1.3 - Spearman's Rho

A relationship between co-variables

Spearman's rho measures the direction and strength of a monotonic relationship between two rankable co-variables.

In a correlational study, every person supplies a pair of measurements, such as weekly rehearsal hours and recall score. The aim concerns a relationship, rather than a difference between conditions. Both measurements must be at least ordinal so their order is meaningful. Interval or ratio measurements can also be ranked. Different people's measurement pairs should be independent; repeatedly measuring one person does not create independent participants.

Spearman does not require a normal distribution or a straight-line relationship. It assesses ordering: if higher rehearsal hours tend to accompany higher recall, rho is positive. If higher hours accompany lower recall, rho is negative. A curved but consistently increasing pattern can therefore have a strong rho. A U-shaped relationship can have a small rho despite a real pattern, because its direction changes.

Rho ranges from −1 to +1. +1 is perfect positive rank agreement, −1 perfect reverse ordering, and values near zero indicate little monotonic association. Strength concerns distance from zero, so −0.9 is stronger than +0.4. A value of zero does not rule out all possible relationships.

Rank each column while preserving pairs

Rank the two variables separately, and keep each person's two ranks on the same row.

Do not pool rehearsal hours and recall scores into one list: they have different meanings and possibly different units. Give the lowest score rank 1 in both columns. Ranking highest as 1 in both columns is also consistent, but reversing only one column reverses the correlation's sign incorrectly. Sort copies if helpful; never detach a participant's X value from their Y value.

This original dataset has no ties:

The differences between ranks, not differences between the raw hours and scores, are squared. Here $\sum d^2=2$. There are six pairs, so $N=6$, rather than 12 individual measurements. A zero rank difference is retained: unlike Wilcoxon, a person with matching ranks contributes a valid zero to the sum and still counts in N.

If equal X scores occupy positions 2 and 3, each receives rank 2.5. Handle Y ties separately using their own occupied positions. Ties do not cause whole people to be discarded. The ranks still total $N(N+1)/2$ in each column.

Substitute into the supplied formula

With no tied scores, the usual shortcut calculates rho from the sum of squared rank differences.

For the worked example:

To three significant figures, report $r_s=+0.943$. To two decimal places it is +0.94; these are different precision instructions. Keep the unrounded result for subsequent comparisons where possible. Rho is dimensionless: neither hours nor recall points are its units. A result outside −1 to +1 indicates an arithmetic or ranking error.

Tied ranks require care. Averaging tied ranks is correct, but this shortcut does not make a full tie correction. The general Spearman coefficient is the Pearson correlation of the two rank columns, using their average ranks. A suitable tie-aware formula or software can calculate it. For example, paired ranks X=(1, 2.5, 2.5, 4) and Y=(1, 2, 3, 4) give a tie-aware rho of approximately 0.949; simply putting their $\sum d^2=0.5$ into the untied shortcut gives 0.950, a slightly different answer. Do not describe the shortcut as exact with ties.

In a supplied task, follow the stated equation and precision, and identify any declared approximation. These lesson calculation tasks use untied data; a tied-data inference task must supply an appropriate tie-aware result or support rather than demanding an untaught software calculation.

Use magnitude, direction and the correct table

Spearman significance depends on the number of pairs, hypothesis direction and selected threshold, while correlation strength and causality are separate questions.

This supplied OCR workbook extract is for a two-tailed test with untied data. Entries are positive critical magnitudes:

For the example, $N=6$. The magnitude of rho is about 0.943, exceeding 0.886, so reject the null of no monotonic association at the 5% level. It is below 1.000, so it does not meet the supplied 1% threshold. State the positive direction: students reporting more rehearsal generally had higher recall scores.

The two-tailed rule is absolute observed rho greater than or equal to critical magnitude. If rho were reported as −0.943 with $N=6$, its magnitude 0.943 would exceed the 5% cutoff too; comparing the signed negative number directly with +0.886 would produce the wrong decision. For a one-tailed positive prediction, the observed association must also be positive; a strong negative result does not support that prediction. Choose direction before examining the results.

Statistical significance is evidence against the null model, not the probability that the null is false. A strong sample correlation may fail to reach significance when N is small. A weaker correlation may reach significance with many pairs. Use the table to decide significance and the coefficient to describe sample strength, rather than equating the two.

The rehearsal study did not manipulate hours. Motivation might affect both rehearsal and recall, or successful recall might encourage more rehearsal. Consequently the association alone cannot establish that rehearsal caused better recall. If the result is non-significant, report insufficient evidence of a monotonic association rather than proving that no relationship of any kind exists. Ties also alter exact-table assumptions: a real analysis needs suitable tie-aware significance support.