Spearman correlation calculator
Enter paired values, get Spearman’s rho.
What is Spearman’s rank correlation?
The Pearson correlation of the ranks instead of the raw values. It measures how consistently one variable rises or falls with the other, even if the relationship is not a straight line.
How are ties handled?
Tied values get the average of the ranks they would occupy — two values tied for 7th and 8th both get 7.5 (NIST). The calculator then correlates the ranks directly.
What about the shortcut formula?
ρ = 1 − 6Σd² ÷ (n(n² − 1)) gives the same answer only when there are no ties. With ties it can be off, which is why the calculator uses the ranks directly.
Example
For x = 86, 97, 99, 100, 101, 103, 106, 110, 112, 113 and y = 2, 20, 28, 27, 50, 29, 7, 17, 6, 12, the sum of squared rank differences is 194, so ρ = 1 − 6 × 194 ÷ 990 = −0.176 — a weak negative relationship.
How is the p-value found?
With the usual t approximation, t = ρ√((n − 2)/(1 − ρ²)) on n − 2 degrees of freedom. For very small samples, exact tables are more accurate.
When should I use Spearman instead of Pearson?
Use Spearman when the data are ranks or ordinal ratings, when the relationship rises or falls steadily but not along a straight line, or when a few extreme values would distort Pearson's r. For the eight points y = x³ at x = 1 to 8, every increase in x comes with an increase in y, so Spearman's rho is 1.000, while Pearson's r is only 0.932 because the points are curved.
Does a high rho show that one variable causes the other?
No. Like any correlation it only describes how two variables move together. A strong rank correlation can come from a third factor or from coincidence, especially in small samples.
How many pairs do I need?
The calculator needs at least three pairs, but that is only enough to compute a number. With n = 5 pairs, a rho of 0.8 (ranks 1-2-3-4-5 against 2-1-4-3-5) gives p ≈ 0.104 under the t approximation, so it is not significant at the 5% level. Small samples need exact tables or far larger effects.
How the calculation works
Each list is replaced by ranks: the smallest value gets rank 1, the next rank 2, and so on, with tied values sharing the average of the ranks they cover. Spearman's rho is then the ordinary Pearson correlation of the two rank lists, so it always lies between −1 and +1. A value of +1 means that the larger x is, the larger y is, in every pair (a perfectly increasing, or monotonic, relationship); −1 means perfectly decreasing; and 0 means no consistent upward or downward tendency. Only the order of the values matters, so the actual spacing between them, units, and any monotonic rescaling (such as taking logarithms) leave rho unchanged.
Worked example with ties
Take x = 3, 1, 4, 1, 5, 9 and y = 2, 7, 1, 8, 2, 8. In x the two 1s occupy ranks 1 and 2, so each gets 1.5, and the ranks of x are 3, 1.5, 4, 1.5, 5, 6. In y the two 2s share ranks 2 and 3 (2.5 each) and the two 8s share ranks 5 and 6 (5.5 each), giving 2.5, 4, 1, 5.5, 2.5, 5.5. Correlating those two rank lists gives rho = -0.1343 with p ≈ 0.7997, a weak, non-significant relationship. The shortcut formula 1 − 6Σd² ÷ (n(n² − 1)) would give a slightly different number here because of the ties.
Common mistakes
Entering the x and y lists in a different order, so the pairs do not line up, gives meaningless results; the n-th number in one list must belong to the same person or item as the n-th number in the other. Reading rho as the slope of a line is another trap: Spearman says nothing about how much y changes per unit of x. A relationship that goes up and then down (a U or an arch) can have a rho near zero even though the variables are strongly related, so look at a scatter plot as well. Finally, the p-value uses a large-sample t approximation, so treat it as rough for fewer than about 10 pairs.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, the sections on rank correlation and on ranks of tied data; OpenStax Introductory Statistics, chapter 12 (linear regression and correlation) for the t statistic on n − 2 degrees of freedom. Related: correlation coefficient calculator, linear regression calculator and standard deviation calculator.