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Do Two Things Move Together?
When study hours rise, do test scores rise too? When temperature rises, do ice cream sales rise? The Pearson correlation coefficient, called r, answers this kind of question with a single number between −1 and +1. A value near +1 means the two variables rise together in a straight-line pattern. A value near −1 means one rises as the other falls. A value near 0 means no straight-line pattern at all. The square of r, written r², tells you what share of the variation in one variable is linearly tied to the other.
Correlation is one of the most used and most misused numbers in science and business. It is powerful because it summarises a whole cloud of points in one number, and it is dangerous for exactly the same reason: one number cannot tell you whether the pattern is real, whether it is straight, or whether one thing causes the other. This tool computes r correctly; interpreting it wisely is still your job.
How to Use This Calculator
Paste one x, y pair per line, separated by a comma or a space, then press Calculate Pearson r. You need at least three pairs, and both columns must vary: a constant column has no correlation, because there is nothing to move with anything. The tool shows r, the two means, the full formula with your sums filled in, a plain reading of strength and direction, and r².
Strength is read in rough bands: below 0.2 is very weak, 0.2 to 0.4 is weak, 0.4 to 0.7 is moderate, and above 0.7 is strong. These bands are conventions, not laws. In physics, 0.9 might count as weak; in social science, 0.4 might count as strong. Always judge strength against what is normal in your field.
A Worked Example
Take seven pairs: (1, 2), (2, 4), (3, 5), (4, 4), (5, 6), (6, 8), (7, 9). The means are 4 and about 5.43. Plugging the sums into the formula gives r ≈ 0.95, a strong positive correlation, and r² ≈ 0.90, meaning about 90% of the variation in y is linearly tied to x. Try it in the tool and check the sums. Notice the pair (4, 4): it sits below the trend, but one off-trend point does not break a strong correlation.
Now flip the second column: (1, 9), (2, 8), and so on. The correlation flips sign to about −0.95. Same strength, opposite direction. That symmetry is a good check that you understand the sign: r measures direction as well as strength.
The Formula, in Plain Parts
Pearson r divides the covariance of x and y by the product of their standard deviations. In plain terms, the numerator adds up, for each point, how far x sits from its mean times how far y sits from its mean. Points where both sit on the same side of their means push the sum up; points where they sit on opposite sides push it down. The denominator rescales that sum so the result always lands between −1 and +1, whatever the units of x and y. That rescaling is why r is unit-free: measuring x in metres or centimetres gives the same r.
Because r only measures straight-line association, a perfect curve can have r near zero. If y = x² for x from −3 to 3, the points form a U shape, and Pearson r is 0, even though x determines y exactly. Always plot your data before trusting r. A scatter plot takes seconds and catches what r cannot.
Correlation Is Not Causation
The most important sentence in this whole article is this: correlation does not prove cause and effect. Ice cream sales and drowning deaths correlate strongly, because both rise in summer. Neither causes the other; heat causes both. Confounding variables, reverse causation, and coincidence all produce correlations, and only careful study design, not a bigger r, can separate them.
Outliers deserve special suspicion. A single extreme point can drag r from weak to strong or flip its sign. If removing one point changes your conclusion, your conclusion rests on that point, and you should say so. The tool computes r for the data you paste, so check your data for outliers before you quote the result.
Common Mistakes
Beyond the causation trap, common mistakes include computing r on fewer than a handful of points, where any value is possible by chance; computing r when one variable is constant, which is undefined; and quoting r without n, the number of pairs. An r of 0.9 from five points is far weaker evidence than an r of 0.9 from five hundred. Always report n alongside r.
Another mistake is treating r as a percentage. An r of 0.5 does not mean 50% related. The percentage-like number is r²: 0.25, or 25% of the variation linearly tied. The tool reports both so the distinction stays visible.
Try It Yourself
Paste a data set you care about, perhaps hours studied against scores, or advertising spend against sales, and read the strength, direction and r² together. Then plot the points by hand or in a spreadsheet and check whether the picture matches the number. When the picture and the number agree, you have a result worth reporting. When they disagree, trust the picture and investigate.
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