Spearman Rank Correlation#

The Spearman rank correlation (also: Spearman's rho, $r_s$ ) is a non-parametric measure of the strength and direction of the monotonic relationship between two variables. Instead of raw values, the ranks of the observations are used.

When to Use#

Use the Spearman correlation when you want to:

Measure the monotonic relationship (not necessarily linear) between two variables
The data are at least ordinal scaled
The assumptions of the Pearson correlation are not met (e.g., no normal distribution)
Outliers are present that could bias the results

Assumptions#

Both variables are at least ordinal scaled
Monotonic relationship between the variables (does not need to be linear)
Independence of observation pairs
No tied ranks – if ties are present, a correction is applied

Formula#

The Spearman rank correlation is calculated using rank differences:

r_s = 1 - \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 - 1)}

where $d_i = \text{Rank}(x_i) - \text{Rank}(y_i)$ is the difference in ranks for each observation pair.

Alternatively, Spearman's $r_s$ can be calculated as the Pearson correlation of the ranks:

r_s = r_{\text{Pearson}}(\text{Rank}(X), \text{Rank}(Y))

Example#

Practical Example: Customer Satisfaction and Return Intention

A market researcher investigates the relationship between customer satisfaction (Likert scale: 1–5) and return intention (Likert scale: 1–5) in a restaurant. Both variables are ordinal scaled.

Variable X: Customer satisfaction (1 = very dissatisfied, 5 = very satisfied)
Variable Y: Return intention (1 = very unlikely, 5 = very likely)

The Spearman correlation yields $r_s$ = 0.65, p < 0.001. There is a moderate to strong positive monotonic relationship: More satisfied customers show a higher intention to return.

Effect Size#

The Spearman coefficient $r_s$ is itself a measure of effect size and is interpreted analogously to the Pearson coefficient:

| Effect Size | | $r_s$ | | |---|---| | Small | 0.10 | | Medium | 0.30 | | Large | 0.50 |

Advantage over Pearson: The Spearman correlation is more robust to outliers and does not require a linear relationship – a monotonic relationship is sufficient.