Skip to content
Hard

The Correlation Coefficient

Simple Explanation

The correlation coefficient, r, is a single number between βˆ’1 and 1 that precisely measures both the direction and strength of a linear relationship. r close to +1: strong positive correlation. r close to βˆ’1: strong negative correlation. r close to 0: little or no linear relationship.

Why Do We Need It?

A scatter diagram alone is subjective β€” the correlation coefficient gives an exact, comparable number, letting you say precisely HOW strong a relationship is, not just its rough direction.

Formula

Pearson's Correlation Coefficient

r = Ξ£(x βˆ’ xΜ„)(y βˆ’ Θ³) / √[Ξ£(x βˆ’ xΜ„)Β² Β· Ξ£(y βˆ’ Θ³)Β²]

A single number, always between βˆ’1 and 1, that measures both the strength and direction of the linear relationship between two variables.

r
β€” the correlation coefficient
x, y
β€” the paired data values for each variable
xΜ„, Θ³
β€” the means of the x-values and y-values respectively

When to use it: Whenever you need a precise numerical measure of how strongly two variables are linearly related, beyond just a visual scatter diagram impression.

Worked Example

Interpret a correlation coefficient

A study finds r = βˆ’0.85 between hours of TV watched and exam scores. Interpret this value.

    Why Does This Work?

    The formula's numerator, Ξ£(xβˆ’xΜ„)(yβˆ’Θ³), is positive when x and y tend to be above/below their means together (both high or both low at the same time β€” positive relationship) and negative when they move oppositely β€” the denominator simply rescales this to always land between βˆ’1 and 1, regardless of the original units or the size of the dataset.

    Real-Life Example

    Quantifying the strength of a marketing effect

    A company wants to know precisely how strongly advertising spending predicts sales, not just whether a rough trend exists.

    Computing r gives an exact, comparable strength value, letting the company compare this relationship's strength against other business factors on the same βˆ’1 to 1 scale.

    Practice

    Which value of r represents the STRONGEST correlation?

    Hard

    Common mistake

    Assuming a negative r means a "weak" or "bad" correlation β€” the SIGN of r shows direction only; the STRENGTH is shown by its magnitude, |r|, regardless of sign.

    Quick Review

    • r ranges from βˆ’1 to +1.
    • Sign shows direction (positive/negative); magnitude |r| shows strength.
    • |r| close to 1: strong relationship. |r| close to 0: weak or no linear relationship.
    ← Back to Statistics