Correlation Calculator (Pearson and Spearman)

Calculate Pearson's r or Spearman's rank correlation ρ for paired data, with a confidence interval by the Fisher transform, a test of zero correlation and a scatter plot.

Result

Deterministic calculation

Pearson r

0.999

95% confidence interval: 0.994 to 1.000
Test that it is zero: t = 54.129, 6 df, p < 0.0001
r² (shared variance): 0.998
Sample size n: 8

Data

Note

The p-value is the probability, if the null hypothesis and every assumption of the test were true, of a result at least as extreme as the one observed. It is not the probability that the hypothesis is true, and it says nothing about how large or important an effect is. Judge the effect size and its interval, the design and the assumptions along with it, and do not treat a p-value on one side of 0.05 as proof of an effect or of its absence.

Note

A correlation measures how closely points follow a line (Pearson) or a monotonic curve (Spearman). It does not show that one variable causes the other, and a coefficient near zero does not mean there is no relationship: a curved one can have r = 0. Always look at the plot.

Note

Pearson's r is strongly affected by outliers and by a restricted range. Spearman's ρ is more robust.

Note

With fewer than ten pairs the coefficient is very uncertain; look at the width of the interval.

Significant figuresScientific notation

Calculation

r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² Σ(y − ȳ)²)
t = r √(n − 2) / √(1 − r²); interval by the Fisher z transform, z = atanh(r), with standard error 1/√(n − 3)

Formula

r = \dfrac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2\ \sum (y - \bar{y})^2}}

t = \dfrac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}

\text{interval:}\quad \tanh\!\left(\operatorname{atanh} r \pm \dfrac{z}{\sqrt{n - 3}}\right)

$r,\ \rho$: the coefficient, from −1 (perfect inverse) to +1 (perfect direct)
$z$: 1.96 for a 95% interval

How it works

Pearson's r measures how closely two variables follow a straight line. Spearman's ρ is the same calculation on the ranks, so it detects any monotonic relationship, straight or curved, and is much less affected by outliers. The interval is computed on the Fisher z scale, where r is approximately normal, then transformed back.

A coefficient near zero does not mean the variables are unrelated, since a curved relationship can give r = 0, and a large one does not mean one causes the other. Plot the data first; Anscombe's famous examples show very different patterns sharing the same r.

Worked example

The same eight points as the regression example.

r = 0.999, r² = 0.998.
Fisher z interval: 0.994 to 1.000; t = 54.1 with 6 degrees of freedom.

A near-perfect linear correlation, r = 0.999 (95% CI 0.994 to 1.000).

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

Paired, independent observations.
For Pearson, a roughly linear relationship and no extreme outliers, with bivariate normality for the interval.
For Spearman, a monotonic relationship; its p-value and interval are approximate for small n.

Common mistakes

Inferring causation from correlation.
Calculating r on a range restricted by selection, which lowers it.
Correlating two quantities that share a component, or pooling distinct groups so that the group difference creates the correlation.

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Correlation Calculator (Pearson and Spearman)

Result

Calculation

Formula

How it works

Worked example

Assumptions

Common mistakes

Related tools

Result

Calculation