What does Spearman's rho measure?

Spearman's rho (ρ) measures the strength and direction of the monotonic relationship between two variables — whether one tends to increase or decrease as the other does, without requiring linearity. It ranges from -1 (perfect negative monotonic) through 0 (no monotonic relationship) to +1 (perfect positive monotonic).

How is rho calculated?

Each dataset is ranked from lowest to highest (ties get the average rank). Rho is then the Pearson r of the two rank vectors. Equivalently, if all ranks are distinct, rho = 1 - (6 * sum(d²)) / (n * (n² - 1)), where d is the rank difference for each pair.

How is the p-value computed?

The test converts rho to a t-statistic using t = rho * sqrt((n-2) / (1 - rho²)) with n-2 degrees of freedom, then evaluates the two-tailed p-value via the regularized incomplete beta function — the same approach used by statistical software.

When should I use Spearman instead of Pearson?

Use Spearman when data are ordinal, when the relationship is monotonic but not linear, when outliers are present, or when you cannot assume the variables follow a normal distribution. Pearson r is more powerful when the linearity and normality assumptions are met.

How are tied ranks handled?

Ties receive the average of the ranks they would occupy. For example, if two values tie for 3rd and 4th place, both get rank 3.5. This is the standard mid-rank correction and is built into the calculator.

How many data points do I need?

You need at least 3 pairs (n ≥ 3) so that degrees of freedom n-2 ≥ 1. For reliable power the test typically needs 10 or more observations. Both variables must have the same number of values.

Spearman Rank Correlation Calculator

Name: Spearman Rank Correlation Calculator
Availability: InStock
Author: Nham Vu

Paste two datasets and instantly get Spearman's rho, p-value, and a rank-scatter plot — no install needed.

Enter Your Data

X Variable (comma-separated numbers)

Y Variable (same length as X)

Significance Level (α)

Enter data on the left and click Calculate to see results.

Rho (ρ)

—

p-value

—

t-stat

—

Rank Scatter Plot

Ranked X vs Ranked Y

Rank Table

#	X	Rank X	Y	Rank Y	d²

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Summary

Paste two datasets and instantly get Spearman's rho, p-value, and a rank-scatter plot — no install needed.

How it works

Enter the X variable values as comma-separated numbers in the first field.
Enter the matching Y variable values in the second field — order must correspond to the same observations.
Select a significance level (alpha): 0.10, 0.05, or 0.01.
Click "Calculate" to rank both datasets and compute Spearman's rho, t-statistic, and two-tailed p-value.
Read the significance verdict and inspect the rank scatter plot to visualize the monotonic relationship.
Copy the results summary to clipboard for use in a report or assignment.

Use cases

Test whether student rankings on two exams agree (ordinal agreement).
Measure the monotonic relationship between income level and life satisfaction ratings.
Assess correlation between two judges' ordinal scores in a competition.
Analyze the association between hours studied and test rank without assuming normality.
Explore whether customer satisfaction scores follow the same trend as net-promoter scores.
Quickly check rank correlation in small samples where Pearson's assumptions may not hold.
Compare survey Likert-scale responses between two groups.
Validate sensor readings that follow a monotonic but non-linear relationship.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu