about how heavy the tails and how sharp the peak are compared with a bell curve. It turns each into a score and combines them into one number, K-squared. If K-squared is too big, the data probably aren't normal, and you should look at both scores separately to see which feature caused it. Passing the test doesn't prove data are normal, because other shapes can have the same lopsidedness and tails as a bell curve.

Skewness–Kurtosis Normality Test

D'Agostino's K-squared test is a test of whether data come from a normal distribution. It computes two summaries from the sample: skewness (the third standardized moment, measuring asymmetry) and kurtosis (the fourth standardized moment, related to tail heaviness and peakedness). Each is transformed so that, if the data really are normal, it behaves approximately like a standard normal variable, and the statistic K² is the sum of their squares. Under the null hypothesis of independent, identically distributed normal data, with a large enough sample, K² is compared with a chi-squared distribution with 2 degrees of freedom. Because the sum hides which part caused a rejection, you should also check the two separate values. The test only looks at skewness and kurtosis, so a non-normal distribution that happens to match normal values of both won't be detected; very large samples can flag tiny, unimportant departures, and small samples or outliers can make it unstable.

D'Agostino's K-squared test is an omnibus test of normality built on the third and fourth standardized moments. It computes sample skewness √b₁ and sample kurtosis b₂, applies transformations to each so that under normality they approximate standard normal variables Z₁ and Z₂, and forms K² = Z₁² + Z₂². Under the null hypothesis of i.i.d. Gaussian data and adequate sample size, K² is referred to a chi-squared distribution with two degrees of freedom. Because the sum hides which component drove a rejection, the individual Z-values should be inspected to tell whether the departure is asymmetry, tail weight or peakedness, or both. Its power is targeted rather than universal: distributions with Gaussian-like skewness and kurtosis but different shape can go undetected. Large samples may reject for negligible departures, and small samples or outliers can make the moment estimates unstable. For regression residuals or time series, the fitted model and serial dependence violate the i.i.d. assumption, so suitable diagnostics or simulation-based calibration are needed rather than naive application.

Scope of Application¶

The test is used in exploratory statistics, model diagnostics, quality control, scientific measurement, simulation validation, teaching, and software normality-testing libraries. Use it with sample and n, missing/outlier policy, independence/identical-distribution assessment, transformations and fitted parameters, skewness and kurtosis definitions, exact software/formula, both component statistics, K-squared/reference/p-value, alpha and multiplicity, Q–Q/effect diagnostics, sensitivity and downstream consequence. Its power targets skewness and kurtosis departures; non-rejection is not proof of normality, and dependent/discrete/censored data need different calibration.

Exploration. Flags moment departures.
Regression. Diagnoses residuals cautiously.
Quality control. Checks distribution assumptions.
Simulation. Audits generated samples.
Teaching. Separates skewness and kurtosis contributions.

Clarity¶

Report sample definition and n, missing/outlier handling, independence and identical-distribution evidence, transformations and parameter estimation, skewness/kurtosis definitions and software/version, component statistics, K-squared and reference calibration, exact/asymptotic/simulated p-value, alpha and multiplicity, effect plots/quantiles, sensitivity analyses, and the downstream decision for which normality matters. The closest near miss sets the boundary: Jarque–Bera is nearest because it also combines skewness and kurtosis, but it uses a different statistic and calibration. A positive case must satisfy this test: D'Agostino's K-squared test is the declared skewness/kurtosis transformation and squared-component omnibus test applied to an appropriate sample under an i.i.d. normal null.

Manages Complexity¶

The test compresses two interpretable moment departures into one omnibus number while leaving dependence, practical size, and non-moment departures outside the statistic. The central omnibus compression–diagnostic direction tradeoff is this: One p-value is convenient while it hides which component drives departure. A second statistical sensitivity–practical relevance tension matters because Large n detects tiny deviations while the downstream method may be robust. The simple null–data dependence tension adds that The reference law is tractable while real residuals can be correlated.

Abstract Reasoning¶

Use three linked moves: verify sampling assumptions and intended downstream use; compute skewness and kurtosis under declared conventions; apply the specified finite-sample transforms and combine squares. As a collapse test, the inference exits when dependence, discrete/censored data, strong parameter selection, or insufficient sample conditions make the stated null reference inappropriate. A fourth check is to inspect component direction and graphical/effect diagnostics. A final check is to report non-rejection or rejection without overclaiming distribution identity.

Knowledge Transfer¶

The moment-combination idea transfers to other reference families only with newly derived transforms and calibration; the Gaussian chi-squared approximation cannot be copied. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Broader method class requiring exact signature review. General purpose rather than exact statistic.

Relationships to Other Abstractions¶

Current abstraction D'Agostino's K-squared test Domain-specific

Parents (1) — more general patterns this builds on

D'Agostino's K-squared test is a kind of Statistical Test Domain-specific

D'Agostino's K-squared test satisfies the defining boundary of Statistical Test: A statistical test is a formally specified procedure that compares observed data or a derived statistic with a sampling distribution, randomization distribution, or model under a null hypothesis to quantify incompatibility and apply a declared decision rule or evidential interpretation.

Hierarchy path (1) — routes to 1 parentless root

D'Agostino's K-squared test → Statistical Test

Neighborhood in Abstraction Space¶

D'Agostino's K-squared test sits in a crowded region of the domain-specific corpus (38^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Empirical Measurement & Statistical Inference Methods (50 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08