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 based on the third and fourth standardized moments. It computes sample skewness and kurtosis, transforms each to improve its approximation to a standard normal variable, and adds the squared transformed components.

Under the declared i.i.d. Gaussian null and suitable sample-size conditions, the combined statistic is compared with a chi-squared reference with two components. The separate transformed values should also be inspected because the sum obscures whether rejection is driven by asymmetry, tails/peakedness, or both.

The test has targeted rather than universal power: distributions can be non-normal while matching Gaussian skewness and kurtosis. Large samples may flag negligible departures; small samples and outliers can be unstable. For regression residuals or time series, model fitting and dependence require suitable diagnostics or simulation rather than naïve application.

How would you explain it like I'm…

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.

Structural Signature¶

Sig role-phrases:

numeric sample. Provides observations assumed independent, identically distributed, and sufficiently large for approximation. Constitutive input. If altered: Dependence or mixtures invalidate the nominal null.
Gaussian null. Specifies the distributional family whose skewness and kurtosis are expected values. Constitutive hypothesis. If altered: Failing to reject does not prove normality.
sample skewness transform. Converts estimated asymmetry into an approximately normal statistic. Constitutive component. If altered: Formula and finite-sample correction matter.
sample kurtosis transform. Converts tail/peakedness departure into a second approximately normal statistic. Constitutive component. If altered: Kurtosis convention must be stated.
sum-of-squares decision. Combines components, compares with reference distribution, and reports p-value/effect diagnostics. Identity-bearing verdict. If altered: Sign/direction is lost in the sum and must be recovered from components.

What It Is Not¶

Not proof of normality. Non-rejection is limited evidence.
Not universal non-normality detection. Moment-matched alternatives may escape.
Not Shapiro–Wilk. The statistic and power profile differ.
Not valid for arbitrary dependence. The null calibration assumes sampling conditions.

Scope of Application¶

The test is used in exploratory statistics, model diagnostics, quality control, scientific measurement, simulation validation, teaching, and software normality-testing libraries.

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.

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.

Abstract Reasoning¶

Verify sampling assumptions and intended downstream use.
Compute skewness and kurtosis under declared conventions.
Apply the specified finite-sample transforms and combine squares.
Inspect component direction and graphical/effect diagnostics.
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.

Examples¶

Canonical¶

An i.i.d. continuous sample is analyzed with a documented implementation; transformed skewness and kurtosis, their squared sum, p-value, Q–Q plot, and practical departure are reported together.

Mapped back: numeric sample → documented i.i.d. observations; Gaussian null → specified normal family; sample skewness transform → reported z-skewness; sample kurtosis transform → reported z-kurtosis; sum-of-squares decision → K2, calibration, and interpretation.

Applied / In Practice¶

A simulation study tests generated innovations before using Gaussian-based intervals, repeats the analysis across seeds and sample sizes, and avoids applying the nominal p-value to autocorrelated output.

Mapped back: numeric sample → independent innovation draws; Gaussian null → generator target; sample skewness transform → replicate asymmetry diagnostics; sample kurtosis transform → replicate tail diagnostics; sum-of-squares decision → power/calibration across n.

Structural Tensions¶

T1: omnibus compression vs. diagnostic direction. One p-value is convenient while it hides which component drives departure. Diagnostic: Were component statistics retained?

T2: statistical sensitivity vs. practical relevance. Large n detects tiny deviations while the downstream method may be robust. Diagnostic: What consequence follows from the effect size?

T3: simple null vs. data dependence. The reference law is tractable while real residuals can be correlated. Diagnostic: How was dependence handled?

Structural–Framed Character¶

The test is structural. Sample moments, transforms, squared combination, and null calibration determine the statistic; analyst choices frame use but not identity. Evaluative weight is low; statistical practice matters; origin is statistics; vocabulary travels with formulas; transfer recognizes the same calculation. Its portable skeleton is Standardized-Component Omnibus Test, a prospective future-prime candidate. Its character: two moment diagnostics normalized and combined into one null-calibrated departure measure.

Structural Core vs. Domain Accent¶

Skeletal core. Transform distinct deviations to comparable null scales and aggregate their squared magnitudes.

Domain-bound accent. Gaussian skewness, kurtosis, i.i.d. sampling, K-squared, and chi-squared calibration define the test.

Why not prime. Component aggregation travels; this is one named normality test.

This entry is a kind of Statistical Test.

Normality test. Broader method class requiring exact signature review.
Goodness of fit. 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

Not to Be Confused With¶

Jarque–Bera. Tell: Same moments but different statistic/calibration?
Shapiro–Wilk. Tell: Order-statistic or moment omnibus?
Raw skewness/kurtosis. Tell: Descriptive components or calibrated test?
Normality proof. Tell: Decision under a null or distribution identity?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/D%27Agostino%27s_K-squared_test (revision 1309757832).
Preserved source candidate: http://www.cee.mtu.edu/~vgriffis/CE%205620%20materials/CE5620%20Reading/DAgostino%20et%20al%20-%20normaility%20tests.pdf
Preserved source candidate: https://web.archive.org/web/20120325140006/http://www.cee.mtu.edu/~vgriffis/CE%205620%20materials/CE5620%20Reading/DAgostino%20et%20al%20-%20normaility%20tests.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.