Spearman's Rank Correlation Calculator

Background.

Two judges score the same ten wines. A teacher wants to know whether pupils who rank high in maths also rank high in music. A biologist suspects that bigger nests hold more eggs but doubts the relationship is a straight line. In each case the question is not whether one quantity rises by a fixed amount for each step in the other, but simply whether the two tend to move in the same order. Spearman's rank correlation measures exactly that.

The method throws away the actual values and keeps only their order. Each list is replaced by ranks — 1 for the smallest, 2 for the next, and so on — and the ordinary Pearson correlation is calculated on those ranks. The result, rho, runs from −1 to +1. A value of +1 means the two lists are in exactly the same order; −1 means one is the exact reverse of the other; values near 0 mean the order of one says little about the order of the other.

Working with ranks buys two useful properties. Any relationship that only ever goes up, or only ever goes down, scores a perfect ±1 even if it curves sharply: y = x³ over the numbers 1 to 5 gives rho of exactly 1, where Pearson's coefficient on the raw values is only about 0.943. And one wild value cannot dominate, because the largest number in a list is simply ranked last however extreme it is.

The price is that rho ignores how far apart the values are, and it cannot see a relationship that rises and then falls. It is also a description of the data in front of you; it does not tell you whether the pattern would hold in a larger sample, and it says nothing about cause.

What is spearman's rank correlation calculator?

Spearman's rank correlation coefficient, written ρ or r_s, is the Pearson correlation coefficient computed on the ranks of two paired variables rather than on their values. It measures the strength and direction of a monotonic association: how consistently one variable increases, or decreases, as the other increases. Values tied within a list share the average of the ranks they would otherwise occupy.

How to use this calculator.

1. 01Type or paste the first variable into X values as numbers separated by commas, for example 12, 15.5, 9, 20. Line breaks or spaces alone will not separate values.
2. 02Enter the second variable into Y values in the same order, so that the first Y belongs to the same person, item or occasion as the first X.
3. 03Check that both lists have the same length and at least three pairs, and that neither list consists of a single repeated value, since a variable with no variation cannot be ranked meaningfully.
4. 04Read rho and the pair count. The sign gives the direction of the association and the size its consistency; the page does not test whether it is statistically significant.

The formula.

Each list is sorted and every value replaced by its position, starting at 1. Where several values are equal, each receives the average of the positions they occupy, so two values tied for 1st and 2nd both get 1.5. Rho is then the Pearson correlation of the two rank lists: the sum of the products of each pair's deviations from the mean rank, divided by the square root of the product of the two sums of squared deviations. When there are no ties this equals the textbook shortcut ρ = 1 − 6Σd² ÷ [n(n² − 1)], where d is the difference between a pair's two ranks and n the number of pairs. With ties the shortcut drifts slightly, which is why the full Pearson form is used here.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook, Rank Correlation.
2. [2]OpenStax, Introductory Statistics 2e.
3. [3]NIST/SEMATECH e-Handbook of Statistical Methods.