Pearson Correlation Coefficient

Investigating whether two continuous variables move together is a standard analytical task in econometrics, psychological studies, epidemiological tracking, and industrial quality control. Analysts examine paired numerical measurements, such as advertising expenditure versus monthly sales revenue or ambient outdoor temperature versus municipal power grid demand, to quantify linear dependency. You provide paired lists of numeric observations representing the independent predictor variable and the dependent response variable to evaluate bivariate correlation.

The computed correlation coefficient r ranges from negative one to positive one, where values near positive one indicate strong positive linear alignment and values near negative one indicate strong inverse trends. The square of r, known as the coefficient of determination, reveals the percentage of total variance in one variable explained by linear changes in the other. Always remember that Pearson correlation only measures straight-line linear associations and does not establish causal relationships or detect curved non-linear patterns.

The calculation takes the sum of cross-products of deviations from their respective means for both variables. It divides that numerator by the product of the square roots of the individual sum of squared deviations, normalizing the resulting coefficient between negative one and positive one.