Linear Regression Analysis Tool

R² represents the proportion of variance in y that is explained by the linear model. An R² of 0.85 means 85% of the variability in y is accounted for by x. It ranges from 0 (no explanatory power) to 1 (perfect fit). However, a high R² does not imply causation, and with few data points, R² can be misleadingly high.

Pearson's r measures the strength and direction of a linear relationship between two variables, ranging from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation). It is related to R² by $$R^2 = r^2$$. Unlike R², r preserves the sign, indicating whether the relationship is positive or negative.

Two points always determine a unique line with R²=1, regardless of the actual relationship. Three or more points allow the line to "miss" some points, revealing the quality of fit. Statistically, regression with n points has n-2 degrees of freedom for testing, so n=3 gives 1 degree of freedom — the minimum for any meaningful residual analysis.

The method of least squares minimizes $$\sum_{i=1}^n (y_i - (mx_i + b))^2$$ — the sum of squared vertical distances from data points to the fitted line. Squaring ensures positive and negative errors do not cancel, and the squared penalty penalizes large errors more heavily. Setting partial derivatives to zero yields the normal equations that give the slope and intercept formulas.

Standard linear regression assumes a linear relationship. For non-linear data, you can: (1) transform variables (log, square root) to linearize the relationship; (2) use polynomial regression ($$y = ax^2 + bx + c$$); (3) use non-linear regression methods. Always check residual plots to assess linearity assumptions.

Correlation measures statistical association — two variables move together. Causation means one variable directly influences the other. High correlation does not imply causation; confounding variables, reverse causation, or coincidence can produce strong correlations without causal links. Establishing causation requires controlled experiments or rigorous causal inference methods.

The slope represents the expected change in y for a one-unit increase in x. For example, if x is study hours and y is exam score, a slope of 5 means each additional hour of study is associated with a 5-point score increase. Always state the units: "5 points per hour" is more informative than just "slope = 5."

Residuals are the differences between observed y-values and predicted y-values: $$e_i = y_i - (mx_i + b)$$. They represent the error or unexplained variation. Good regression models have residuals that are randomly scattered around zero with no pattern. Systematic patterns in residuals suggest the linear model is inadequate.

Yes, the same formulas extend to any number of points by summing over all n data pairs. More points generally give more reliable estimates. For large datasets, the formulas become: $$m = \frac{n\sum x_iy_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$$. This calculator uses n=3 for simplicity but demonstrates the exact same principles.