Theil index calculator

Enter a positive income for every equally weighted person, household, region, or other unit, and the calculator returns the Theil T index together with the count, total income, and mean income used in the calculation. A result of zero represents perfect equality, while larger values indicate greater concentration. The deterministic calculation is suitable for individual analysis, reproducible reports, data pipelines, and automated comparisons.

What the Theil index tells you

The Theil index condenses the distribution of income into one non-negative inequality measure. When every observation has exactly the same income, each income-to-mean ratio equals one, every logarithmic contribution is zero, and the final index is zero. As income becomes concentrated among fewer observations, the index rises. Unlike a currency total or an average, the result has no unit, which makes it useful for comparing distributions expressed in different currencies or scales when their populations are defined consistently. The measure is sensitive to changes across the entire distribution, including the upper tail. However, it does not identify which particular households are poor or rich, and two visibly different distributions can produce the same index. Interpret it alongside counts, means, quantiles, and domain context. This calculator reports the Theil T form, sometimes called the mean log deviation weighted by relative income, rather than the separate Theil L measure. Stating that convention matters whenever results are published or compared with another source.

How the calculation works

For n incomes, the calculator first finds their arithmetic mean. It then divides every income by that mean, takes the natural logarithm of the ratio, multiplies the ratio by its logarithm, adds all contributions, and divides the sum by n. In symbols, Theil T is the average of (income divided by mean income) times the natural logarithm of that ratio. Natural logarithms are essential; using base-ten or base-two logarithms changes the numerical scale. Every input must be a finite number strictly greater than zero. This requirement avoids undefined logarithms and follows the capability’s explicit contract, so zero, negative, missing, infinite, and non-numeric values return an input error instead of being silently discarded. Each array entry has equal population weight. If an income record represents several people, expand it into individual observations or transform the data with appropriate frequency weights before using this unweighted calculator. The returned total and mean let you verify that the submitted population was interpreted as intended.

Comparing groups and interpreting results

Use the same population definition, accounting period, income concept, and treatment of taxes or transfers for every comparison. Mixing monthly household income with annual personal income produces a numerical answer but not a meaningful comparison. The Theil family is especially useful because overall inequality can be decomposed into within-group and between-group components, but that decomposition requires group populations and totals and is not performed by this single-array calculator. You can still calculate a separate index for each consistently defined group to explore patterns, provided you do not mistake those independent results for an additive decomposition. There is no universal threshold that makes a Theil value good or bad; interpretation depends on the society, market, sample design, and policy question. Also check whether a small number of extreme observations dominate the result, whether inflation adjustments are consistent, and whether the sample is representative. For automation, validate upstream data before submission and preserve the exact input array with the output so analysts can reproduce the result later.