Spearman’s Rank Correlation Calculator

How do you calculate Spearman’s rank correlation coefficient?

Rank each variable separately, 1 for the smallest, giving tied values the average of the ranks they span. For each pair, subtract one rank from the other to get d, then square and add: rs = 1 − 6Σd²/(n(n² − 1)), where n is the number of pairs. rs runs from −1, opposite orders, to 1, the same order.

Worked example: Anscombe’s first data set

The calculator opens on the first of Anscombe’s four sets of eleven points (Anscombe 1973, The American Statistician 27: 17 to 21). Ranked separately and paired up again, its ranks give:

• n = 11
• Σd² = 40, where d = rank of x − rank of y
• rs = 1 − 6Σd² / (n(n² − 1)) = 1 − 6 × 40 / (11 × 120) = 0.81818
• t = rs √((n − 2) / (1 − rs²)) = 0.81818 × √(9 / (1 − 0.81818²)) = 4.2691
• p, two-tailed = P(|rs| ≥ 0.81818), by the t approximation on 9 degrees of freedom = 0.00208
• p = 0.00208 ≤ 0.05: significant at α = 0.05

An APA-style report would read rs(9) = .82, p = .002, with n − 2 = 9 in the brackets. With eleven pairs the p-value comes from the t approximation. At 10 pairs or fewer the calculator counts every ordering of the ranks instead, as the section on exact p-values explains.

Tied values

Equal values share a rank: each takes the average of the positions they occupy, so 10, 20, 20, 30 rank 1, 2.5, 2.5, 4. Anscombe’s fourth set is the extreme case. 10 of its eleven x values are 8, so each ranks 5.5, and the one value of 19 ranks 11.

The short formula assumes the ranks are 1 to n with no repeats, so with ties it is only an approximation. On the fourth set it gives 0.625, while Pearson’s r calculated on the averaged ranks, the definition the short formula comes from, gives 0.500. The calculator reports the second, which is what R and SciPy report, and shows the first in the working for comparison with a mark scheme.

Spearman’s rs and Pearson’s r

Pearson’s r measures how close the points lie to a straight line, and rs measures only whether the two orders agree. That makes rs the better choice for a relationship that rises steadily but not in a straight line, for data that are themselves ranks or ordered categories, and for data with an outlier. Anscombe’s third set is ten points almost exactly on a straight line and one far above it. The outlier holds Pearson’s r to 0.816, but it moves only one place in the ranking, so Σd² = 2 and rs = 0.991. The linear regression calculator gives Pearson’s r with the line itself.

Exact p-values and the t approximation

If the two variables are unrelated, every ordering of one set of ranks against the other is equally likely, so the exact p-value is a count: the share of all n! orderings that give an rs at least as extreme as the one observed. Up to 10 pairs the calculator makes that count, ties included, which is how printed tables of critical values for small samples are worked out. Above 10 it uses t = rs √((n − 2)/(1 − rs²)) on n − 2 degrees of freedom, and the result names the method it used.

The approximation runs slightly low with few pairs, so check a borderline result against the table your course uses. At 11 pairs, an rs of 0.618 has an exact two-tailed p-value of 0.0478 and the next attainable value below it, 0.609, has 0.0519, so 0.618 is the exact critical value at 0.05. The t approximation gives 0.609 a p-value of 0.0467 and calls anything from 0.602 upward significant. Ranks in perfect order are the one case above 10 pairs that stays exact, because the orderings that reach rs = 1 or −1 can be counted directly.

Critical values of Spearman’s rs

A result is significant when rs, or −rs for a predicted negative correlation, is at or above the value for its number of pairs. Each value is the smallest rs whose exact probability is at most α, from the same count the calculator makes, and the tests check them against Zar (1972, Journal of the American Statistical Association 67: 578 to 580). The table starts at 5 pairs. Where it says n/a, no rs is that rare, not even 1.

Common mistakes

• Ranking the two variables in opposite directions. Rank both from smallest to largest, or both from largest to smallest. Mixing the two flips the sign of rs.
• Giving tied values different ranks. Equal values must share the average rank. Ranking them 2 and 3 in the order they happen to be written makes rs depend on the order of the rows.
• Taking n as the number of values. n is the number of pairs, so ten pairs are twenty values and n = 10.
• Reading rs as cause and effect. A strong rank correlation says the two orders agree, not that one variable drives the other. Both can follow a third.
• Choosing a one-tailed test after seeing the sign of rs. Predict the direction before collecting the data, or use the two-tailed test.

What this calculator leaves out

It gives Spearman’s rs and its test, and nothing else: not Kendall’s tau, a confidence interval for rs, or a partial correlation that holds a third variable fixed. To ask whether two groups’ means differ, rather than whether two variables move together, use the t-test calculator. Counts in categories need the chi-square calculator, and the spread of a single variable the standard deviation calculator.