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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
Read r from the plot, test it against its bar, and never let it claim cause.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.