# 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.

## Definitions and Properties

• 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.

## Assumptions

• 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.

## Applications

• 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.

## Practical Example

• 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.