Correlation Calculator
With number of pairs 10, sum of x 55, sum of y 120, sum of xy 800 and 2 more fields, correlation comes to 0.9559 — correlation r. It is reached in 10 steps, the last of which is 140 / sqrt(82.5 * 260), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST/SEMATECH e-Handbook of Statistical Methods, not an approximation fitted to it.
Pearson correlation from the sums a class is given, with r², the slope it implies, and the sample size at which it stops being noise.
Formula and sources checked · How we check
Number of pairs 10, Sum of x 55, Sum of y 120, Sum of xy 800
0.9559
Correlation r for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Mean of x
55 / 105.5- Mean of y
120 / 1012- Σxy − (Σx)(Σy)/n
800 - (55 * 120) / 10140- Σx² − (Σx)²/n
385 - (55 ^ 2) / 1082.5- Σy² − (Σy)²/n
1700 - (120 ^ 2) / 10260- Correlation coefficient r
140 / sqrt(82.5 * 260)0.956- r², the share of variance explained
0.9559042 ^ 20.914- Slope of the line of best fit
140 / 82.51.697- Where that line crosses y
12 - 1.6969697 * 5.52.667- t statistic for r ≠ 0
abs(0.9559042) * sqrt((10 - 2) / (1 - 0.9559042 ^ 2))9.206
Worked example
Ten pairs with these sums give r = 0.9559: the two variables move together closely. r² of 0.9138 says about 91% of the variation in y is accounted for by x — and the other 9% is not, which is the half of the sentence usually left off. The line through them has slope 1.697.
How to work it out yourself
- 1.Collect the five sums the formula needs: n, Σx, Σy, Σxy, Σx² and Σy².
- 2.Compute Sxy = Σxy − ΣxΣy/n, and Sxx and Syy the same way with the squares.
- 3.r is Sxy over the square root of Sxx times Syy. It always lands between −1 and 1.
- 4.Square it for r², the share of the variance in y that the line accounts for.
The formula
- Mean of x
55 / 10 - Mean of y
120 / 10 - Σxy − (Σx)(Σy)/n
800 - (55 * 120) / 10 - Σx² − (Σx)²/n
385 - (55 ^ 2) / 10 - Σy² − (Σy)²/n
1700 - (120 ^ 2) / 10 - Correlation coefficient r
140 / sqrt(82.5 * 260) - r², the share of variance explained
0.9559042 ^ 2 - Slope of the line of best fit
140 / 82.5 - Where that line crosses y
12 - 1.6969697 * 5.5 - t statistic for r ≠ 0
abs(0.9559042) * sqrt((10 - 2) / (1 - 0.9559042 ^ 2))
Source: NIST/SEMATECH e-Handbook of Statistical Methods, NIST SP 811 — expressing measurement results
Questions people actually ask
- What does the correlation coefficient mean?
- How closely the points sit to a straight line, and in which direction. r = 1 is a perfect rising line, −1 a perfect falling one, 0 no linear relationship at all. It says nothing about how steep the line is: a slope of 0.01 and a slope of 100 can both have r = 0.99.
- What is the difference between r and r²?
- r carries the direction and r² does not — both −0.9 and 0.9 give 0.81. r² is read as the share of the variance in y explained by x, which is why it is the figure quoted in a report. Watch the size of the drop: r = 0.7 sounds strong and explains under half the variance.
- Is a correlation of 0.9 significant?
- It depends entirely on how many pairs produced it. With 4 points r = 0.9 is unremarkable; with 40 it would be very hard to get by chance. The t statistic above is the test: compare it against a t table with n − 2 degrees of freedom.
- Does correlation prove causation?
- No, and the failure has three named shapes: the arrow could point the other way, something else could drive both, or the pairing could be coincidence in a small sample. A correlation is a reason to look for a mechanism, not a substitute for finding one.
- What if the relationship is curved?
- Pearson r will understate it, possibly to zero. A perfect parabola through the origin has r = 0 while being entirely predictable. Plot the points before quoting the coefficient — the classic demonstration is Anscombe’s quartet, four data sets with identical r and nothing else in common.
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