# Pearson’s Product-Moment Correlation Coefficient

## Introduction

• Pearson’s Product-Moment Correlation Coefficient, denoted as r, is a statistical measure that calculates the strength and direction of the relationship between two variables.
• N.B: It is appropriate for linear relationships only.

## Definitions and Properties

• It is denoted as r, and the formula for calculation is r = ∑((x_i - x̄)(y_i - ȳ))/ √(∑(x_i - x̄)^2∑(y_i - ȳ)^2
• Where x_i and y_i are individual data points, is the mean of the x-values, and ȳ is the mean of the y-values.
• The range of r is between -1 and 1, inclusive.
• Positive r indicates a direct linear relationship, where increasing x values correspond to increasing y values.
• Negative r indicates an inverse linear relationship, where increasing x values correspond to decreasing y values.
• The absolute value of r indicates the strength of the relationship. Closer to 1 or -1 indicates a stronger relationship.

## Rules and Assumptions

• Pearson’s Product-Moment Correlation Coefficient measures only linear relationships. It may mislead for non-linear relationships.
• Both variables should be continuous and quantitative. They must not be categories or ranks.
• Outliers can greatly affect r, so it’s necessary to check your data for outliers before calculation.
• It assumes that each pair (x,y) is an independent observation, drawn from a bivariate normal distribution.
• The relationship doesn’t imply causation; so even if r is close to 1 or -1, it shouldn’t automatically be assumed that changes in one variable cause changes in the other.

## Applications

• Used in a broad range of fields including mathematics, statistics, physics, social sciences, and ecology.
• Enables data analysis for regression, enables prediction based on relationship between variables.
• Offers a quick initial view of whether an apparent relationship is potentially valid.

## Practical Example

• In a study looking at height and weight, you may find a positive correlation coefficient, suggesting that as height increases, weight does too. However, one can’t claim that height gains cause weight gains based on this correlation.

In conclusion, understanding Pearson’s Product-Moment Correlation Coefficient is crucial for analysing and interpreting various statistical scenarios, helping you draw meaningful conclusions from a set of data points.