Non-Parametric Rank Correlation
Spearman Rank-Order Correlation
Measure the strength and direction of a monotonic relationship between two variables. Suitable for ordinal data or continuous data failing normality assumptions.
Configure data
n = 0
n = 0
Data & Assigned Ranks
Tied values receive the average of their respective ranks. The difference d = Rank(X) − Rank(Y).
Scatterplot of Ranks
Effect Size (rs²)
APA 7th edition reporting statement
Copy-paste template
Calculation Details
Interpretation Report
Step-by-step analysis & plain-language conclusions
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Hypotheses
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Assumptions check
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Computed statistical values
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Statistical decision
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Effect size & practical significance
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Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How Spearman Correlation Works
Complete theoretical foundation, assumptions, effect size interpretation, handling of ties, and primary references for Spearman's Rank-Order Correlation.
What is Spearman's rho (rs)?
Developed by Charles Spearman (1904), Spearman's rank-order correlation coefficient (often denoted as $r_s$ or $\rho$) is a non-parametric measure of statistical dependence between two variables. Instead of measuring linear relationships with raw data (like Pearson's $r$), Spearman assesses how well the relationship can be described using a monotonic function by applying the Pearson correlation formula to the ranks of the data.
Simplified Formula (No Ties):
r_s = 1 - ( 6 × Σd² ) / ( n × (n² - 1) )
where:
d = Difference between the ranks of corresponding variables
n = Number of observations
*Note: When tied values exist, the standard Pearson correlation equation is applied directly to the ranked variables, as the simplified formula becomes inaccurate. This calculator uses the precise covariance-of-ranks method to properly handle ties.*
Assumptions
- Ordinal, Interval, or Ratio variables — data must be at least ordinal (can be ranked). It is ideal for continuous data that severely violates Pearson's normality assumption.
- Paired observations — each participant must have a pair of values (one for X, one for Y).
- Monotonic relationship — the variables must tend to move in the same relative direction, but not necessarily at a constant linear rate. Inspect a scatterplot to verify monotonicity.
Effect size measures
- Correlation Coefficient (rs) — Used directly as an effect size. Cohen (1988) suggested conventions: |rs| = 0.10 (small), |rs| = 0.30 (medium), |rs| = 0.50 (large).
- Squared Coefficient (rs²) — Represents the proportion of variance in the ranks of one variable that can be explained by the ranks of the other variable.
APA 7th edition reporting format
Template: "A Spearman's rank-order correlation was run to assess the relationship between [Variable X] and [Variable Y]. There was a [positive/negative], [weak/moderate/strong] monotonic correlation between the two variables, rs([df]) = .xx, p = .xx."
Report the $r_s$ statistic to two decimal places without a leading zero. Report exact p-values (e.g., p = .032) unless p < .001. State the degrees of freedom ($N - 2$) in parentheses.
Primary references
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72-101.