Pearson correlation measures how closely two variables move together on a scale from minus one to plus one. It is the covariance divided by the product of the two standard deviations, which strips out the units and leaves a pure measure of linear association.
Two cautions apply throughout. First, r only detects straight-line relationships: a perfect parabola can return a correlation of zero. Second, and more famously, correlation is not causation — two series can move together because one drives the other, because something else drives both, or by sheer coincidence over a short window. The coefficient of determination, r squared, is often the more honest figure to quote, since it states the share of variance actually accounted for.
Below the calculator sits the what each value of r accounts for table: 8 rows — 0.1 at one end, 0.99 at the other — which is usually faster than entering a value you have seen before. The page opens with x values already set to 1, 2, 3, 4, 5, 6, 7, 8, so there is a finished result on screen before anything is typed. The 4 bands (strong negative at one end, strong positive at the other) exist to answer the question a number on its own cannot: is this high, low or ordinary? Independent Events Probability Calculator and Margin of Error Calculator are the nearest neighbours to this one.
The formula
x, y- The paired observations
x̄, ȳ- The two means
r- Pearson correlation coefficient
r²- Share of variance explained
How it works, step by step
- Paste your x values, separated by commas or spaces.
- Paste the matching y values in the same order.
- The gauge shows Pearson r.
- The r squared value and best-fit slope appear below.
Worked examples
Eight near-linear pairs
The correlation is 0.999143, so r squared is 0.998286 — about 99.83% of the variance in y is explained. The best-fit slope is 2.0095.
Where r misses the pattern
For x = −3 to 3 against y = x squared, the covariance sum is exactly 0, so r is 0 despite a perfect deterministic relationship. Pearson only sees straight lines.
Checking the 0.8 row
The reference table lists Pearson r = 0.8, giving r squared: 0.6400, Variance explained: 64.00%, Variance unexplained: 36.00%. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
What counts as a strong correlation?
It depends on the field. Physics might treat 0.95 as poor while social science treats 0.4 as notable. Judge against typical results in your own domain.
Should I report r or r squared?
r squared is usually more informative, because it states the share of variance explained. An r of 0.5 sounds substantial until you note it accounts for only a quarter of the variation.
Does a correlation of zero mean the variables are unrelated?
No. It means no linear relationship. Curved, cyclical or threshold relationships can all produce an r near zero, which is why plotting the data first matters.
How many pairs do I need?
At least two for the arithmetic to work, but small samples produce wildly unstable correlations. Thirty pairs is a reasonable minimum before taking a value seriously.
What do the bands under the pearson r mean?
They label the figure so it can be read without a reference point of your own: Strong negative up to -0.5; Weak or none from -0.5 to 0.3; Moderate positive from 0.3 to 0.7; Strong positive from 0.7 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Correlation Coefficient Calculator use?
It computes r = Σ((x−x̄)(y−ȳ)) ÷ √(Σ(x−x̄)^2 Σ(y−ȳ)^2), where x, y is the paired observations; x̄, ȳ is the two means; r is pearson correlation coefficient; r² is share of variance explained. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Correlation and variance explained
| Pearson r | r squared | Variance explained | Variance unexplained |
|---|---|---|---|
| 0.1 | 0.0100 | 1.00% | 99.00% |
| 0.3 | 0.0900 | 9.00% | 91.00% |
| 0.5 | 0.2500 | 25.00% | 75.00% |
| 0.7 | 0.4900 | 49.00% | 51.00% |
| 0.8 | 0.6400 | 64.00% | 36.00% |
| 0.9 | 0.8100 | 81.00% | 19.00% |
| 0.95 | 0.9025 | 90.25% | 9.75% |
| 0.99 | 0.9801 | 98.01% | 1.99% |
A correlation of 0.7 still leaves half the variation unaccounted for.