Pearson Correlation Calculator

The Pearson correlation coefficient measures the direction and strength of a linear relationship between two quantitative variables. It is commonly written as r and ranges from negative one to positive one. A positive coefficient means larger X values tend to occur with larger Y values, while a negative coefficient means larger X values tend to occur with smaller Y values. A value near zero indicates little linear association, although a strong nonlinear pattern may still be present. The calculation centers each observation by subtracting its variable's mean. It sums the products of paired deviations and divides by the square root of the product of the two sums of squared deviations. This standardization makes the coefficient independent of measurement units. Multiplying every X value by a positive constant, for example, does not change r. The order of paired observations still matters, so the first X value must correspond to the first Y value, the second to the second, and so forth. The calculator also reports r squared, often called the coefficient of determination in a simple linear-regression setting. It is the proportion of variance represented by the fitted linear relationship in that setting. Squaring removes direction, so r values of 0.8 and -0.8 both produce an r squared value of 0.64. This descriptive percentage should not automatically be interpreted as the percentage of one variable caused by the other. Pearson correlation works best when observations are independent, the relationship is approximately linear, and extreme outliers are understood. A single unusual pair can substantially change the coefficient. Restricted ranges can make a relationship appear weaker, while combining distinct groups can create or reverse a trend. Always inspect a scatterplot alongside the number when possible. Correlation does not establish causation, and this calculator does not provide a p-value or confidence interval. Those inferential results require sample-size assumptions and a defined sampling process. Use this tool for transparent descriptive calculation, checking coursework, and quickly comparing paired data before applying a fuller statistical model.