Correlation Coefficient Calculator
Paste two lists of numbers to find Pearson’s r, the strength of the relationship and a scatter summary.
Calculate Pearson’s r
Your results
Paired values and deviations
Each pair with its distance from the mean of X and the mean of Y, and the product of the two deviations. Positive products push r up, negative products push it down.
How the correlation coefficient works
This free calculator finds Pearson’s correlation coefficient for any two lists of numbers. It is for students, teachers, researchers and anyone checking whether two things move together. There is no sign-up, and everything runs in your browser, so your data never leaves your device.
What is the correlation coefficient?
The correlation coefficient, written as r, is a number between minus 1 and plus 1 that measures how strongly two variables move together. A value near plus 1 means they rise together, a value near minus 1 means one rises as the other falls, and a value near 0 means there is no straight-line relationship between them.
As a rough guide, an absolute value below 0.2 is very weak, 0.2 to 0.4 is weak, 0.4 to 0.6 is moderate, 0.6 to 0.8 is strong, and above 0.8 is very strong. The sign tells you the direction, and the size tells you the strength.
How is Pearson’s r calculated?
Pearson worked out the formula in 1895, building on earlier work by Francis Galton. For each pair, subtract the mean from each value to get the deviations, multiply the two deviations together, and add the products. Divide that total by the square root of the two sums of squared deviations multiplied together. The result is r.
In symbols, r equals the sum of (x minus mean of x) times (y minus mean of y), divided by the square root of the sum of (x minus mean of x) squared times the sum of (y minus mean of y) squared.
What does a correlation of 0.8 mean?
A correlation of 0.8 is a strong positive relationship, which means the two variables usually rise together. The value of r squared is 0.64, so about 64 percent of the variation in one variable is explained by the other in a straight-line model. It does not mean one variable causes the other, only that they move together.
For the example in this calculator, X is 1, 2, 3, 4, 5 and Y is 2, 4, 5, 4, 5. The correlation is 0.7746, which is a strong positive relationship, and r squared is 0.60.
Does correlation prove causation?
No, correlation never proves causation on its own. Two variables can move together because one causes the other, because a third factor drives both, or purely by coincidence. Ice cream sales and drownings rise together in summer, but the real cause is hot weather. Always look for a plausible mechanism and controlled evidence before claiming a cause.
A famous example is the strong correlation between the number of films Nicolas Cage appears in and the number of people who drown in swimming pools each year. Both series moved together for a while, yet nobody believes one causes the other.
What is r squared?
R squared is the square of the correlation coefficient, and it tells you what share of the variation in one variable is explained by the other. An r of 0.7746 gives an r squared of 0.60, so about 60 percent of the variation is explained. A higher value means a tighter straight-line fit.
The table below was worked out with the same formulas this calculator uses. A perfect straight line gives r of 1, while a falling line gives a negative r.
| Example | X values | Y values | Pearson r | Reading |
|---|---|---|---|---|
| Perfect rise | 1, 2, 3, 4, 5 | 55, 60, 65, 70, 75 | 1.00 | Perfect positive |
| Mostly rising | 1, 2, 3, 4, 5 | 50, 65, 60, 75, 70 | 0.82 | Very strong positive |
| Mostly falling | 1, 2, 3, 4, 5 | 70, 60, 65, 55, 50 | -0.90 | Very strong negative |
Frequently asked questions
What if my X and Y lists have different lengths?
Every X value needs a matching Y value, so the lists must have the same count. Add or remove values until they match, and the calculator will show the result.
What happens if one list never changes?
If all the X values are the same, or all the Y values are the same, Pearson’s r is not defined, because there is no variation to compare. The calculator will tell you the correlation cannot be worked out.
What is the difference between Pearson and Spearman correlation?
Pearson measures straight-line relationships using the actual values, while Spearman ranks the values first and measures whether one rises as the other rises, even on a curve. Use Pearson for linear patterns and Spearman for ranked or curved patterns.
What does a negative correlation mean?
A negative correlation means that as one variable goes up, the other tends to go down. For example, more hours of sleep the night before an exam is often linked with fewer mistakes, which is a negative relationship between sleep and errors.
How many pairs do I need?
You need at least two pairs for the formula to run, but a correlation from two or three pairs is almost meaningless. Aim for at least 10 pairs, and more is better, because small samples can show a strong r by pure chance.
Can one outlier change the correlation?
Yes, a single extreme pair can pull r sharply up or down, especially in a small sample. Always look at the paired-values table and check whether one pair is doing most of the work.
Is my data sent anywhere?
No. This calculator runs entirely in your browser. Nothing you paste is uploaded, stored or shared with anyone.
More statistics tools
Working with averages? Try the Mean Median Mode Calculator. Working with chance? Try the Probability Calculator. Ranking a score? Try the Percentile Calculator. For spread in more depth, see the Standard Deviation Calculator, and for standardised scores use the Z-Score Calculator, the P-Value Calculator and the Confidence Interval Calculator.
