# Pearson's product-moment correlation coeffecient

## 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.