Linear Regression Calculator

Linear regression fits a straight line of the form:

$$\hat{y} = a + bx$$

Where b (slope) and a (y-intercept) are determined by the ordinary least squares formulas:

$$b = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$$

$$a = \bar{y} - b\bar{x}$$

The slope b represents the average change in Y for each one-unit increase in X. The intercept a is the predicted Y value when X equals zero. Together, they define the regression equation that best fits the observed data.

The Pearson correlation coefficient measures the linear association strength:

$$r = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{\sqrt{[n\sum x_i^2 - (\sum x_i)^2][n\sum y_i^2 - (\sum y_i)^2]}}$$

The coefficient of determination R² = r² indicates the proportion of variance in Y explained by the linear relationship with X. An R² of 0.85 means 85% of Y's variability is accounted for by the regression model, while 15% remains unexplained. The OLS method guarantees that no other straight line produces a smaller total squared error for the given data, making it the optimal linear unbiased estimator under standard assumptions of homoscedasticity and normally distributed errors.

Understanding your linear regression results requires examining several outputs together:

• Slope (b): A positive slope indicates Y increases as X increases; a negative slope means Y decreases as X increases. The magnitude tells you the rate of change — a slope of 2.0 means Y increases by 2 units for every 1-unit increase in X.
• Intercept (a): The starting value of Y when X = 0. In many practical contexts, the intercept may not have a meaningful physical interpretation (e.g., predicting weight at zero height).
• Correlation (r): Values near +1 or -1 indicate strong linear relationships. Values near 0 suggest weak or no linear association. The sign matches the slope direction.
• R-Squared: Higher values indicate better model fit. In social sciences, R² > 0.5 is often considered good. In physical sciences, R² > 0.95 is typically expected. Always consider the context of your field.

Key assumptions of linear regression include: (1) linearity — the relationship between X and Y is approximately linear; (2) independence of observations; (3) homoscedasticity — constant variance of residuals; (4) normally distributed residuals. Violating these assumptions may lead to biased or inefficient estimates. When the linearity assumption fails, consider polynomial, exponential, or logarithmic regression models instead.