Home › Core Math & Algebra › Spearman's Rank Correlation Spearman's Rank Correlation Calculator (ρ) Paste your paired data, compute Spearman's rank correlation coefficient ρ, and see the full ranking table with d and d². Non-parametric stats 1. Enter paired data (X, Y) Enter each pair on a new line, separated by a space, comma, or tab. Example: 10 12 . Paired observations Load example data Clear Minimum 3 pairs are required. The calculator assigns average ranks when there are ties. 2. Options Show full ranking table (ranks, d, d²) Also show classic Σd² formula (no ties) Calculate Spearman's ρ 3. Results Spearman's coefficient ρ – Sample size n: – Strength & direction: ρ ranges from -1 (perfect decreasing monotonic relationship) to +1 (perfect increasing monotonic relationship). Test statistic (approximate) t statistic: – Degrees of freedom: – For moderate to large n, you can approximate a significance test using a t distribution with n − 2 degrees of freedom. This calculator does not compute p-values; use statistical software for formal hypothesis tests. Classic Σd² formula (valid when there are no ties) Σd² = – , n = – . If there are no ties, ρ can also be computed as ρ = 1 − 6 Σd²

(n (n² − 1)) . When ties are present, using the Pearson correlation of the ranks (the method used by this calculator) is recommended. 4. Ranking table (X, Y, ranks, d, d²) i Xᵢ Yᵢ rank(Xᵢ) rank(Yᵢ) dᵢ = Rₓ − Rᵧ dᵢ² Spearman's rank correlation: definition and intuition Spearman's rank correlation coefficient, usually denoted by ρ (rho), is a non-parametric measure of the strength and direction of a monotonic relationship between two variables. Instead of using the original values, it works on their ranks . This makes Spearman's ρ particularly useful when: your variables are ordinal (e.g. Likert scales); the relationship is monotonic but not necessarily linear; there are outliers that could distort Pearson correlation. Spearman's ρ formula (no ties) For n pairs of data \((x_i, y_i)\), after ranking the X and Y values, define \(d_i = R(x_i) - R(y_i)\). If there are no ties , Spearman's ρ can be computed as: \( \displaystyle \rho = 1 - \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 - 1)} \) Spearman's ρ with ties (what this calculator does) In real data, ties are common. The standard approach is: Assign ranks to X and Y separately, using average ranks for ties. Compute Spearman's ρ as the Pearson correlation between the two rank variables. If \(R_x\) and \(R_y\) are the rank variables and \(\bar{R}_x\), \(\bar{R}_y\) their means, then: \( \displaystyle \rho = \frac{\sum_{i=1}^{n} \bigl(R_{x,i} - \bar{R}_x\bigr)\bigl(R_{y,i} - \bar{R}_y\bigr)} {\sqrt{\sum_{i=1}^{n} \bigl(R_{x,i} - \bar{R}_x\bigr)^2} \sqrt{\sum_{i=1}^{n} \bigl(R_{y,i} - \bar{R}_y\bigr)^2}} \) This is exactly what the calculator does: it always uses rank-based Pearson correlation, which is valid with or without ties. Interpreting Spearman's ρ There is no universal rule, but a common interpretation is: |ρ| < 0.2 – very weak or no monotonic association 0.2 ≤ |ρ| < 0.4 – weak 0.4 ≤ |ρ| < 0.6 – moderate 0.6 ≤ |ρ| < 0.8 – strong 0.8 ≤ |ρ| ≤ 1.0 – very strong The sign of ρ indicates the direction: positive values mean that higher X tends to be associated with higher Y, while negative values indicate that higher X tends to correspond to lower Y. Step-by-step example The example included in the calculator is adapted from a classic illustration of Spearman's rank correlation. It uses paired data on two ranked variables. The calculator: ranks X and Y from smallest to largest, averaging ties; computes d and d² for each pair; calculates ρ from the ranks using Pearson correlation; (optionally) shows Σd² and the classic formula if you tick the corresponding option. This explicit ranking table is often required in reports, lab assignments and exam solutions, because it shows each intermediate step. Spearman vs Pearson correlation Pearson correlation measures the strength of a linear relationship between two continuous variables, under assumptions of approximate normality and homoscedasticity. It is sensitive to outliers and to departures from linearity. Spearman's correlation instead: is based on ranks rather than raw values; detects monotonic (always increasing or always decreasing) relationships, even if non-linear; is robust to outliers in the original scale; is appropriate for ordinal variables (e.g. 1–5 satisfaction scores). When to prefer Spearman's ρ Survey research with Likert-type response scales; Relationships that look curved but monotonic in a scatterplot; Presence of a few extreme values that might unduly influence Pearson's r; Any situation where parametric assumptions of Pearson correlation are doubtful. Limitations and good practice Spearman's ρ only captures monotonic relationships: if the relationship is strongly non-monotonic, ρ may be near 0 even when there is structure. Very small sample sizes (e.g. n < 8) can yield unstable coefficients and unreliable inference. For formal hypothesis testing (p-values, confidence intervals), use dedicated statistical software or libraries that implement exact or high-precision methods. Related Core Math & Statistics tools Statistics overview Correlation coefficient (Pearson's r) t-test calculator p-value calculator Kruskal–Wallis test Interquartile range (IQR) z-score calculator Sample size calculator Core Math & Algebra hub Explore more core math and algebra calculators in this family. Variance & standard deviation Mean calculator One-way ANOVA Central limit theorem visualizer All Core Math & Algebra tools → Tip for reporting results In academic and professional reports, Spearman's ρ is often reported as: ρ = 0.74, n = 24, t(22) = 4.5 Optionally add a p-value from a statistical package and a brief interpretation in context.