Correlation and Regression Analysis

Correlation analysis measures the strength and direction of a linear relationship between two variables using the correlation coefficient $r$, which ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation). The Pearson correlation coefficient quantifies this linear dependency with the formula $\displaystyle r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2}\sum{(y_i - \bar{y})^2}}}$. Regression analysis models the functional relationship between an independent variable and a dependent variable using the linear equation , where is the intercept, is the slope, and is the error term. The slope is estimated by and represents the expected change in for a unit change in . The coefficient of determination , equal to , indicates the proportion of variance in explained by through the regression model. These techniques are fundamental in engineering for data analysis, prediction, quality control, and decision-making.