Goodman and Kruskal's Gamma Calculator

Gamma is computed from the counts of concordant and discordant pairs in the data:

$$\gamma = \frac{C - D}{C + D}$$

Where:

• C (Concordant pairs) — pairs of observations where the ranking on both variables agrees. If observation i ranks higher than observation j on variable X, it also ranks higher on variable Y.
• D (Discordant pairs) — pairs where the ranking disagrees. If i ranks higher on X, it ranks lower on Y (or vice versa).

Pairs that are tied on either or both variables are excluded from the computation entirely. This is both a strength (simplicity) and a limitation (information loss when ties are common).

The probabilistic interpretation of Gamma is elegant: it represents the probability that a randomly chosen pair is concordant minus the probability that it is discordant, conditional on the pair being untied. Formally:

$$\gamma = P(\text{concordant}) - P(\text{discordant}) \quad \text{among untied pairs}$$

The approximate standard error for large samples is:

$$SE \approx \frac{2\sqrt{1 - \gamma^2}}{\sqrt{C + D}}$$

A significance test can be performed using z = γ/SE, compared to the standard normal distribution. However, this approximation works best when the number of untied pairs is large.

Gamma has a clear and direct interpretation:

• γ = +1: Perfect positive ordinal association (all untied pairs are concordant)
• γ = −1: Perfect negative ordinal association (all untied pairs are discordant)
• γ = 0: No ordinal association (equal numbers of concordant and discordant pairs)

For magnitude interpretation: |γ| < 0.10 (negligible), 0.10–0.30 (weak), 0.30–0.60 (moderate), 0.60–0.80 (strong), > 0.80 (very strong). Note that Gamma tends to have larger absolute values than Kendall's Tau or Somers' D for the same data because it ignores ties, so direct comparison with these measures requires caution.