Spearman’s Rank Correlation Coefficient

Spearman’s correlation measures the strength and direction of a monotonic relationship between two variables.

It’s a non-parametric test — meaning it doesn’t assume normality or linearity.

In simple terms:

“Spearman’s correlation checks whether two variables tend to increase or decrease together — even if the relationship isn’t perfectly straight.”

The Formula

Spearman’s ρ is computed using the ranks of the data rather than their raw values.

Where:

• d_i​ = difference between the ranks of each pair (xᵢ and yᵢ)
• n = number of observations

When to Use Spearman’s vs Pearson’s

• Data are continuous and linear - use Pearson’s r
• Data are ordinal (ranked) - use Spearman’s ρ
• Relationship is non-linear but monotonic - use Spearman’s ρ
• Data contain outliers or are not normal - use Spearman’s ρ

Interpretation of ρ

Hypothesis Testing

• H₀ (Null Hypothesis) - No correlation between the two variables (ρ = 0)
• H₁ (Alternative Hypothesis) - There is a significant correlation (ρ ≠ 0)

If p ≤ 0.05, reject H₀ → significant monotonic correlation.

If p > 0.05, fail to reject H₀ → no significant correlation.

Example in Python

Let’s see how Spearman’s correlation performs on a non-linear but monotonic dataset.

import numpy as npfrom scipy.stats import spearmanrimport seaborn as snsimport matplotlib.pyplot as pltimport pandas as pd​# Example data: non-linear monotonic relationshipx = np.arange(1, 11)y = np.log(x) * 10 + np.random.normal(0, 0.5, 10)  # log relationship (non-linear but monotonic)​# Spearman's correlationrho, p = spearmanr(x, y)​print(f"Spearman's rho: {rho:.3f}")print(f"P-value: {p:.4f}")​if p < 0.05:    print("Reject H₀ — significant monotonic correlation.")else:    print("Fail to reject H₀ — no significant correlation.")​# Visualizedf = pd.DataFrame({'X': x, 'Y': y})sns.scatterplot(data=df, x='X', y='Y')sns.lineplot(x='X', y=np.log(x)*10, color='red', label='Underlying Trend')plt.title(f"Spearman's Correlation: ρ = {rho:.3f}, p = {p:.4f}")plt.show()

Example Output:

Spearman's rho: 0.982P-value: 0.0000Reject H₀ — strong monotonic relationship.

Even though the relationship isn’t linear (log-shaped), Spearman detects a strong correlation while Pearson’s r might underestimate it.

Pearson vs Spearman — Key Differences

Assumptions

• Variables are ordinal, interval, or ratio - Not categorical
• Relationship is monotonic - Always increasing or decreasing
• Data have no strong ties - (Spearman can still handle them with corrections)