OneTaskBy Tim King · Updated
Correlation Coefficient Calculator
Pearson r, Spearman Rho and Significance From Paired Data, Working Shown
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How the Correlation Coefficient Calculator works
This correlation coefficient calculator measures how closely two sets of numbers move together. Paste paired values and it gives Pearson's r, a plain-English strength rating, r², Spearman's rank correlation and a p-value telling you whether the relationship could be chance.
A correlation coefficient runs from −1 to +1. Plus one means the points sit on a rising straight line, minus one a falling line, and zero means no straight-line relationship at all. The sign gives the direction and the size gives the strength.
How the correlation coefficient calculator works
The formula for Pearson's r is Sxy ÷ √(Sxx × Syy), the covariance of x and y scaled by both spreads. Spearman's ρ applies the same formula to the ranks of the values, so it captures any steady rise or fall, not just straight lines. Significance uses t = r√(n − 2) ÷ √(1 − r²).
First example. The default data, daily temperature against ice cream sales over 12 days, gives r ≈ 0.96, a very strong positive correlation.
Second example. Values (1, 10), (2, 8), (3, 6), (4, 4) give r = −1 exactly.
How many pairs make r significant? With 10 pairs r must exceed 0.632 at the 5% level; with 100 pairs, just 0.197.
Smallest |r| that is significant at the 5% level (two-tailed) by sample size
| Pairs n | Critical r (5%) | Critical r (1%) |
|---|---|---|
| 5 | 0.878 | 0.959 |
| 6 | 0.811 | 0.917 |
| 7 | 0.754 | 0.875 |
| 8 | 0.707 | 0.834 |
| 9 | 0.666 | 0.798 |
| 10 | 0.632 | 0.765 |
| 12 | 0.576 | 0.708 |
| 15 | 0.514 | 0.641 |
| 20 | 0.444 | 0.561 |
| 25 | 0.396 | 0.505 |
| 30 | 0.361 | 0.463 |
| 35 | 0.334 | 0.430 |
| 40 | 0.312 | 0.403 |
| 50 | 0.279 | 0.361 |
| 60 | 0.254 | 0.330 |
| 70 | 0.235 | 0.306 |
| 80 | 0.220 | 0.286 |
| 100 | 0.197 | 0.256 |
| 120 | 0.179 | 0.234 |
| 150 | 0.160 | 0.210 |
| 200 | 0.139 | 0.182 |
| 300 | 0.113 | 0.149 |
| 500 | 0.088 | 0.115 |
| 1,000 | 0.062 | 0.081 |
Tips
- Plot first. A curve or a single outlier can make r misleading; Spearman is more robust.
- Correlation is not causation. Ice cream and sunburn rise together because both follow the weather.
- Equal list lengths. Every x needs a matching y in the same position.
FAQ
What is a strong correlation coefficient?
Rules of thumb vary by field, but a common scale is: below 0.3 weak, 0.3 to 0.7 moderate and above 0.7 strong, ignoring the sign. In physics a correlation of 0.9 may be disappointing, while in social science 0.4 can be a notable finding, so context matters as much as the number itself.
When should I use Spearman instead of Pearson?
Use Spearman when the data is ranked, has outliers, or rises steadily but not in a straight line. Pearson measures straight-line association and assumes roughly normal data. When both give similar answers you can quote Pearson; when they differ a lot, look at a scatter plot before deciding.
What does r² mean?
r² is the share of the variation in one variable that a straight line explains from the other. An r of 0.8 gives r² = 0.64, so 64% of the variation is shared. It is the same R² that the linear regression calculator reports for a line of best fit.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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