How to calculate the correlation coefficient
Pearson's r divides the covariance of x and y by the product of their standard deviations, giving a value from −1 to +1. For x = 1, 2, 3, 4 and y = 2, 4, 6, 8 every point lies on a perfectly rising straight line, so r = 1. A value of 0 means no linear relationship. Squaring r gives the share of variation one variable explains, which is what the coefficient of determination calculator reports.
How to use this calculator
Enter up to four matching x and y pairs and leave unused rows blank. Both columns of a pair must be filled to count. The badge labels the strength and direction. For the underlying numerator, see the covariance calculator.
- At least two complete pairs are needed; with a single point there is no variation to compare.
- A constant column breaks the formula. If every x (or every y) is identical, its standard deviation is zero and r is undefined.
- Order the pairs consistently — x1 must belong with y1, or the association is scrambled.
Interpreting Pearson's r
Values near +1 mean strong positive association, near −1 strong negative, and near 0 little linear link. Correlation does not prove causation. For the precision of a sample mean, see the standard error calculator.
Frequently asked questions
- How do I calculate the correlation coefficient?
- Divide the covariance of x and y by the product of their standard deviations. For x = 1, 2, 3, 4 and y = 2, 4, 6, 8 the result is 1.
- What does a correlation of 1 mean?
- It means a perfect positive linear relationship: every point lies exactly on a rising straight line.
- Can r be negative?
- Yes. A negative r means that as one variable rises the other tends to fall, with −1 being a perfect inverse line.
- Does correlation imply causation?
- No. A high correlation shows association only; a third factor or coincidence can produce it.
- What if the relationship is curved rather than straight?
- Pearson's r only measures linear association, so a strong curved pattern can still give r near 0. Plot the points first; if they bend, consider Spearman's rank correlation or transform the data before trusting r.