---
title: "Correlation and the Product-Moment Coefficient"
description: "Summarise the strength and direction of a linear relationship in one number between -1 and 1, judge it against a critical value for the sample size, and keep it apart from any claim about cause."
canonical: https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ
source: https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ.md
retrieved: 2026-09-12
---

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# Correlation and the Product-Moment Coefficient

Summarise the strength and direction of a linear relationship in one number between -1 and 1, judge it against a critical value for the sample size, and keep it apart from any claim about cause.

Subject: Mathematics · Area: Data & Statistics · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ

## Ready when they can

- Describe what values of r near 0, 1 and -1 look like on a scatter plot
- Test a value of r against a critical value for the sample size
- Give a case where two correlated quantities share an outside cause

## Lesson: Reading r without being fooled

Start with the scatter plot, never the formula. Points that climb uphill from left to right mean a positive link. Points that slide downhill mean a negative one. The tighter they hug a straight line, the closer r sits to 1 or to minus 1. A shapeless cloud means r near zero.

A big r from a tiny sample may be luck. You test r against a critical value set by the sample size. With ten pairs, the bar at the 5 percent level is 0.632. Beat it and the link counts as significant. Larger samples face a lower bar, so always report n beside r.

A significant r still never proves cause. Two linked quantities often share an outside cause. Towns with high ice cream sales record more drownings, and hot weather drives both. Always ask what third factor could move the pair together.

**Tip.** Plot first and look before you compute. A strong curve can score r near zero. One wild point can fake a strong r. Describe the plot in words, then trust the number only if the plot backs it.

**Recap.** Read r from the plot, test it against its bar, and never let it claim cause.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Scatter Graphs & Correlation](https://lightmysky.com/learn/mathematics/scatter-graphs-and-correlation-mt_8atyuvPUZc)
- [Testing a Binomial Proportion](https://lightmysky.com/learn/mathematics/testing-a-binomial-proportion-mt_ZJ8MTLYYKE)

## Opens up

- [Linear Regression by Least Squares](https://lightmysky.com/learn/computing/linear-regression-by-least-squares-mt_QRRkpIrwWs)
- [The Regression Line and Making Predictions](https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE)
- [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF)