Calculation Of Rank Correlation Coefficient

The rank correlation coefficient is computed under three situations: (1) the ranks are already given, (2) the ranks must be worked out from the data, and (3) some ranks are repeated.

Case 1 — ranks are given (Example 3). Five competitors in a beauty contest are ranked by three judges A, B and C; we find which pair of judges agrees most. Rank correlation is computed for each of the three pairs using $r_s = 1 - \dfrac{6\sum D^2}{n^3 - n}$ with $n = 5$.

For judges A and B, the differences in ranks give $\sum D^2 = 14$:

For A and C, $\sum D^2 = 10$, giving $r_s = 1 - \dfrac{60}{120} = 0.5$. For B and C it works out to $0.9$. So judges A and C are the closest in taste, while B and C differ the most.

Case 2 — ranks not given (Example 4). Five students' marks in Statistics (X) and Economics (Y) are given. Each variable is first ranked (highest mark = rank 1), producing pairs of ranks $R_X$ and $R_Y$; formula (4) is then applied exactly as in Case 1.

Case 3 — ranks repeated (Example 5). When items tie, each tied item gets the average of the ranks they would occupy. In the example Y takes the value 50 at the 9th, 10th and 11th positions, so all three receive the average rank 10. With repeated ranks a correction factor is added for every group of ties, and the formula becomes

where $m_1, m_2, \dots$ are the sizes of the tied groups. For this data ($n = 12$, $\sum D^2 = 198$) the correction from the ties works out to

so that

Thus X and Y show a positive rank correlation — they move in the same direction — but the relationship is not strong.