In this article
You have two lists of numbers for the same ten people: hours studied and exam score. Do they move together? The Pearson correlation coefficient, written r, answers that in one number between −1 and +1. It is one of the most used statistics there is, and one of the most misunderstood.
What r actually measures
Pearson's r measures how close the points would sit to a straight line. Positive means both rise together. Negative means one rises as the other falls. The closer to 1 or −1, the tighter the points hug the line. Near zero, there is no straight-line pattern to speak of.
These cut-offs are a convention, not a law. In physics, 0.9 can be disappointing. In social science, 0.4 can be notable. What counts as strong depends on the field.
A worked example
Ten students record the hours they studied the week before an exam and their score out of 100.
| Student | Hours | Score |
|---|---|---|
| 1 | 1 | 48 |
| 2 | 2 | 52 |
| 3 | 2 | 55 |
| 4 | 3 | 60 |
| 5 | 4 | 62 |
| 6 | 4 | 70 |
| 7 | 5 | 68 |
| 8 | 6 | 78 |
| 9 | 7 | 82 |
| 10 | 8 | 88 |
Paste the two columns as pairs, for example 1, 48, one pair per line, into the calculator.
The result is r = 0.985. The points sit very close to a rising line. In rough terms each extra hour goes with about 5.7 extra marks in this small group.
r² : the share that lines up
Square r and you get r², sometimes called the coefficient of determination. It is the share of the variation in one list that lines up with the other through a straight line. Here r² = 0.985² ≈ 0.97, so about 97 percent of the spread in scores matches the spread in hours.
- Matches the hours studied97%
- Everything else (sleep, luck, topic, nerves)3%
'Lines up with' is careful wording on purpose. It describes the pattern in this group. It does not say the hours caused 97 percent of the result.
One outlier can break the number
Suppose student 10 studied for 8 hours but scored 30, perhaps because they fell ill on the day. Nothing else changes. Watch r:
One unusual point dragged r from strong to weak. This is why you should always plot your data before trusting r. A scatter plot would show that nine students follow a clear line and one does not.
r = 0 does not mean 'no relationship'
Pearson's r only detects straight-line patterns. Take x from −3 to 3 and let y = x². Every y is fully determined by x, yet r is exactly 0, because the left half of the curve cancels the right half.
- Straight line
- r captures it fully
- Easy to summarise
- Perfect relationship, but curved
- Up on one side, down on the other
- r sees nothing
Moral: r = 0 says there is no straight-line link. Curves, clusters and thresholds can all hide behind a small r.
Correlation is not causation
If two lists are correlated there are at least four stories that could explain it. The number cannot tell you which one is true.
The ice cream and drowning pair is the classic case of a third factor. Summer drives both. Neither causes the other.
Small samples fool you easily
With only a handful of pairs, a high r can appear by luck. The table shows roughly how large |r| must be before it would be unusual if the two lists were really unrelated, using the usual 5 percent test.
Standard critical values for Pearson's r. With 5 pairs even r = 0.8 is not convincing. With 100 pairs, r = 0.25 is statistically clear, though it may still be a weak, small effect.
A short checklist
- Plot first. Draw the scatter. Look for outliers, curves and separate clusters.
- Compute r. Use it as a summary of the straight-line part only.
- Check the sample size. Few pairs mean wide uncertainty.
- Think about a third factor. Ask what else could drive both lists.
- Claim carefully. Say 'is associated with', not 'causes', unless an experiment supports it.
Questions people ask
What does the correlation coefficient r tell you?
Pearson's r measures how closely two sets of numbers follow a straight line. It runs from −1 (perfect downward line) through 0 (no straight-line link) to +1 (perfect upward line).
What is a good correlation coefficient?
It depends on the field. A common rough guide calls 0.7 or above strong, 0.3 to 0.7 moderate and below 0.3 weak, but these cut-offs are conventions, not rules.
What is the difference between r and r squared?
r² is r multiplied by itself. It shows the share of the variation in one variable that lines up with the other through a straight line. An r of 0.98 gives an r² of about 0.97.
Does correlation mean causation?
No. Two things can be correlated because one causes the other, because the effect runs the other way, because a third factor drives both, or by chance.
Can r be zero even when two variables are related?
Yes. Pearson's r only detects straight-line patterns. A perfect U-shaped relationship such as y = x² can give r = 0.