Kendall's ฯ measures rank correlation by counting concordant and discordant pairs โ more robust than Spearman when ties are common or n is very small.
Kendall's ฯ counts the number of *concordant* pairs (xแตข < xโฑผ โ yแตข < yโฑผ) minus the number of *discordant* pairs, normalised by all possible pairs. The result is a directly interpretable probability: ฯ = 0.4 means a random pair is 40 percentage points more likely to be concordant than discordant.
Different conventions handle ties differently: ฯ-a (no tie adjustment), ฯ-b (handles ties; the default and our pick), ฯ-c (for kรk contingency tables). All run from โ1 to +1.
Vs Spearman: ฯ is more robust with many ties or very small samples (n < 30) and has a slightly more interpretable definition. Spearman is mildly more powerful with continuous data and no ties. For most practical purposes they tell the same story; pick ฯ when you want the probability-of-concordance interpretation or when ties matter.
Two numeric or ordinal columns.
ฯ, T statistic, p, Fisher-z-style CI.
|ฯ| is typically smaller than |ฯ| or |r| on the same data โ about 0.7 times. A ฯ of 0.3 corresponds roughly to a Spearman ฯ of 0.4โ0.45.
Sign and significance match Spearman's; the magnitude is on a different scale, so always report which ฯ variant you used.