Once you have fit a multiple regression model, the next question is how well it actually describes the data. Measures of model fit summarize how much of the variation in the response variable the model accounts for, and they give you a basis for comparing competing models built from different sets of predictors. In this topic we focus on the coefficient of determination, R-squared, and its companion statistic, adjusted R-squared.
R-squared is easy to compute and easy to over-trust. A key fact in multiple regression is that R-squared can never decrease when you add another predictor, even a useless one, so a high R-squared alone does not mean you have a good or parsimonious model. Adjusted R-squared addresses this by penalizing the model for the number of predictors relative to the sample size, so it can go down when an added variable does not earn its place. Understanding the difference is what lets you use these numbers responsibly when refining a model.
As you watch, pay attention to three things: the formulas and where the degrees of freedom enter, the behavior of R-squared versus adjusted R-squared as predictors are added, and how these statistics appear alongside the ANOVA table, the F-test, and the standard error of the estimate in typical regression output. Try to connect each number on the output to the sums of squares it comes from.
A long-form lecture in Brandon Foltz's Statistics 101 series that works through adjusted R-squared in the multiple regression setting. Based on the title and description, it builds up from the regression model and ordinary R-squared to explain why an adjustment is needed and how the penalty for additional predictors works. Expect a slow, worked treatment with discussion of how the statistic behaves as variables are added to a model.
The extended pace and step-by-step style make this a good primary explanation of the concept of fit, the limitations of R-squared, and the correction adjusted R-squared provides, covering several subtopics in one pass.
A short, focused screencast from LearnChemE, produced by faculty in the Department of Chemical and Biological Engineering at the University of Colorado Boulder. In about six minutes it introduces adjusted R-squared, likely showing the formula with its degrees-of-freedom terms and illustrating how the value changes relative to ordinary R-squared when predictors are added.
It is a concise, engineering-oriented refresher that pairs well with the longer lecture and directly targets the comparison between R-squared and adjusted R-squared.
A compact problem-solving video that works a multiple regression example with 40 observations, filling in an ANOVA table and computing the F-test, R-squared, and the standard error of the estimate. The emphasis is on reading and completing regression output rather than on theory.
It shows where fit statistics sit within standard regression output and connects R-squared to the sums of squares and the overall F-test, which is exactly the context in which you will interpret model fit on exams and in practice.
Together these videos center on MO3 by defining R-squared, exposing its limitation of never decreasing as predictors are added, and showing how adjusted R-squared corrects for model size. The Joshua Emmanuel example reinforces MO4 by placing R-squared in the ANOVA table alongside the overall F-test and standard error, and it assumes the fitted model framework from MO1 and MO2. The discussion of how adjusted R-squared responds to adding or dropping variables gives you the reasoning tool needed for MO5, where you must justify keeping or removing predictors during model refinement.
These videos were chosen from YouTube on October 2, 2026. Each has captions; the player turns them on. Videos on YouTube can be changed or removed — if one no longer plays, tell your instructor. If a player does not appear, use its Watch on YouTube link.