What Is the Pearson Correlation Coefficient?
The Pearson correlation coefficient, denoted r, measures the strength and direction of the linear relationship between two numeric variables. Its value always falls between −1 and +1: a value of +1 means a perfect positive linear relationship, −1 means a perfect negative relationship, and 0 means no linear correlation at all. It is one of the most widely used statistics in research, finance, and data science.
How to Use This Calculator
Enter your X values and Y values as comma- or space-separated lists. Each X must pair with the Y in the same position, so both lists should contain the same number of values. The calculator returns r, the coefficient of determination r², the number of paired observations, and the mean of each variable.
The Formula Explained
The formula subtracts each value from its mean to get deviations, multiplies the paired X and Y deviations and sums them (the numerator), then divides by the square root of the product of the summed squared deviations (the denominator):
The numerator captures how the variables move together (covariance), while the denominator normalises the result so it stays in the −1 to +1 range.
Worked Example
Suppose X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 5. The means are \bar{x} = 3 and \bar{y} = 4. The sum of deviation products is 6, \sum (x_i - \bar{x})^2 = 10, and \sum (y_i - \bar{y})^2 = 6. So
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