Pearson’s product-moment correlation coeffecient

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, x̄ 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.