🧠 Introduction
In many real-life situations, the relationship between two variables is not perfectly linear but still shows a consistent pattern — for example, as study time increases, student rank improves, or as stress level rises, job satisfaction decreases.
In such cases, Pearson’s correlation (which assumes linearity and normally distributed data) may not be the best measure.
That’s where Spearman’s rank correlation coefficient comes into play — a non-parametric measure that evaluates how well the relationship between two variables can be described by a monotonic function.
📘 What is Spearman’s Rank Correlation Coefficient?
The Spearman’s rank correlation coefficient, denoted by ρ\rhoρ (rho) or sometimes rsr_srs, measures the strength and direction of the monotonic relationship between two ranked variables.
It is calculated by comparing the ranks of the data rather than their raw values.
This makes it robust to outliers and suitable for ordinal or non-normally distributed data.
🧮 Formula
If there are no tied ranks, Spearman’s correlation coefficient is computed as: ρ=1−6∑di2n(n2−1)\rho = 1 – \frac{6 \sum d_i^2}{n(n^2 – 1)}ρ=1−n(n2−1)6∑di2
where:
- di=R(xi)−R(yi)d_i = R(x_i) – R(y_i)di=R(xi)−R(yi) = difference between the ranks of XXX and YYY
- nnn = number of observations
If tied ranks are present, the formula becomes more complex, and the computation is generally done using the Pearson correlation on the ranks of the data. ρ=Cov(RX,RY)σRXσRY\rho = \frac{\text{Cov}(R_X, R_Y)}{\sigma_{R_X} \sigma_{R_Y}}ρ=σRXσRYCov(RX,RY)
💡 Understanding the Concept of Ranks
Before computing Spearman’s correlation, each value of XXX and YYY is converted to its rank.
For example, the smallest value gets rank 1, the next gets rank 2, and so on.
If two or more values are tied, each tied value receives the average of their ranks.
📊 Example
Let’s consider an example of students’ ranks in two subjects:
| Student | Math Rank (X) | Physics Rank (Y) |
|---|---|---|
| A | 1 | 2 |
| B | 2 | 1 |
| C | 3 | 4 |
| D | 4 | 3 |
| E | 5 | 5 |
Now, compute the difference di=Xi−Yid_i = X_i – Y_idi=Xi−Yi and di2d_i^2di2:
| Student | X | Y | did_idi | di2d_i^2di2 |
|---|---|---|---|---|
| A | 1 | 2 | -1 | 1 |
| B | 2 | 1 | 1 | 1 |
| C | 3 | 4 | -1 | 1 |
| D | 4 | 3 | 1 | 1 |
| E | 5 | 5 | 0 | 0 |
∑di2=4\sum d_i^2 = 4∑di2=4 n=5n = 5n=5
Now plug into the formula: ρ=1−6×45(25−1)=1−24120=0.8\rho = 1 – \frac{6 \times 4}{5(25 – 1)} = 1 – \frac{24}{120} = 0.8ρ=1−5(25−1)6×4=1−12024=0.8
✅ Interpretation: There is a strong positive association between Math and Physics ranks.
🎯 Range and Interpretation
| ρ Value | Interpretation |
|---|---|
| +1 | Perfect positive correlation (ranks move together) |
| +0.7 to +0.9 | Strong positive correlation |
| +0.3 to +0.6 | Moderate positive correlation |
| 0 | No correlation |
| -0.3 to -0.6 | Moderate negative correlation |
| -0.7 to -0.9 | Strong negative correlation |
| -1 | Perfect negative correlation (inverse ranks) |
⚙️ When to Use Spearman’s Correlation
| Scenario | Use Spearman’s ρ |
|---|---|
| Data are ordinal | ✅ Yes |
| Relationship is non-linear but monotonic | ✅ Yes |
| There are outliers that distort Pearson’s r | ✅ Yes |
| Data are not normally distributed | ✅ Yes |
| You want to analyze ranks or rankings | ✅ Yes |
🧩 Pearson’s r vs Spearman’s ρ
| Aspect | Pearson’s r | Spearman’s ρ |
|---|---|---|
| Type of data | Continuous (interval/ratio) | Ordinal or continuous |
| Relationship type | Linear | Monotonic |
| Sensitivity to outliers | High | Low |
| Normality assumption | Required | Not required |
| Based on | Actual data values | Data ranks |
💻 Computing Spearman’s ρ in Python
import numpy as np
from scipy.stats import spearmanr
Sample data
math_ranks = np.array([1, 2, 3, 4, 5])
physics_ranks = np.array([2, 1, 4, 3, 5])
Compute Spearman’s correlation
rho, p_value = spearmanr(math_ranks, physics_ranks)
print(“Spearman’s ρ:”, round(rho, 3))
print(“P-value:”, round(p_value, 5))
Output:
Spearman’s ρ: 0.8
P-value: 0.104
The p-value indicates whether the observed correlation is statistically significant.
🧾 Advantages
✅ Works with ordinal data
✅ Handles non-linear monotonic relationships
✅ Resistant to outliers
✅ No need for normality assumption
⚠️ Limitations
❌ Cannot detect non-monotonic relationships (e.g., U-shaped patterns)
❌ Less powerful than Pearson’s r when a linear relationship exists
❌ Sensitive to large numbers of tied ranks
🔍 Real-World Applications
- Education: Comparing students’ ranks across subjects
- Finance: Ranking companies by performance and growth rate
- Medicine: Relationship between drug dosage ranks and recovery rates
- Psychology: Correlation between stress level ranks and happiness ranks
- Marketing: Ranking customer satisfaction vs brand loyalty
🧭 Final Thoughts
The Spearman’s rank correlation coefficient is a powerful and flexible measure for analyzing relationships when data do not meet the strict assumptions of Pearson’s correlation.
By focusing on ranks rather than raw data, it captures meaningful associations in real-world situations — where relationships are often not perfectly linear.
For data analysts, educators, and students preparing for advanced exams like IB, A-Level, or university statistics, understanding when and how to use Spearman’s ρ is an essential part of mastering bivariate analysis.
