Kendall Coefficient of Concordance Calculator

Enter S (the sum of squared deviations of the item rank-sums from their mean), the number of judges, and the number of items into the calculator to determine Kendall’s W.

Related Calculators

Kendall’s W Formula

The following formula is used to calculate Kendall’s W from S (the sum of squared deviations of the item rank-sums from their mean), the number of judges, and the number of items. This standard form assumes complete rankings with no ties; when ties occur, a tie correction is required.

W = \frac{12 \cdot S}{J^2 \cdot (N^3 - N)}

Variables:

W is Kendall’s coefficient of concordance
S is the sum of squared deviations of each item’s total rank from the mean total rank: S = Σ(R_i − R̄)², where R_i is the sum of ranks for item i across judges
J is the number of judges
N is the number of items

To calculate Kendall’s W (with no ties), first compute each item’s total rank R_i by summing its ranks across the J judges, compute the mean total rank R̄, then compute S = Σ(R_i − R̄)². Finally, multiply 12 by S and divide by J²(N³ − N). If your rankings include ties, the denominator must be adjusted using a standard tie correction term.

What is Kendall’s Coefficient of Concordance?

Kendall’s coefficient of concordance, also known as Kendall’s W, is a measure of agreement among raters or judges. It assesses the degree to which different judges rank a set of items in the same order. The value of Kendall’s W ranges from 0 to 1, where 0 indicates no agreement among the judges and 1 indicates complete agreement. This statistic is commonly used in studies where multiple judges are ranking or rating the same set of items, such as in psychology, marketing research, and other fields that require inter-rater reliability.

How to Calculate Kendall’s W?

The following steps outline how to calculate Kendall’s W.

For each item, sum its ranks across all judges to get its total rank (R_i).
Compute the mean of the total ranks (R̄) and then compute S = Σ(R_i − R̄)².
Next, determine the number of judges (J).
Next, determine the number of items (N).
Finally, calculate Kendall’s W using the formula W = (12 * S) / (J² * (N³ – N)). (If there are ties, apply a tie correction.)
After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem :

Use the following variables as an example problem to test your knowledge.

S = Σ(R_i − R̄)² = 150

Number of judges (J) = 5

Number of items (N) = 10