Spearman Rank Correlation Calculator
Spearman's rank correlation coefficient (rho) measures the strength and direction of the association between two ranked variables. It is a non-parametric alternative to Pearson correlation, applicable when data are ordinal, when the relationship is monotonic rather than strictly linear, or when outliers are present. The method works by replacing each observation with its rank within its dataset, then computing Pearson correlation on those ranks. Enter two comma-separated lists of numbers (the same length) to compute rho.
Spearman rank correlation formula
rho = 1 - (6 × ∑d²) / (n × (n² - 1))
Where d is the difference between the rank of each X value and the rank of its paired Y value, and n is the number of pairs. This simplified formula is exact when there are no ties; the calculator uses the general Pearson formula on ranks (which handles ties correctly).
Interpreting Spearman rho
- Rho close to +1: strong positive monotonic relationship (both variables tend to increase together).
- Rho close to -1: strong negative monotonic relationship (one increases as the other decreases).
- Rho near 0: little or no monotonic relationship.
- Guideline (Cohen, 1988): 0.10 small, 0.30 medium, 0.50 large effect for behavioral sciences.
- Test significance: for n greater than 10, t = rho times sqrt((n-2) / (1 - rho squared)) follows a t-distribution with n-2 degrees of freedom.
Frequently asked questions
What is Spearman's rank correlation?
Spearman's rank correlation coefficient (rho) measures the strength and direction of the monotonic relationship between two variables. Unlike Pearson correlation, it is based on the ranks of the data rather than the actual values, making it robust to outliers and applicable to ordinal data.
What does the formula rho = 1 - 6 sum(d squared) / (n(n squared - 1)) mean?
In the formula, d is the difference between the ranks of each paired observation, and n is the number of pairs. You rank each dataset independently, compute the rank differences, square them, sum them, and plug into the formula. The result ranges from -1 to +1.
When should I use Spearman instead of Pearson?
Use Spearman when data are ordinal, when the relationship is monotonic but not linear, or when the data contain outliers that would distort the Pearson coefficient. Spearman is also preferred for small samples where normality cannot be assumed.
What values does Spearman rho take?
Rho ranges from -1 to +1. A value of +1 means a perfect positive monotonic relationship (as one variable increases, the other always increases). A value of -1 means a perfect negative monotonic relationship. A value of 0 means no monotonic relationship.
How do I handle tied ranks?
When two or more observations have the same value, they are assigned the average of the ranks they would have received. For example, if values at positions 3 and 4 tie, both get rank 3.5. This calculator uses average ranks for ties.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Measures of Correlation.
- NIST/SEMATECH e-Handbook: Engineering Statistics Handbook.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.