concordance_kendall calculates Kendall’s coefficient of concordance on independent rankings of objects or individuals.

For full information please refer to the NAG Library document for g08da

xfloat, array-like, shape

must be set to the value of object in comparison , for , for .


The value of Kendall’s coefficient of concordance, .


The approximate significance, , of .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .


No equivalent traditional C interface for this routine exists in the NAG Library.

Kendall’s coefficient of concordance measures the degree of agreement between comparisons of objects, the scores in the th comparison being denoted by

The hypothesis under test, , often called the null hypothesis, is that there is no agreement between the comparisons, and this is to be tested against the alternative hypothesis, , that there is some agreement.

The scores for each comparison are ranked, the rank denoting the rank of object in comparison , and all ranks lying between and . Average ranks are assigned to tied scores.

For each of the objects, the ranks are totalled, giving rank sums , for . Under , all the would be approximately equal to the average rank sum . The total squared deviation of the from this average value is, therefore, a measure of the departure from exhibited by the data. If there were complete agreement between the comparisons, the rank sums would have the values (or some permutation thereof). The total squared deviation of these values is .

Kendall’s coefficient of concordance is the ratio

and lies between and , the value indicating complete disagreement, and indicating complete agreement.

If there are tied rankings within comparisons, is corrected by subtracting from the denominator, where , each being the number of occurrences of each tied rank within a comparison, and the summation of being over all comparisons containing ties.

concordance_kendall returns the value of , and also an approximation, , of the significance of the observed . (For approximately follows a distribution, so large values of imply rejection of .) is rejected by a test of chosen size if . If , tables should be used to establish the significance of (e.g., Table R of Siegel (1956)).


Siegel, S, 1956, Non-parametric Statistics for the Behavioral Sciences, McGraw–Hill