Multiple Linear Regression Calculator

Multiple linear regression models a numerical outcome as a linear combination of two or more explanatory variables. The fitted equation has an intercept plus one coefficient for each predictor. Each coefficient estimates how much the dependent variable changes when that predictor increases by one unit while the other included predictors are held constant. This calculator uses ordinary least squares, the standard method that selects coefficients minimizing the sum of squared residuals. Data must be arranged by observation. Enter dependent values as one ordered list. In the independent-variable field, enter one row for each matching observation and separate predictor columns with commas. If Y contains five observations, X must contain five rows. The first Y value corresponds to the first X row, and so forth. Misaligned records can produce a mathematically valid but substantively meaningless model, so preserve the original row order. The intercept is the predicted value of Y when every predictor equals zero. Whether that interpretation is useful depends on whether zero is meaningful and within the observed predictor range. A positive coefficient indicates that predicted Y rises as its predictor rises, conditional on the other variables. A negative coefficient indicates the reverse. Coefficients depend on measurement units, so rescaling a predictor changes its numerical coefficient without changing the fitted predictions. R-squared measures the proportion of variation in Y explained by the fitted model relative to simply predicting the mean. It usually ranges from zero to one for a model with an intercept. An R-squared of one means the fitted values reproduce every observed outcome, while zero means the model explains no more variation than the mean. A high value does not establish causation, guarantee useful predictions, or prove that assumptions are satisfied. Ordinary least squares relies on important conditions for statistical inference. The relationship should be reasonably linear, observations should be appropriately independent, residual variance should be stable, and predictors should not be exact linear combinations of one another. Severe multicollinearity can make individual coefficients unstable even when predictions appear accurate. This calculator rejects a singular design matrix because unique coefficients cannot be recovered from perfectly redundant predictors. You also need more observations than fitted coefficients, including the intercept. A model with two predictors estimates three coefficients and therefore requires at least four rows here. In practical analysis, substantially more data is desirable. Inspect residual plots, influential points, uncertainty intervals, adjusted R-squared, and out-of-sample performance before making important decisions. The equation and R-squared are useful starting summaries, not a complete regression diagnosis.