Evaluate ranked dependence with flexible inputs and summaries. Compare pair structure, strength, and directional consistency. See meaningful ordinal patterns before making analytical judgments today.
The sample below compares two ordered three-level ratings. Higher row and column numbers indicate stronger or better ordered outcomes.
| Rating group | Low outcome | Medium outcome | High outcome | Row total |
|---|---|---|---|---|
| Low predictor | 12 | 5 | 1 | 18 |
| Medium predictor | 6 | 14 | 4 | 24 |
| High predictor | 2 | 7 | 16 | 25 |
| Column total | 20 | 26 | 21 | 67 |
Gamma = (C − D) / (C + D), where C is the number of concordant pairs and D is the number of discordant pairs.
For each cell n(i,j), count all cells strictly down-right for concordant contributions and all cells strictly down-left for discordant contributions.
C = Σ n(i,j) × Σ n(k,l) for k > i and l > j. D = Σ n(i,j) × Σ n(k,l) for k > i and l < j.
For a quick analytical interval, the page uses an approximate standard error:
SE ≈ sqrt((1 − Gamma²) / (C + D)), then Z = Gamma / SE, followed by a normal-approximation confidence interval and p value.
Gamma ranges from -1 to +1. Positive values show aligned ordering. Negative values show reverse ordering. Zero suggests balance between concordant and discordant evidence. Ties are ignored, so very tie-heavy data may suit tau-b or Somers' D better.
It measures ordinal association between two ranked variables. The coefficient compares concordant versus discordant pairs and ignores tied pairs in the main ratio.
Use gamma when both variables are ordinal rather than continuous. It is especially useful for survey scales, rank categories, ordered clinical scores, and preference levels.
Gamma excludes tied pairs from its denominator. If ties are common, the remaining untied pairs can make the association look stronger than tie-sensitive measures.
A pair is concordant when the observation ranked higher on one variable also ranks higher on the other variable. Their ordering agrees in direction.
A pair is discordant when the observation ranked higher on one variable ranks lower on the other variable. Their ordering moves in opposite directions.
Yes. The 2 by 2 mode calculates concordant and discordant pairs directly from the ordered table, then applies the same gamma formula and interval logic.
No. This page uses a normal-approximation method for quick analytical reporting. For small samples or complex tie structures, specialist statistical software may provide better inference.
A negative gamma means the ordering tends to reverse across variables. Higher ranks in one variable usually align with lower ranks in the other.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.