Spearman’s Rank Correlation Coefficient

Maths: Statistics for machine learning

2 min read

Published Oct 22 2025, updated Oct 23 2025

Machine LearningMathsNumPyPandasPythonStatistics

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

ρ value	Relationship	Description
+1.0	Perfect positive monotonic	As X increases, Y always increases
+0.7 to +0.9	Strong positive	High ranks of X → high ranks of Y
0	No relationship	No monotonic pattern
–0.7 to –0.9	Strong negative	High X → low Y
–1.0	Perfect negative monotonic	As X increases, Y always decreases

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 np

from scipy.stats import spearmanr

import seaborn as sns

import matplotlib.pyplot as plt

import pandas as pd

# Example data: non-linear monotonic relationship

x = np.arange(1, 11)

y = np.log(x) * 10 + np.random.normal(0, 0.5, 10) # log relationship (non-linear but monotonic)

# Spearman's correlation

rho, 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.")

# Visualize

df = 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: 0.982

P-value: 0.0000

Reject 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

Feature	Pearson’s r	Spearman’s ρ
Relationship type	Linear	Monotonic (increasing/decreasing)
Data type	Continuous	Ordinal or continuous
Assumes normality?	Yes	No
Sensitive to outliers?	Yes	No
Uses	Raw values	Ranked values
Test type	Parametric	Non-parametric

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)