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.
More to read
13.1 The Correlation Coefficient r
The correlation coefficient r squeezes a whole scatter plot into one number between -1 and 1. Sign gives direction: positive climbs uphill, negative slides downhill. Size gives tightness: near the ends the points hug a line, near zero they spray everywhere. Train your eyes first by guessing r from plots, then compute.
13.2 Testing the Significance of the Correlation Coefficient
A big r from a tiny sample may be pure luck, so tests compare r against a critical value for the sample size. Beat the bar and the correlation counts as significant at your level. Larger samples face a lower bar because flukes average out. Always report n beside r, since the same r convinces with thirty points and shrugs with six. Significance asks whether the pattern is real, not whether it is strong.
12.2 Scatter Plots
Every correlation argument starts with a scatter plot, never with a formula. Plot first and look: uphill drift, downhill drift, curve, clusters, or one wild point dragging everything. A strong curve can score r near zero, and a single outlier can fake a strong r. The plot shows what the number means and warns when the number lies. Draw it, describe it in words, and only then compute.
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2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 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.