Spearman’s Rank Correlation Coefficient

Introduction

Spearman’s Rank Correlation Coefficient, often symbolised as ρ, is a non-parametric measure of statistical gauging the strength and direction of monotonic relationships between paired data.
This measure is named after the British psychologist Charles Spearman.
Unlike Pearson’s correlation, Spearman’s is used when the relationship between the variables is non-linear and the variables measured are ordinal.

Spearman’s rank correlation coefficient ρ is calculated using this formula: ρ = 1- [(6∑d_i^2) / (n(n^2 - 1))].
In this formula, d represents the differences between the ranks of two variables, and n is the number of observations.
The result will always fall between -1 and 1, inclusive.
A positive ρ indicates a positive relationship between the variables; as one increases, the other also increases.
A negative ρ indicates a negative relationship between the variables; as one increases, the other decreases.
The absolute value of ρ indicates the strength of the relationship, with values closer to 1 or -1 indicating a stronger correlation.

Spearman’s Rank Correlation Coefficient can be applied to ordinal variables and does not require the assumption of normality in the dataset.
It is used when we cannot assure the relationship between variables is linear and variables may not be normally distributed.
An important assumption is that there are no tied ranks.

This coefficient is commonly used in fields such as psychology, education, and social sciences to analyse human behaviour.
It’s most often used when determining the correlation between variables that are non-linearly related or if data has outliers that could skew results of a Pearson’s correlation.
It can be central to hypothesis testing or to establish the validity of experimental results.

For example, one might use Spearman’s rank correlation coefficient to test the relationship between people’s subjective rankings of a public speaker (from 1st place to 100th place) and the number of pauses the speaker made during their speech. Regardless of whether the relationship is linear or not, a ρ close to 1 might suggest that the number of pause a speaker made has a strong positive influence on viewers’ perception of them.