Regression Calculator

A linear regression model describes the relationship between a predictor variable (X) and a response variable (Y) using a straight line: Y = a + bX. The slope (b) shows how much Y changes for each unit increase in X, while the intercept (a) is the expected value of Y when X equals zero. It's one of the most widely used statistical techniques for prediction and explanation.

R² (the coefficient of determination) measures the proportion of variance in Y that is explained by X, ranging from 0 to 1. An R² of 0.90 means 90% of the variation in Y is explained by the model. What counts as 'good' depends on the field — social sciences may accept 0.3–0.5, while engineering or physics may require 0.99+.

Pearson's r is the correlation coefficient and ranges from −1 to +1, indicating both the strength and direction of the linear relationship. R² is simply r squared and ranges from 0 to 1, representing the proportion of variance explained. For example, r = 0.9 gives R² = 0.81, meaning 81% of Y's variance is explained by X.

Simple linear regression assumes: (1) a linear relationship between X and Y, (2) independence of observations, (3) homoscedasticity — residuals have constant variance across all values of X, (4) normally distributed residuals, and (5) no significant outliers. Violating these assumptions can lead to biased or unreliable estimates.

MSE (Mean Squared Error) is the average of the squared differences between observed and predicted Y values — it penalises large errors heavily. RMSE (Root Mean Squared Error) is the square root of MSE, putting the error back in the same units as Y, making it easier to interpret. Lower MSE and RMSE indicate a better-fitting model.

You should only force the line through the origin (set intercept = 0) when your theory or physical law explicitly requires that Y must equal zero when X equals zero. In most practical situations, including the intercept produces a more accurate and unbiased fit, even if the intercept value seems unintuitive.

At minimum, you need at least 3 data points, but meaningful regression typically requires 10 or more. A common rule of thumb is at least 10–20 observations per predictor variable. With too few points, estimates of slope, intercept, and R² can be highly unstable and unreliable.

The regression equation Ŷ = a + bX lets you predict Y for any new X value. For example, if a = 1.5 and b = 2.0, then for X = 5 you would predict Ŷ = 1.5 + (2.0 × 5) = 11.5. The residuals shown in the table (Y − Ŷ) tell you how far each actual observation is from the model's prediction.