Spearman's Rank-Order Correlation (monotonic / nonparametric)
VerifiedAdvanced & specialized
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Measures the strength and direction of the MONOTONIC relationship between two rankable (ordinal, interval, or ratio) variables on the same cases.
Captures linear and non-linear-monotone associations alike. Operates on ranks rather than raw values, so it is robust to outliers and to non-normal distributions. Reports ρ on a [-1, +1] scale, Fisher-z 95% CI, t-approximation, p-value, and ρ².
Worked example
Does job rank relate to satisfaction, when both are ordinal?
For 60 employees, job rank (1–5) and satisfaction (1–7) were recorded. Both are ordinal, so Spearman's rho measures the monotonic association.
Rank and satisfaction were positively associated, rs(58) = .58, p < .001 — higher-ranked staff reported more satisfaction.
A Spearman correlation showed a significant positive association between job rank and satisfaction, rs(58) = .58, p < .001.
Try it yourself: Load this ready-made sample and follow the run above.
When to use it
- Two ordinal variables (Likert × Likert, ranking × ranking)A market researcher measures customer satisfaction (1–7 Likert) and recommendation intent (1–10 Likert) for 200 customers.
- Two continuous variables with non-normal distributions or outliersAn economist measures household income and annual savings in 500 households.
- Monotone but non-linear relationships (exponential, logarithmic, growth curves)A pharmacologist measures drug dose (mg) and plasma concentration (ng/mL) over a wide dose range.
When NOT to — use instead
- Two binary variablesSpearman on 0/1 vs 0/1 collapses to phi (≈ Pearson on the binary pair) — use the proper categorical test. → Chi-Square Test of Independence
- Heavy-tied ordinal data with ≤ 3 distinct ranks per variableTie correction inflates the standard error; Kendall's τ-b is more honest in this regime. → Kendall's Tau (\u03c4-b concordance / tie-aware)
- Continuous variables with linear relationshipWhen data is continuous and the X-Y relationship is linear, Pearson r is more powerful and directly interpretable as a linear-correlation coefficient. → Pearson Correlation (linear association)
- Need to control for a confounding variableSpearman is bivariate. → Partial Correlation (linear, controlling for covariates)
Hypotheses
Parameter tested: population Spearman rank correlation ρ_s
Assumptions (and what to do if they fail)
Check: Inspect the scatterplot; look for a consistently up or consistently down pattern.
If violated: Non-monotonic relationships are under-represented by ρ; the test loses power and can point in the wrong direction.
Check: Report the largest tie block.
If violated: Heavy ties reduce power and can bias the large-sample approximation; exact p-values are preferable.
Ready to run a Spearman's Rank-Order Correlation (monotonic / nonparametric) on your own data?
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