Ordinal association
Goodman & Kruskal's Gamma Calculator (γ)
Goodman and Kruskal's gamma (γ) measures the strength and direction of association between two ordinal variables using only concordant and discordant pairs. It ranges from −1 to +1, where the sign shows direction and the magnitude shows strength.
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Open the workspace →What is Goodman and Kruskal's gamma?
Gamma is a measure of association for two ordinal variables — variables whose categories have a natural low-to-high order. It looks at every pair of cases and asks whether they are ordered the same way on both variables (a concordant pair) or ordered oppositely (a discordant pair). Pairs that are tied on either variable are simply ignored.
Because gamma is built entirely from concordant and discordant pairs, it has an intuitive reading: a positive value means higher values on one variable tend to go with higher values on the other, while a negative value means higher on one goes with lower on the other.
Formula
Definition
γ = (C − D) / (C + D)
- C
- = the number of concordant pairs — pairs of cases ordered the same way on both variables
- D
- = the number of discordant pairs — pairs of cases ordered oppositely on the two variables
- ties
- = tied pairs (equal on either variable) are ignored entirely
Worked example
Worked example
Consider this ordinal 3×3 table, where both the rows and the columns run from low to high:
[[20, 10, 5], [10, 20, 10], [5, 10, 20]]
Counting across all pairs of cases, the concordant pairs clearly outnumber the discordant pairs — the mass of the table sits along the low-low to high-high diagonal. Plugging the concordant and discordant totals into γ = (C − D) / (C + D) gives:
γ = 0.558.
A gamma of 0.558 is a moderate-to-strong positive ordinal association: higher values on one variable tend to go with higher values on the other.
When to use it
Use it when
- Both variables are ordinal (categories have a natural low-to-high order).
- You want a simple, strongly-signed measure of direction and strength.
- You want an intuitive reading based on whether pairs agree or disagree in their ordering.
Not the right tool when
- The data are nominal (unordered categories) — use lambda or Cramér's V instead.
- There are many ties — gamma overstates strength because it ignores them, so prefer Kendall's tau-b or Somers' d.
- You need a measure that accounts for tied pairs in the denominator — gamma does not.
How to interpret it
Rule of thumb
The sign gives the direction of association and the magnitude gives its strength. Gamma is usually larger in magnitude than Kendall's tau because it ignores ties, so treat its value as an upper bound on the ordinal association rather than a conservative one.
Frequently asked questions
- Gamma vs Kendall's tau — what's the difference?
- Both are ordinal measures built from concordant and discordant pairs, but gamma ignores tied pairs entirely while Kendall's tau-b includes ties in its denominator. As a result gamma tends to be larger in magnitude, and tau is the more conservative choice when ties are common.
- Why is gamma larger than tau?
- Gamma divides the net agreement (C − D) by only the untied pairs (C + D), whereas tau divides by a larger total that also accounts for ties. With the same numerator-style logic but a smaller denominator, gamma's magnitude is generally larger than tau's.
- Can gamma be negative?
- Yes. Gamma ranges from −1 to +1. A negative gamma means discordant pairs outnumber concordant ones — higher values on one variable tend to go with lower values on the other — while a positive gamma means the opposite.
References & further reading
- Goodman, L. A., & Kruskal, W. H. (1954). Measures of association for cross classifications. JASA.
- Goodman and Kruskal's gamma — Wikipedia
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