Generalized entropy index¶

A parameterized family of decomposable inequality measures computed from powers or logarithms of each observation's ratio to the population mean.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4694
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Inequality Measurement → Economics & Finance

Core Idea¶

The generalized entropy index is a family of scale-invariant inequality measures indexed by alpha, including mean-log-deviation and Theil limiting cases. Individual mean-relative shares are transformed by a power or logarithm and averaged; alpha shifts sensitivity toward the lower or upper tail, and additive decomposition allocates total inequality across groups. 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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The Fair-Share Score Dial

Imagine kids sharing cookies, and some get many while some get few. This is a way to give a score for how unfair the sharing is. It has a dial: turn it one way to care more about kids with very few cookies, turn it the other way to care more about kids with lots. If everyone's cookies doubled, the score would stay the same.

Tunable Unfairness Meter

The Generalized entropy index is a family of scores economists use to measure how unequal things like incomes are. It compares each person's amount to the average amount, changes those comparisons with a math step, and averages the results. A setting called alpha decides whether the score pays more attention to the poorest people or the richest people. Doubling everyone's money doesn't change the score. You can also split the total score into the inequality inside groups and the inequality between groups.

Tail-Tunable Inequality Measure

The Generalized entropy index is a family of inequality measures labeled by a parameter alpha. For each person you compute their income divided by the mean income, transform that ratio with a power (or a logarithm in special cases), and average across the population. Low alpha values make the index more sensitive to the bottom of the distribution, and high values make it more sensitive to the top. Two well-known limiting cases are the mean log deviation and the Theil index. The index is scale-invariant and additively decomposable, meaning total inequality splits cleanly into within-group and between-group parts. For comparisons to be valid, the inputs, weights, alpha and the treatment of zero incomes must be fixed and applied to the same population.

The Generalized entropy index GE(α) is a one-parameter family of scale-invariant inequality measures. Individual outcomes are normalized by the mean, y_i/μ, then transformed by a power function for general α or by a logarithm in the limits, and averaged with population weights. The parameter α tunes tail sensitivity: small or negative α emphasizes the lower tail, large α the upper tail. Limiting cases include the mean log deviation (α→0) and the Theil index (α→1). A key property is additive decomposability: total inequality splits exactly into between-group and within-group components. Because the measure depends on conventions, inputs, weights, mean normalization, α and treatment of zero values (problematic for log or negative-power forms) must be fixed. Results should be interpreted only within the same population basis.

Scope of Application¶

Generalized entropy index belongs to economics and is useful where the analyst can specify a nonnegative distribution such as incomes, population weights, arithmetic mean, sensitivity parameter alpha, within-group and between-group components and comparison population, then evaluate inputs, weights, mean normalization, alpha and zero-value conventions are fixed and the result is interpreted within the same population basis. The scope is broad within that domain but bounded by the need for inputs, weights, mean normalization, alpha and zero-value conventions are fixed and the result is interpreted within the same population basis. 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 inputs, weights, mean normalization, alpha and zero-value conventions are fixed and the result is interpreted within the same population basis 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 Generalized entropy index 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 Generalized entropy index. Generalized entropy index 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: a nonnegative distribution such as incomes, population weights, arithmetic mean, sensitivity parameter alpha, within-group and between-group components and comparison population. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express inputs, weights, mean normalization, alpha and zero-value conventions are fixed and the result is interpreted within the same population basis independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of economics because they reuse a nonnegative distribution such as incomes, population weights, arithmetic mean, sensitivity parameter alpha, within-group and between-group components and comparison population, Individual mean-relative shares are transformed by a power or logarithm and averaged; alpha shifts sensitivity toward the lower or upper tail, and additive decomposition allocates total inequality across groups., and type the carrier, state every parameter and convention in the definition, test that inputs, weights, mean normalization, alpha and zero-value conventions are fixed and the result is interpreted within the same population basis, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Generalized entropy index Domain-specific

Parents (1) — outgoing typed relations

Generalized entropy index is a kind of Measurement Prime

The index measures distributional inequality under a tunable sensitivity rule.

Children (1) — abstractions linked here by typed relations

Theil Index Domain-specific is a kind of Generalized entropy index

Theil T and Theil L are the named α = 1 and limiting α = 0 members of the parameterized generalized-entropy family.

Hierarchy path (1) — routes to 1 parentless root

Generalized entropy index → is a kind of Measurement

Neighborhood in Abstraction Space¶

Generalized entropy index sits in a crowded region of the domain-specific corpus (33^rd 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