Spearman Rank Correlation Calculator

Spearman Rank Correlation

Variable X Data Separate paired values with commas, spaces, or newlines

Please enter at least 3 valid numeric values for Variable X.

Variable Y Data Must match the number of observations in Variable X

Please enter at least 3 valid numeric values for Variable Y.

Hypothesis Direction

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Non-Parametric Correlation

What Is Spearman Rank Correlation?

Spearman's Rank Correlation Coefficient ($r_s$ or $\rho$) is a non-parametric statistical measure that assesses the strength and direction of the monotonic relationship between two ranked or continuous variables.

Unlike Pearson Correlation ($r$), which strictly evaluates linear relationships and assumes bivariate normality, Spearman's rho converts raw values into ranks ($R_i$). It remains robust to extreme outliers, non-linear monotonic curves, and ordinal Likert-scale data.

When to Use It

Non-Normal Continuous Data: Evaluating associations when variables violate normality or contain outliers.
Ordinal & Survey Scales: Correlating Likert items, customer satisfaction scores, or competition rankings.
Monotonic Non-Linear Trends: Measuring relationships where $Y$ consistently increases (or decreases) with $X$ without requiring a straight line.
Non-Parametric Alternative: Employing a robust alternative when Pearson's linearity assumptions fail.

The Mathematical Formulas

$$\text{Standard Formula (No Ties): } r_s = 1 - \frac{6 \sum_{i=1}^n d_i^2}{n(n^2 - 1)}$$ $$\text{General Formula (With Ties): } r_s = \frac{\operatorname{Cov}(R_X, R_Y)}{\sigma_{R_X} \sigma_{R_Y}}$$ $$t = r_s \sqrt{\frac{n - 2}{1 - r_s^2}}, \quad df = n - 2$$

Where $d_i = R(x_i) - R(y_i)$ represents the difference between the assigned ranks of observation $i$, and $t$ tests the significance against the null hypothesis $H_0: \rho = 0$.

Worked Example

Scenario: 10 students are evaluated on Exam 1 ($X$) and Exam 2 ($Y$) with pairs: $[(56,66), (75,70), (45,40), (71,60), (62,65), (64,56), (58,59), (80,77), (76,67), (61,63)]$.

Step 1: Rank Observations Separately:
$R_X = [2, 8, 1, 7, 5, 6, 3, 10, 9, 4]$
$R_Y = [7, 9, 1, 4, 6, 2, 3, 10, 8, 5]$

Step 2: Calculate Rank Differences ($d_i$) & Sum of Squares ($\sum d_i^2$):
$d = [-5, -1, 0, 3, -1, 4, 0, 0, 1, -1]$
$\sum d_i^2 = 25 + 1 + 0 + 9 + 1 + 16 + 0 + 0 + 1 + 1 = 54.0$.

Step 3: Compute Spearman's Rho ($r_s$):
$r_s = 1 - \frac{6(54)}{10(100 - 1)} = 1 - \frac{324}{990} = 1 - 0.3273 = 0.6727$.

Step 4: Hypothesis Testing ($t$-statistic & $p$-value):
$t = 0.6727 \sqrt{\frac{8}{1 - (0.6727)^2}} = 2.5717, \quad df = 8 \implies p = 0.0330$.

Conclusion: Since $p = 0.0330 \le 0.05$, reject $H_0$. There is a statistically significant positive monotonic correlation.

Correlation Computation Complete

Spearman Correlation Results

Spearman Rank Correlation Coefficient

rₛ = 0.6727

Statistically Significant Moderate-to-Strong Positive Monotonic Association (p = 0.0330)

Spearman's Rho (rₛ) 0.6727

t-Statistic 2.5717

p-value 0.0330

Sample Size (n) 10

Degrees of Freedom df = 8

Sum of d² (Σdᵢ²) 54.00

Rank Assignment & Squared Differences Table

Observation (#)	Value X	Rank R(X)	Value Y	Rank R(Y)	Difference dᵢ = R(X) - R(Y)	Squared Diff (dᵢ²)

Spearman Rank Correlation Calculator