Theil Index¶
An additively decomposable generalized-entropy inequality measure comparing population shares with shares of a positive resource, with Theil T weighting log relative shares by the resource and Theil L weighting log shortfalls by population.
Core Idea¶
The Theil Index is a family of inequality measures derived from generalized entropy and information theory. It compares each unit's share of a positive resource such as income with its share of the population. Equality occurs when the two distributions coincide; divergence increases as resource shares become more concentrated. For N units with positive values x_i and mean μ, Theil T is T = (1/N) Σ (x_i/μ) ln(x_i/μ).
Scope of Application¶
The Theil Index is a precondition-bounded distributional measure: it applies where aligned population shares and shares of a declared positive additive resource are compared through the exact Theil T or Theil L logarithmic formula.
- Individual income inequality. Persons' population shares are compared with their shares of disposable, market, gross, or another explicitly defined income concept under a common period and price basis.
- Household income inequality. Households enter with declared equivalence scales, member weights, taxes and transfers, survey coverage, and treatment of zero or negative income.
- Earnings and wage dispersion. Workers, jobs, hours, or full-time equivalents form the population while pay period, labor-force selection, top coding, and nonemployment boundaries remain explicit.
- Wealth-component inequality. Net worth or a positive asset component can be measured only under a policy for debts, negative net wealth, missing high-wealth observations, valuation dates, and household units.
Clarity¶
Naming the Theil Index makes inequality legible as a logarithmic divergence between population shares and shares of a positive resource. It prevents “Theil” from hiding two different measures: Theil T weights relative shares by the resource and emphasizes the upper tail, whereas Theil L weights units by population and is especially sensitive near zero.
Manages Complexity¶
The Theil Index compresses a full distribution into the summed logarithmic divergence between population shares and shares of a declared positive resource. The analyst tracks the T or L variant, unit and survey weights, log base, normalization, treatment of zeros and negatives, and any exhaustive partition. Those coordinates yield a total inequality value and unit or group contributions; additive decomposition then makes readable how much of that measured divergence lies within groups and how much remains between their means, including nested regional or organizational levels.
Abstract Reasoning¶
The analyst begins with a declared population, a positive additive resource, and aligned population and resource weights, then normalizes both over the same units. Equality predicts coincident shares and a value of zero; departures are accumulated as logarithmic share ratios. Choosing Theil T or Theil L determines the direction of weighting and therefore the inference: T gives greater leverage to resource-rich units, whereas L gives equal population leverage and becomes increasingly sensitive near zero.
Knowledge Transfer¶
Within distributional economics, the Theil Index transfers literally across individual or household income, wages, wealth components, regional disparity, productivity dispersion, survey-weighted estimates, and nested group decompositions when the population, positive resource, weights, and T or L variant are fixed. The carried method normalizes population and resource shares, accumulates the appropriate logarithmic divergence, and partitions total inequality into within- and between-group terms under variant-specific weights.
Relationships to Other Abstractions¶
Current abstraction Theil Index Domain-specific
Parents (1) — more general patterns this builds on
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Theil Index is a kind of Generalized entropy index Domain-specific
Theil T and Theil L are the named
α = 1and limitingα = 0members of the parameterized generalized-entropy family.
Hierarchy path (1) — routes to 1 parentless root
- Theil Index → Generalized entropy index → Measurement
Neighborhood in Abstraction Space¶
Theil Index sits in a sparse region of the domain-specific corpus (65th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Gini Coefficient — 0.87
- Foster–Greer–Thorbecke Poverty Measures — 0.85
- Genetic Load — 0.84
- Class stratification — 0.84
- Divisia Index — 0.83
Computed from structural-signature embeddings · 2026-10-08