Correlation Coefficient Calculator

How to use the calculator

1. Enter one pair per row, X first and Y second, separated by a space, a tab or a comma. Pasting two spreadsheet columns works.
2. Choose the coefficient: Pearson's r for a linear relationship between measured values, Spearman's r or Kendall's τ for ranks or a monotonic but not necessarily straight relationship.
3. Set the confidence level (95% by default) and press Calculate.

At least four pairs are needed, and each row must contain exactly two numbers. The output lists the coefficient, a Z score, a one-sided p-value, a two-sided confidence interval, right-sided and left-sided intervals, the number of pairs and the least-squares regression line. For Spearman and Kendall the regression line is fitted to the ranks of the data, not the raw values.

The p-value tests the hypothesis that the true coefficient is zero or in the opposite direction to the one observed. A positive coefficient is tested against "≤ 0" and a negative one against "≥ 0", so the p-value is always the probability in the direction of the effect.

What a correlation coefficient tells you

A correlation coefficient condenses the relationship between two variables into one number between −1 and +1. The sign gives the direction: positive means that X and Y rise together, negative means that one rises as the other falls. The distance from zero gives the strength, and 0 means no association of the kind the coefficient measures.

For Pearson's r, the square of the coefficient, r², is the share of the variance in Y that a straight line on X accounts for. A correlation can be statistically significant and still too small to matter, so read the size and the interval, not only the p-value.

Pearson, Spearman or Kendall?

None of the three requires both variables to be normally distributed for the coefficient itself to be valid. The Fisher-transformation intervals and p-values do lean on normal-theory approximations, so for very small samples treat them as rough guides. You can screen a variable with the normality test calculator.

Formulas

Pearson's r is the covariance of the two variables divided by the product of their standard deviations:

r = Σ (xᵢ − x̄)(yᵢ − ȳ) ÷ √[ Σ (xᵢ − x̄)² × Σ (yᵢ − ȳ)² ]

Spearman's r is Pearson's r applied to the ranks of the data. Tied values share the average of the ranks they span. With no ties it reduces to 1 − 6 Σ d² ÷ (n(n² − 1)), where d is the difference between the two ranks of a pair.

Kendall's τ compares every pair of observations. The version used here, τ-b, adjusts for ties:

τ = (C − D) ÷ √[ (N − T_x)(N − T_y) ]C and D are concordant and discordant pairs, N = n(n − 1) ÷ 2, and T_x and T_y count pairs tied in X or in Y.

Confidence intervals for Pearson's and Spearman's coefficients use the Fisher transformation z′ = ½ ln[(1 + r) ÷ (1 − r)], which is approximately normal with standard error 1 ÷ √(n − 3) for Pearson. The interval is computed on the z′ scale and transformed back with the hyperbolic tangent:

r_low, r_high = tanh( z′ ∓ z_(α/2) ÷ √(n − 3) )

For Spearman the standard error is inflated to √(1.06 ÷ (n − 3)). The Z score is z′ divided by that standard error, and the p-value is the one-sided normal tail beyond it. Kendall's τ uses a normal approximation of its own, with bounds capped at −1 and +1.

Worked example

Suppose eight students record weekly study hours (X = 1, 2, 3, 4, 5, 6, 7, 8) and their test scores (Y = 52, 55, 61, 60, 68, 72, 70, 81). The means are 4.5 and 64.875. The sums of squares and cross-products are Sxx = 42, Syy = 648.875 and Sxy = 159.5.

r = 159.5 ÷ √(42 × 648.875) = 159.5 ÷ 165.09 ≈ 0.9662

The Fisher transform is z′ = atanh(0.9662) ≈ 2.0313 and the standard error is 1 ÷ √5 ≈ 0.4472. A 95% interval on that scale is 2.0313 ± 1.96 × 0.4472, which is 1.1548 to 2.9078. Transforming back gives 0.8193 to 0.9941. The Z score is 2.0313 × √5 ≈ 4.54, with a one-sided p-value of about 0.0000028.

The regression line is y = 3.7976 x + 47.786, so each extra study hour goes with about 3.8 more points. The same data give Spearman's r = 0.9524 and Kendall's τ = 0.8571, both a little lower because they only use the order of the values, and r² = 0.934 says that about 93% of the score variation lines up with study time.

Pitfalls

• Correlation is not causation. Two variables can move together because of a third factor or by coincidence.
• Outliers can make or break Pearson's r. Plot the data first; one extreme point can create a strong correlation or hide one.
• Restricted ranges shrink r. Looking only at students who scored between 60 and 70 would show a much weaker relationship than the full range.
• Non-linear patterns can hide. A curve can have a Pearson correlation near zero while the two variables are tightly linked; a Spearman or Kendall coefficient will catch a monotonic curve.
• Do not extrapolate the line. The regression line describes the range of your data only.

Small samples give wide intervals. With only four or five pairs a high coefficient can arise by chance, so look at the confidence interval as well as the coefficient. For the spread of a single variable use the standard deviation calculator.