# How To Compute & Interpret Spearman’S Rank Order Correlation?

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- Spearman’s Rank Order Correlation Page 1 How to Compute and Interpret Spearman’s Rank Order Correlation Objective: Learn how to compute, interpret and use Spearman’s rank order correlation. Keywords and Concepts 1. Spearman’s rank order correlation 4. Rank order 5. Difference between ranks 2. rho 3. Ordinal data 6. Degrees of freedom (df) Spearman’s rank order correlation (? or rho) determines the relationship between two sets of ordinal data (usually paired) that initially appear in rank order or have been converted to rank order. It uses the item’s position in a rank-ordered list as the basis for assessing the strength of the association. Data in Kinesiology and sports competition frequently appear as ranked data. For example, a coach may rank his players’ skill level from 1 (highest skill), 2 (next best) on down to the last rank (lowest skill). Baseball leagues and ladder and round-robin tournaments rank individuals or teams. Even when data have been collected on a parametric variable, the raw data can be converted to rankings and the rank order correlation method applied, although at the expense of some loss in mathematical precision. Rank Order Correlation Formula 6? 2d ? =1 - 2 N(N - 1)
- Spearman’s Rank Order Correlation Page 2 where, ? (rho) is Spearman’s rank order correlation coefficient, d is the difference between ranks for the two observations within a pair (see table 1 below), and N represents the total number of subjects (i.e., number of data pairs ). The number in the numerator is always 6. The degrees of freedom (df) for rho calculates as (df = Npairs - 2.) Example The data in Table 1 illustrate 10 major universities’ ranked for research dollars awarded in health sciences and their football team’s conference ranking in 2001. Table 1. Rankings of major U.S. universities (A-J) for research dollars and football rankings. School A B C D E F G H I J Research 1 2 4 6 3 5 9 7 10 8 dollars Football 4 5 3 1 9 7 6 8 2 10 rankings 3 3 1 5 6 2 3 1 8 2 d 9 9 1 25 36 4 9 1 64 4 d2 Solution When two scores tie in rank, both are given the mean of the two ranks they would occupy and the next rank is eliminated to keep N consistent. For example, if two schools tied for 4th place, both would receive a rank of 4.5 (4 + 5 ÷ 2), and the next school would be ranked number 6. The following equation computes Spearman’s rank order correlation:
- Spearman’s Rank Order Correlation Page 3 6? 2d ? =1 - 2 N(N - 1) 6(162) r =1 - 10(10 2 - 1) 972 r =1 - 990 r = 0.02 Interpretation To determine if the rho coefficient is statistically significant (e.g., reject the Null hypothesis that the real rho is zero), compare the magnitude of rho versus the value found in Table 2. The degrees of freedom (df) equal: df = Npairs - 2 = 10 - 2 = 8 From table 2, at the 0.05 level of significance, with df = 8, a rho correlation coefficient of 0.74 is required for statistical significance. Thus, the observed rho of 0.02 indicates that there is no relationship between rankings of college football programs and the amount of research dollars generated in the health sciences. Note that the table only goes up to N = 30. If N > 30, then the following formula computes the critical value to assess the statistical significance of the rho coefficient.
- Spearman’s Rank Order Correlation Page 4 ?z ?= N- 1 where the value of z corresponds to the significance level. For example, if the significance level is 0.05, z will equal 1.96. If rho exceeds the computed critical value, it is statistically significant.
- Spearman’s Rank Order Correlation Page 5 Table 2. Critical values of Spearman's Rank Correlation Coefficient (rho). n alpha = 0.01 alpha = 0.05 alpha = 0.02 alpha = 0.01 5 0.900 -- 6 0.829 0.886 0.943 -- 7 0.714 0.786 0.893 -- 8 0.643 0.738 0.833 0.881 9 0.600 0.683 0.783 0.833 10 0.564 0.648 0.745 0.794 11 0.523 0.623 0.736 0.818 12 0.497 0.591 0.703 0.780 13 0.475 0.566 0.673 0.745 14 0.457 0.545 0.646 0.716 15 0.441 0.525 0.623 0.689 16 0.425 0.507 0.601 0.666 17 0.412 0.490 0.582 0.645 18 0.399 0.476 0.564 0.625 19 0.388 0.462 0.549 0.608 20 0.377 0.450 0.534 0.591 21 0.368 0.438 0.521 0.576 22 0.359 0.428 0.508 0.562 23 0.351 0.418 0.496 0.549 24 0.343 0.409 0.485 0.537 25 0.336 0.400 0.475 0.526 26 0.329 0.392 0.465 0.515 27 0.323 0.385 0.456 0.505 28 0.317 0.377 0.448 0.496 29 0.311 0.370 0.440 0.487 30 0.305 0.364 0.432 0.478 “Distribution of sums of squares of rank differences to small numbers of individuals,quot; The Annals of Mathematical Statistics, Vol. 9, No. 2. Reprinted with permission of the Institute of Mathematical Statistics.