Pearson Correlation Calculator

The Pearson correlation coefficient (r) measures the strength and direction of a linear relationship between two quantitative variables. It ranges from -1 to 1. A value of r=1 means a perfect positive correlation, r=-1 a perfect negative correlation, and r=0 suggests no linear relationship between the variables.

Common thresholds: |r| >= 0.9 is a very strong correlation, 0.7-0.9 strong, 0.5-0.7 moderate, 0.3-0.5 weak, below 0.3 very weak or no correlation. The sign of r indicates direction: positive means both variables increase together, negative means one increases as the other decreases.

The coefficient of determination r2 is the square of the correlation coefficient. It expresses what percentage of the variance in one variable is explained by the other. For example, r=0.8 gives r2=0.64, meaning 64% of the variance in Y is explained by changes in X. The value of r2 always falls between 0 and 1.

No — correlation does not imply causation. Two variables may be correlated by chance or through a third hidden variable (confounder). A classic example is the correlation between ice cream sales and drowning rates — both increase in summer, but ice cream does not cause drowning. Causal conclusions require additional experimental or controlled research.

Pearson correlation is appropriate when both variables are quantitative (numerical), the relationship between them is linear, the data do not contain many outliers, and the distributions are approximately normal. If the relationship is nonlinear or the data are ordinal, Spearman or Kendall rank correlation is more suitable.

The formula is: r = sum[(xi - x_mean)(yi - y_mean)] / sqrt(sum[(xi - x_mean)^2] * sum[(yi - y_mean)^2]), where x_mean and y_mean are the arithmetic means of each series. The numerator is the (unscaled) covariance and the denominator is the product of the standard deviations, making r dimensionless and scale-independent.

A minimum of 2 pairs of observations is needed mathematically, but results are very unstable with so little data. In practice, statisticians recommend at least 20-30 pairs for results to be interpretable. The larger the sample, the more reliable the result and the easier it is to assess the statistical significance of r.

The calculator matches the series to the shorter one, meaning it uses only as many pairs as the length of the shorter series. Extra values from the longer series are ignored. The number of pairs used (n) is displayed in the results. We recommend ensuring both series have the same number of elements.

Enter values separated by commas (e.g. 1,2,3,4,5), semicolons (e.g. 1;2;3;4;5) or spaces (e.g. 1 2 3 4 5). You can also mix separators. The calculator automatically skips empty entries and non-numeric values. Decimal numbers can be entered with a dot or comma.

Yes — the calculator accepts any real numbers including negatives (e.g. -3, -1.5) and decimals (e.g. 0.75 or 2.5). The Pearson correlation is correctly computed for all such inputs. Negative numbers in a series do not change the calculation method — what matters is the linear relationship between pairs of values.