Several predictors, one equation
Multiple regression extends the straight line into a plane (or a hyperplane): ŷ = b0 + b1X1 + b2X2 + … Each coefficient is the predicted change in Y for one extra unit of its predictor with every other predictor held still. That holding-still clause is the entire value of the method: it separates overlapping influences that simple regressions would double-count.
A retailer models weekly sales (in thousands) on advertising spend X1 and the number of in-store displays X2, over ten weeks. The fit: ŷ = 2.11 + 3.30 X1 + 1.48 X2, with R² = 0.996 and adjusted R² = 0.995. Each extra thousand of advertising predicts about 3.30 thousand of extra sales at the same display count; each extra display predicts 1.48 thousand at the same spend. A week with spend 6 and 2 displays predicts 2.11 + 3.30 × 6 + 1.48 × 2 = 24.87. Both slopes clear their t tests comfortably (t = 19.6 and 3.9).
Reading the output table
Each coefficient gets a standard error, a t statistic and a p-value: the same machinery as the single slope, per predictor. The model's F test asks a joint question: do the predictors together explain more than chance? Here F = 823.9, overwhelming. R² always grows when predictors are added, even useless ones, so adjusted R² is the fairer gauge: it can fall when a passenger joins.
Multicollinearity: predictors that move together (height in cm and height in inches, roughly) split their shared credit arbitrarily. Coefficients wobble, signs can flip, and p-values inflate, while predictions stay fine. If two predictors correlate above about 0.9 with each other, consider dropping one.
How many predictors can the data carry?
A stubborn rule of thumb wants at least 10 to 15 observations per predictor. Below that, least squares starts memorizing the sample instead of modelling the process, and the beautiful R² evaporates on new data. This calculator accepts up to five predictors and requires at least three spare observations beyond them.
Frequently asked questions
What does 'holding other variables constant' actually mean?
The coefficient is computed from the part of that predictor that does not overlap with the others. It describes the predicted difference between two cases identical on every other predictor, differing by one unit on this one.
Why did a predictor's sign flip when I added another variable?
Usually overlapping predictors: the new variable absorbs the shared part, and the old coefficient now describes only the leftover part, which can genuinely point the other way. It is information, not an error.
R-squared or adjusted R-squared: which do I report?
Report both if space allows. Use adjusted R-squared to compare models with different predictor counts; plain R-squared never punishes extra variables.
How is this different from running several simple regressions?
Simple regressions give each predictor full credit for shared variation, counting it several times. Multiple regression divides the credit. The two approaches can disagree in size and even in sign.