Theil Index Calculator

What Is the Theil Index Calculator?

The Theil Index Calculator computes Theil T and Theil L (the Mean Log Deviation) from any set of income, wage, or wealth values. It also performs an additive decomposition of total inequality into within-group and between-group components when you provide named groups such as regions, sectors, or demographic categories. Enter your data in either mode and the calculator returns the Theil T, Theil L, normalized T (on a 0-to-1 scale), the Atkinson equivalent, and a standard Gini coefficient for comparison.

The Theil index was developed by Dutch econometrician Henri Theil in his 1967 book Economics and Information Theory and draws on Shannon's information theory to measure inequality as an entropic distance from a perfectly equal distribution. The US Census Bureau publishes Theil index values alongside Gini coefficients as part of its official income inequality metrics, precisely because the two measures capture different aspects of the distribution. The primary reason researchers reach for the Theil index rather than stopping at the Gini is additive decomposability: total Theil T can be split exactly into between-group and within-group components, which the Gini coefficient cannot do. For measuring residential or school segregation alongside income inequality, our Index of Dissimilarity Calculator covers the spatial evenness dimension.

Theil T vs Theil L: Which Variant Should You Use?

Both Theil T (GE(1)) and Theil L (GE(0)) are members of the Generalized Entropy family of inequality measures, but they assign different weights to different parts of the distribution. Understanding which to report depends on what part of the income distribution your research question concerns.

Theil T weights each individual's log deviation by their income share: a person earning five times the mean contributes five times as much to T as a person earning exactly the mean. As a result, T is most sensitive to high incomes and is the right choice when the research question is about elite concentration, top-income growth, or executive pay. Theil L assigns equal weight to every individual regardless of income, making it more sensitive to the distance of low earners from the mean. L rises faster than T when poverty deepens or when the bottom of the distribution falls further behind. The University of Texas Inequality Project's guide to the Theil index recommends reporting both variants together, particularly when studying earnings inequality across countries or sectors, because the T-minus-L gap itself tells you something about where in the distribution inequality is most concentrated.

When T exceeds L substantially, the top earners are the main drivers of inequality in the dataset. When L exceeds T, the lowest earners are the main story. In a dataset with a long upper tail (a few very high earners), T will typically be much larger than L. In a dataset with a compressed top but a cluster of very low earners at the bottom, L may exceed T.

What Theil Scores Tell You: Interpretation Tiers and Country Context

Unlike the Gini coefficient, the raw Theil T is not bounded at 1 -- its maximum is ln(n), where n is the number of observations. This means raw T values from small samples cannot be directly compared with values from large national datasets. The normalized T (T divided by ln(n)) solves this by rescaling to a 0-to-1 range. For large datasets (n above 1,000), normalization matters little in practice, but for samples under 100 it is important.

Country values above are approximate Theil T figures from the UNU-WIDER World Income Inequality Database (WIID), the most comprehensive cross-country source for Theil index data. The WIID covers over 190 countries and allows comparison across multiple inequality measures including Theil T, Theil L, Gini, and the Palma ratio.

The Additive Decomposition: Why Researchers Prefer Theil Over Gini

The single most important property of the Theil index that the Gini coefficient lacks is exact additive decomposability. Given a population divided into groups, total Theil T equals the between-group component plus the within-group component, with no residual. The between-group component is computed from group means and income shares; the within-group component is the income-share-weighted average of each group's internal Theil T. This decomposition is used routinely by labour economists to identify whether overall wage inequality arises from differences between industries, regions, or demographic groups (between-group) or from pay dispersion within those same categories (within-group).

A dominant between-group component (say, above 40% of total T) points to structural factors: regional development gaps, industry wage premiums, or sectoral sorting by skill level. A dominant within-group component suggests that inequality is diffuse and concentrated inside each category, often reflecting differences in individual-level characteristics such as education, experience, or bargaining power. The FAO Policy Impacts on Inequality guide uses Theil decomposition as its primary tool for diagnosing the sources of rural-urban inequality in developing countries. Our Gini Coefficient Calculator complements this tool: use Gini for broad inequality comparison and Lorenz curve visualization, and use Theil T decomposition when you need to attribute that inequality to specific group-level factors.

Accuracy and Limitations of This Calculator

The Theil T formula T = (1/n) times the sum of (x/mu) times ln(x/mu) is computed exactly for the values you enter. There is no sampling approximation or rounding within the calculation. The only sources of error are the quality of the input data: if values represent grouped midpoints rather than individual incomes, the result is an approximation of the true Theil T with a degree of error that grows larger when bracket widths are wide or distributions within brackets are skewed.

The two most important limitations for applied use are the unbounded upper range and the two-group structure of the standard decomposition. Because T is unbounded, you should never compare raw Theil T values across datasets of different sizes without normalizing by ln(n). The decomposition implemented here follows the standard Theil T additive formula; the PMC review on income inequality measures notes that Theil L uses population shares (n_k/n) rather than income shares (s_k) as weights for the between-group component, which means the two variants decompose differently. Both decompositions are exact for their respective indices, but they produce different within-group and between-group percentages and should not be mixed when reporting results.

The Most Common Theil Index Calculation Mistake

The mistake I encounter most often is treating the raw Theil T value as if it were bounded between 0 and 1 and directly comparable to the Gini coefficient. It is not. A Theil T of 0.45 from a dataset with n=10 occupies a different position on the scale than a Theil T of 0.45 from a dataset with n=10,000, because the maximum possible value is ln(10)=2.30 in the first case and ln(10,000)=9.21 in the second. With that in mind, always report the normalized T alongside the raw T when publishing or presenting results, and always state the sample size n explicitly. This confusion turns up most often in policy documents and journalism that quote a country's Theil T as if it were a percentage or a Gini equivalent, leading readers to misinterpret a value of 0.45 as moderate when it might occupy a relatively low position on the correct scale for that sample size. The Gini coefficient, despite its limitations, has the advantage of always being on a 0-to-1 scale regardless of sample size, which is one reason it remains the more widely cited measure in public communication.