Spearman Rank Correlation

Background

Spearman's rho is Pearson's correlation applied to ranks instead of raw values. It captures monotonic relationships and is the canonical calibration target for LLM-as-judge against human gold sets (production target $\rho > 0.8$).

Problem statement

Implement spearman_corr(x, y). Replace each value by its 1-indexed rank (averaging ranks for ties), then apply Pearson:

$$\rho \;=\; \frac{\sum_i (R_i^x - \bar{R}^x)(R_i^y - \bar{R}^y)}{\sqrt{\sum_i (R_i^x - \bar{R}^x)^2}\,\sqrt{\sum_i (R_i^y - \bar{R}^y)^2}}$$

Input

• x, y - array-like of floats, same length $\ge 2$.

Output

• float in $[-1, 1]$.

Examples

Input: x=[1,2,3,4,5], y=[2,4,6,8,10]
Output: 1.0       # perfectly monotonic

Input: x=[1,2,3,4,5], y=[5,4,3,2,1]
Output: -1.0

Constraints

• Use average ranks for ties (the standard convention).
• Lengths must match and be $\ge 2$. Raise ValueError otherwise.

Notes

• vs Pearson. Pearson is sensitive to outliers; Spearman is robust because it only uses ranks. Use Spearman whenever the relationship is monotonic but not linear.