Spearman's Rho (ρ) Rank Correlation Calculator
This Spearman's rho calculator provides full details of the rank correlation calculation, including rank assignments, differences, and the step-by-step computation of ρ and significance testing.
The calculator offers exact permutation and approximate p-values. Exact inference assumes observations are exchangeable under the null hypothesis and supports tied ranks. It enumerates up to 8 paired observations; automatic mode uses the t approximation above that limit. The result identifies the method actually used.
Further Information
Spearman's rank correlation coefficient (ρ or rho) is a non-parametric measure of the monotonic relationship between two variables. Unlike Pearson's correlation, it does not assume linearity or normality, making it useful for ordinal data or when assumptions of Pearson's correlation are violated.
Requirements
- Two variables measured at least at the ordinal scale
- Monotonic relationship between variables
- No assumption of normality required
- Can handle ordinal data and non-linear monotonic relationships
- At least 3 pairs of observations
Null Hypothesis
The approximate test assesses zero population rank correlation. Exact permutation inference tests independence through exchangeable pairings; zero correlation alone does not guarantee independence.
The null hypothesis states that there is no monotonic relationship between the two variables in the population. A significant result suggests that the observed rank correlation is unlikely to be due to chance.
Equation
ρ = 1 - [6Σd² / n(n² - 1)]
Where n is the sample size, and d is the difference between the ranks of corresponding values. This shortcut applies only without ties. The calculator uses Pearson correlation of average ranks, which also handles ties.
When to Use Spearman vs Pearson
Use Spearman's rho when:
- Data are ordinal (ranks) rather than interval/ratio
- The relationship is monotonic but not linear
- There are significant outliers
- Normality assumptions are violated
Interpretation
- ρ = +1: Perfect positive monotonic relationship
- ρ = 0: No monotonic relationship
- ρ = -1: Perfect negative monotonic relationship
- |ρ| > 0.7: Strong correlation
- |ρ| = 0.3-0.7: Moderate correlation
- |ρ| < 0.3: Weak correlation
