Simple Linear Regression
Enter X and Y values to fit a regression line and calculate R².
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Simple Linear Regression Explained
Simple linear regression fits a straight line (y = a + bx) to a set of data points. The slope (b) tells you how much Y changes for each unit increase in X. The intercept (a) is the Y value when X = 0.
R² (R-squared) measures how well the line fits the data. An R² of 0.85 means 85% of the variation in Y is explained by X. For prediction, plug an X value into the equation to estimate Y.
Simple linear regression fits a line y = mx + b to data, where m (slope) = [Σ(xᵢ−x̄)(yᵢ−ȳ)] / Σ(xᵢ−x̄)² and b (intercept) = ȳ − m × x̄. The line minimizes the sum of squared vertical distances from each data point to the line (least squares criterion).
R² (coefficient of determination) measures how much of the variance in y is explained by x. R² = 0 means x explains nothing; R² = 1 means x perfectly predicts y. R² = 0.75 means 75% of the variation in y is accounted for by the linear relationship with x.
Example: 5 data points — hours studied (x): 1,2,3,4,5 and exam scores (y): 50,55,65,70,75. Regression gives m ≈ 6.5, b ≈ 42.5, so y ≈ 42.5 + 6.5x. Predicted score for 3.5 hours: 42.5 + 6.5×3.5 = 65.25.
Extrapolating beyond the observed x range is risky — the linear relationship may not hold outside your data. Outliers can strongly influence both m and b; inspect your data visually before trusting regression output.
See also: Correlation Calculator · Confidence Interval Calculator