Association measure (entropy)
Uncertainty Coefficient Calculator — Theil's U
Theil's U (the uncertainty coefficient) is an entropy-based measure of how much knowing one categorical variable reduces uncertainty about another. It ranges from 0 (no association) to 1 (one variable completely determines the other).
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| Total | 46 | 54 | 50 | 150 |
Theil's U
0.154
U(row|col) = 0.154 · U(col|row) = 0.154
χ²(4, N = 150) = 50.32, p < .001, V = .41
All statistics
| Pearson chi-square | χ² = 50.322, df = 4, p < .001 |
| G-test (likelihood ratio) | G = 50.805, df = 4, p < .001 |
| Cramér's V | 0.410 (large) |
| Contingency coefficient (C) | 0.501 |
| Lambda (symmetric / row|col / col|row) | 0.357 / 0.370 / 0.344 |
| Goodman–Kruskal gamma (γ) | 0.685 |
| Kendall's tau-b / tau-c | 0.494 / 0.493 |
| Somers' d | 0.494 / 0.494 / 0.493 |
| Theil's U | 0.154 / 0.154 / 0.154 |
What is the uncertainty coefficient?
The uncertainty coefficient comes from information theory. Each categorical variable has an entropy — a measure of how uncertain you are about its value before you look. When two variables are associated, learning one of them shrinks the uncertainty about the other; that shared information is the mutual information between them.
Theil's U expresses that reduction as a proportion: it divides the mutual information by the entropy of the variable you are predicting. The result is a clean 0-to-1 score that reads like “the fraction of uncertainty removed.” Unlike chi-square-based measures, it is directional, so U(X|Y) and U(Y|X) can differ.
Formula
Definition
U(X|Y) = ( H(X) − H(X|Y) ) / H(X) = I(X;Y) / H(X)
- H(X)
- = the entropy of X (uncertainty about X on its own)
- H(X|Y)
- = the conditional entropy of X given Y
- I(X;Y)
- = the mutual information shared by X and Y
- symmetric U
- = 2I(X;Y) / (H(X) + H(Y)): mutual information divided by the average of the two entropies, not the arithmetic mean of U(X|Y) and U(Y|X)
Worked example
Worked example
Take this 3×3 contingency table of two nominal variables:
[[20, 10, 5], [10, 20, 10], [5, 10, 20]]
Compute the entropy of each variable and their conditional entropies, combine them into the mutual information, then divide by the relevant entropy. For the symmetric coefficient, divide the mutual information by the average of the two entropies:
U = 0.101 (≈ 0.10).
In plain terms, knowing one of these variables reduces uncertainty about the other by about 10% — a modest association.
When to use it
Use it when
- Both variables are nominal (unordered categories).
- You want an information-theoretic, prediction-style measure of association.
- You need an asymmetric, directional measure — how much one variable tells you about the other.
Not the right tool when
- The categories have a meaningful order — use an ordinal measure like gamma, Kendall's tau, or Somers' d instead.
- You specifically want a chi-square-based effect size — use Cramér's V.
How to interpret it
Rule of thumb
0 means the variables are independent; 1 means one fully determines the other. For example, U = 0.10 means about a 10% reduction in uncertainty about one variable once you know the other.
Frequently asked questions
- Theil's U vs Cramér's V?
- Both measure association between nominal variables on a 0-to-1 scale, but from different foundations. Theil's U is information-theoretic — it reports the fraction of entropy (uncertainty) removed and can be made directional. Cramér's V is chi-square-based and always symmetric. Use Theil's U for a prediction-style interpretation, Cramér's V when you want an effect size tied to a chi-square test.
- Is the uncertainty coefficient symmetric?
- Not by default. The basic uncertainty coefficient is asymmetric: U(X|Y), the share of X's uncertainty explained by Y, generally differs from U(Y|X). The symmetric version is 2I(X;Y)/(H(X)+H(Y)); it normalizes the shared information by the average entropy and is not the arithmetic mean of the two directional coefficients.
- What does an uncertainty coefficient of 0.1 mean?
- An uncertainty coefficient of about 0.1 means knowing one variable reduces uncertainty about the other by roughly 10% — a weak-to-modest association. It is far from 0 (independence) but well short of 1 (one variable fully determining the other).
References & further reading
- Theil, H. (1970). On the Estimation of Relationships Involving Qualitative Variables. American Journal of Sociology.
- Uncertainty coefficient — Wikipedia
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