Correlation Coefficient Calculator
Pearson's r, r-squared, covariance and the line of best fit for up to 10 (x, y) pairs, with the deviation table and a plain reading of how strong the link is.
Correlation Coefficient Calculator: with the default inputs, pearson's r is 0.996174.
Only the first N (x, y) boxes are used. Pairs 6–10 live under More options.
Between −1 and +1. The sign is the direction, the size is the strength.
- r² (coefficient of determination)
- 0.992363The share of the variation in y that the straight line accounts for.
- Sample covariance
- 26.857143
- Slope of the best-fit line
- 4.47619
- Intercept of the best-fit line
- 46.857143
- Strength
- Very strong positive
Assumptions
- Pearson's r measures linear association only; curved relationships are understated.
- Covariance uses the sample divisor n − 1, matching Excel's COVARIANCE.S.
- The best-fit line is ordinary least squares with y regressed on x; swapping the roles gives a different line (but the same r).
| x | y | x − x̄ | y − ȳ | (x − x̄)(y − ȳ) |
|---|---|---|---|---|
| 1 | 52 | -3.5 | -15 | 52.5 |
| 2 | 55 | -2.5 | -12 | 30 |
| 3 | 61 | -1.5 | -6 | 9 |
| 4 | 64 | -0.5 | -3 | 1.5 |
| 5 | 70 | 0.5 | 3 | 1.5 |
| 6 | 72 | 1.5 | 5 | 7.5 |
| 7 | 79 | 2.5 | 12 | 30 |
| 8 | 83 | 3.5 | 16 | 56 |
Those products sum to 188. When x and y move together the products are mostly positive; when they move oppositely, mostly negative.
How this is worked out
The formula
Pearson's r = Σ(x − x̄)(y − ȳ) ÷ √( Σ(x − x̄)² × Σ(y − ȳ)² ) Sample covariance = Σ(x − x̄)(y − ȳ) ÷ (n − 1) Best-fit slope = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)² Best-fit intercept = ȳ − slope × x̄ r² = r × r
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- How many pairs?
- Only the first N (x, y) boxes are used. Pairs 6–10 live under More options.from 2 to 10 · whole numbers only · defaults to 8
- x1
- A number.defaults to 1
- y1
- A number.defaults to 52
- x2
- A number.defaults to 2
- y2
- A number.defaults to 55
- x3
- A number.defaults to 3
- y3
- A number.defaults to 61
- x4
- A number.defaults to 4
- y4
- A number.defaults to 64
- x5
- A number.defaults to 5
- y5
- A number.defaults to 70
- x6(under More options)
- A number.defaults to 6
- y6(under More options)
- A number.defaults to 72
- x7(under More options)
- A number.defaults to 7
- y7(under More options)
- A number.defaults to 79
- x8(under More options)
- A number.defaults to 8
- y8(under More options)
- A number.defaults to 83
- x9(under More options)
- A number.defaults to 9
- y9(under More options)
- A number.defaults to 88
- x10(under More options)
- A number.defaults to 10
- y10(under More options)
- A number.defaults to 91
What you get back
- Pearson's rmain answer
- Between −1 and +1. The sign is the direction, the size is the strength.
- r² (coefficient of determination)
- The share of the variation in y that the straight line accounts for.
- Sample covariance
- Slope of the best-fit line
- Intercept of the best-fit line
- Strength
What this assumes
- Pearson's r measures linear association only; curved relationships are understated.
- Covariance uses the sample divisor n − 1, matching Excel's COVARIANCE.S.
- The best-fit line is ordinary least squares with y regressed on x; swapping the roles gives a different line (but the same r).
About this calculator
Pearson's correlation coefficient measures how tightly two variables move together in a straight line. It runs from −1 (perfectly opposed) through 0 (no linear relationship) to +1 (perfectly aligned), and it is scale-free: converting inches to centimetres or dollars to euros does not change it.
How to use it
Set the number of pairs, type your x and y values, and read r. Pairs 6–10 are under More options. The deviation table shows exactly where the number comes from — each point's distance from the two means, multiplied together. Points in the upper-right and lower-left quadrants contribute positive products; the other two quadrants contribute negative ones. Correlation is essentially the balance of those two piles, normalised so it can never leave [−1, 1].
How to read the results
- r is direction and strength. The usual verbal bands (Evans, 1996) are: below 0.2 very weak, 0.2–0.4 weak, 0.4–0.6 moderate, 0.6–0.8 strong, above 0.8 very strong. They are conventions, and what counts as "strong" differs wildly between physics and psychology.
- r² is the share of the variation in y explained by the straight-line relationship with x. An r of 0.7 sounds impressive until you notice r² = 0.49 — the line accounts for under half the variation.
- Covariance is the same idea in the original units. It is the raw ingredient of r, and unbounded, which is why r is easier to interpret.
- Slope and intercept define the least-squares line through the points, the same line the regression calculator reports in more detail.
Where it misleads
Correlation is not causation — the point deserves its cliché status, because the usual culprit is a lurking third variable that drives both.
Less famous but more dangerous in practice:
- r only sees straight lines. A perfect parabola, y = x² over a symmetric range, has r = 0. Always look at the chart before trusting the number.
- Outliers dominate. One stray point can drag r from 0.1 to 0.8, or the reverse. This is why the calculator warns you below five pairs.
- Range restriction shrinks r. Correlating SAT score with college GPA using only admitted students understates the relationship, because the low scores were filtered out.
- Aggregated data inflates r. Correlations computed on state or country averages are almost always far stronger than the same correlation at the individual level.
If your relationship is monotone but curved, Spearman's rank correlation is the better tool; it correlates the ranks instead of the values.
Frequently asked questions
▸What counts as a strong correlation?
By the usual convention, |r| above 0.8 is very strong, 0.6–0.8 strong, 0.4–0.6 moderate. Context matters more than the label: 0.3 is a big effect in psychology and a broken instrument in physics.
▸What is the difference between r and r²?
r measures direction and strength on a −1 to +1 scale; r² is the fraction of the variation in y explained by the line, always between 0 and 1. An r of 0.5 means the line explains only 25% of the variation.
▸Can correlation prove causation?
No. A correlation is consistent with x causing y, y causing x, both being driven by something else, or coincidence. Only an experiment or a careful causal design can separate those.
▸Why is my correlation zero when the data clearly has a pattern?
Pearson's r only detects straight-line relationships. A U-shaped or curved pattern can have r near zero. Plot the points, and consider Spearman's rank correlation for monotone curves.
▸How many data points do I need?
Statistically, more than you think — with n = 5 the 95% confidence interval around r is enormous. Thirty pairs is a reasonable minimum for taking r seriously; below ten, treat it as descriptive only.
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