Correlation ratio¶

An effect-size measure equal to the square root of between-category variance divided by total variance, detecting nonlinear mean association.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3924
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics

Core Idea¶

Given a categorical predictor and numerical response, eta-squared is the weighted variance of group means about the grand mean divided by total response variance, and eta is its nonnegative square root. Variance decomposition isolates the share of total dispersion explained by differences among conditional means without requiring those means to vary linearly with category labels. 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.

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How Much Does the Group Matter?

Imagine measuring the heights of kids in a few different classes. Some height differences come from which class you're in, and some are just differences between kids in the same class. The correlation ratio tells you how much of the differences come from the class.

Explained-by-Group Score

The correlation ratio measures how much the group something belongs to explains a number about it. Say you have kids' test scores and you know each kid's class. Find the average score in each class and the average for everyone. If the class averages are very different from each other, most of the spread in scores is explained by class. The correlation ratio squared, called eta-squared, is the share of all the spread in scores that comes from differences between the class averages. It goes from 0, where the class tells you nothing, to 1, where the class tells you everything.

Between-to-Total Variance Ratio

The correlation ratio, written η (eta), measures how strongly a numerical response depends on a categorical predictor. Its square, η², is the variance of the group means around the overall (grand) mean, weighted by group sizes, divided by the total variance of the response. In other words, it's the share of the total spread that is explained by differences between group averages. It ranges from 0 (all group means equal) to 1 (all the variation is between groups), and η is the nonnegative square root. Unlike the ordinary correlation coefficient, it doesn't need the group means to follow a straight line or any order in the categories, so it can pick up any pattern of differences between groups.

The correlation ratio relates a categorical predictor to a numerical response. Eta-squared is the between-group variance, the weighted variance of the group means about the grand mean with weights given by group sizes, divided by the total variance of the response; eta is its nonnegative square root. This follows from decomposing total dispersion into between-group and within-group parts, so eta-squared is the fraction explained by differences among conditional means. It does not require the conditional means to vary linearly with the category labels, which is why it captures relationships that Pearson correlation on coded categories could miss. It lies between 0 (all group means equal) and 1 (no variation within groups). Computing it requires fixed group membership, observation weights, group means, grand mean, and a stated population or sample variance convention.

Scope of Application¶

Correlation ratio belongs to statistics and is useful where the analyst can specify the typed statistics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range. The scope is broad within that domain but bounded by the need for group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range. 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 group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range 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 Correlation ratio 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 Correlation ratio. Correlation ratio 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¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed statistics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of statistics because they reuse the typed statistics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Variance decomposition isolates the share of total dispersion explained by differences among conditional means without requiring those means to vary linearly with category labels., and type the carrier, state every parameter and convention in the definition, test that group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Correlation ratio Domain-specific

Parents (1) — outgoing typed relations

Correlation ratio is a kind of Effect Size Prime

What makes it its own entry: the domain-specific identity determined by group membership, observation weights, group means, grand mean, and the population or sample variance convention are fixed and the between-to-total ratio lies in its valid range.

Hierarchy paths (2) — routes to 2 parentless roots

Correlation ratio → is a kind of Effect Size → presupposes Comparison

Show alternative path (1)

Neighborhood in Abstraction Space¶

Correlation ratio sits in a crowded region of the domain-specific corpus (6^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

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