What r actually measures
Pearson's correlation coefficient, written r, measures the strength and direction of the straight-line relationship between two variables. It lives between −1 and 1. Values near 1 mean the points climb together along a line; near −1 they fall together; near 0 there is no linear pattern at all. The construction is a ratio: the covariance of the two variables divided by the product of their standard deviations. Dividing by the standard deviations strips out the units, so r is the same whether you measure height in centimetres or inches.
Five students log study hours (2, 3, 4, 5, 6) and score 50, 55, 60, 68, 72. The covariance is 14.25, the standard deviations are 1.581 and 9.055, and 14.25 ÷ (1.581 × 9.055) = r = 0.995. That is about as strong as real data gets: hours and marks climb almost in lockstep. It still does not prove the hours caused the marks. Motivated students might both study more and score higher for other reasons.
Reading the size of r
A widely used reading scale: 0.00 to 0.10 negligible, 0.10 to 0.39 weak, 0.40 to 0.69 moderate, 0.70 to 0.89 strong, 0.90 and up very strong. Treat the bands as a rough vocabulary. A correlation of 0.3 can matter a lot in medicine and mean nothing in physics; context decides. Two things do not depend on context. First, r only sees straight-line patterns: a perfect U-shaped relationship can return r near 0. Second, one extreme point can drag r far from where the rest of the cloud sits. This is why the calculator plots your data next to the number.
Quoting r without looking at the scatter plot. Anscombe's famous four datasets all share r = 0.816, and one is a clean line, one is a curve, one is a line with a single wild outlier, one is vertical with one distant point. Same number, four different stories.
Correlation and its square
Square r and you get R², the share of the variation in one variable accounted for by its linear relationship with the other. Here 0.995² = 0.991: 99.1% of the variation in marks is accounted for by hours, in the regression sense. For the test of whether r differs from zero, and for regression slopes, see the linked tools.
Frequently asked questions
What is a good correlation coefficient?
It depends on the field. In physics, 0.9 can be disappointing; in psychology, 0.3 is a solid finding. The bands on this page are a vocabulary, not a grading scheme.
Can correlation be bigger than 1?
No. The formula divides covariance by the product of the standard deviations, and the Cauchy-Schwarz inequality keeps that ratio inside [-1, 1]. A value outside means a computation error.
Does a correlation of 0 mean no relationship?
It means no linear relationship. Curved relationships (a parabola, a circle) can score exactly 0 while being perfectly predictable. Plot the data.
What is the difference between r and R-squared?
r is the correlation itself, signed, between -1 and 1. R-squared is its square in simple regression: an unsigned share of variation, between 0 and 1.