Calculate Kendall’s W with tie correction, rank-sum details, chi-square and an asymptotic p-value for multiple raters.

Rows are items; columns are raters. Enter complete rankings of the same items using exact average ranks for ties.

3–100 item rows, 2–50 rater columns. Separate ranks with spaces, commas or semicolons. No thousands separators or blank cells.

Approximate-test settings

Default 0.05; choose a level strictly between 0 and 1 before interpreting the test.

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 = (12 × S) / (J² × (N³ - 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(Ri − R̄)², where Ri 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 Ri by summing its ranks across the J judges, compute the mean total rank R̄, then compute S = Σ(Ri − R̄)². Finally, multiply 12 by S and divide by J²(N³ − N). For tied average ranks, use denominator J²(N³ − N) − JΣ(t³ − t), summing tie-group sizes t separately within each judge. Each column must be a complete ranking using exact integer or half-integer midranks. The approximate test uses χ² = J(N − 1)W with N − 1 degrees of freedom and assumes independent raters under the null. Small samples and heavy ties may need a permutation procedure; this tool does not perform one.

What is Kendall’s Coefficient of Concordance?

Kendall’s W measures concordance among complete rankings of the same items. It ranges from 0 to 1 when rank variation remains: 1 is maximum concordance, while 0 is zero rank-sum concordance and can coexist with systematic opposition between raters. It is not automatically a measure of validity or general reliability, and there are no universal strength thresholds. With every item tied by every rater, W is undefined.

How to Calculate Kendall’s W?

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

  1. For each item, sum its ranks across all judges to get its total rank (Ri).
  2. Compute the mean of the total ranks (R̄) and then compute S = Σ(Ri − R̄)².
  3. Next, determine the number of judges (J).
  4. Next, determine the number of items (N).
  5. Finally, calculate Kendall’s W using the formula W = (12 * S) / (J² * (N³ – N)). (If there are ties, apply a tie correction.)
  6. 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(Ri − R̄)² = 150

Number of judges (J) = 5

Number of items (N) = 10