R Squared is the fraction of observed outcome sum-of-squares variation accounted for by fitted values from one declared model on one declared sample.

It is an in-sample descriptive ratio unless the predictions come from validation data. The number does not identify causal contributions and does not equal the proportion of individual outcomes predicted correctly.

With an intercept and ordinary least squares on identical rows, SST decomposes into SSR plus SSE. The worked values are 6763.266564 = 5762.640139 + 1000.626425.

Reconstructing the identity is a strong audit because it checks the observed mean, fitted values and residual ledger independently of a software label.

The familiar bounded zero-to-one interpretation assumes a model with an intercept and compares fitted error with an intercept-only baseline.

Without an intercept, the usual decomposition can fail and R Squared can behave differently. The model specification must therefore be reported with the statistic.

On new validation data, predictions can produce more squared error than simply using the validation outcome mean. The resulting out-of-sample R Squared can be negative.

A negative validation value is meaningful: the tested prediction rule performs worse than the stated mean benchmark for those observations.

Adjusted R Squared penalizes the ordinary ratio for estimating additional predictor parameters. It rises only when the added fit is large enough relative to the complexity charge.

The worked value falls from 0.852050 to 0.850200, a small difference because the model gain is large relative to eight fitted predictor parameters and 649 observations.

For an intercept model with nonnegative R Squared, multiple R is the positive square root of R Squared. Here it equals 0.923065.

It is the correlation between observed and fitted values in this setting. It is not a standardized regression coefficient and should not be interpreted as the contribution of one predictor.

R Squared standardizes error by the observed outcome dispersion, whereas RMSE remains in G3 grade points.

Two studies can have similar R Squared but different practical error because their outcome ranges and residual scales differ. Both metrics are required.

The supplied RMSE is sqrt(SSE/N) = 1.241691. Residual standard error uses residual degrees of freedom and equals approximately 1.250391.

The quantities are close but not identical. A careful report labels the denominator rather than calling every square-root error RMSE.

MAE = 0.780275 gives the average absolute fitted error and is less sensitive to a few very large residuals than RMSE.

The gap between RMSE and MAE reflects tail errors such as cases 164 and 173. R Squared alone hides that error-shape information.

A nested change compares a reduced model with a full model fitted to the same outcome rows. The increase is conditional on the predictors already entered.

G2 adds 0.164846 after G1, while age adds only 0.000069 at its position. Entry order therefore defines the sequential change interpretation.

The F-change test evaluates whether the added parameter block reduces SSE more than expected under the reduced-model null.

A detectable change can still be substantively tiny. The final sex step adds only 0.000892 even though its change-test p-value is approximately 0.049871.

Sequential increments depend on entry order. Partial or semi-partial measures answer different conditional questions and should not be silently substituted.

The worked nested sequence is explicitly G1, G2, studytime, failures, absences, age, school and sex. Reordering the terms redistributes the changes.

G2 alone produces R Squared = 0.843730 and G1 alone produces 0.682916. These are separate one-predictor models.

Their values cannot be added because G1 and G2 explain overlapping outcome variation. The full model does not divide its 0.852050 equally among predictors.

school and sex enter as indicator variables with GP and female as the reference groups. One fitted parameter is used for each binary factor.

Changing references changes coefficient signs and labels but does not change fitted values or the complete-model R Squared when the same factor information is retained.

Ordinary R Squared never decreases when predictors are added to an OLS model fitted to the same rows with an intercept.

A higher value therefore does not automatically identify the better model. Use adjusted fit, validation error, theory and parsimony when specifications differ.

R Squared values should not be ranked directly when models use different outcome rows, transformations or populations.

SST changes with the sample. A larger ratio can reflect a narrower or easier outcome distribution rather than a genuinely better prediction rule.

Transforming G3 changes the variance scale and therefore the meaning of R Squared. A model for log outcome variation answers a different question from a model for raw grade variation.

Back-transformed prediction performance should be reported on the scale relevant to decisions rather than relying only on transformed-scale fit.

Logistic, count and survival models do not generally share the OLS sum-of-squares identity. Their pseudo R Squared measures use likelihood or other definitions.

Pseudo values should be named by formula and should not be interpreted as the percentage of raw outcome variance explained.

Validation R Squared should be computed from predictions generated without using the validation outcomes for fitting or model selection.

The current 0.852050 value is fitted-sample evidence. It should not be presented as external predictive performance.

Cross-validation generates held-out predictions across folds and can summarize predictive performance more realistically than fitted R Squared.

All preprocessing and variable selection must occur within each training fold. Otherwise the cross-validated value remains optimistic.

Case 164 has residual about -9.042550, standardized residual about -7.328003 and Cook distance about 0.159844. Case 173 has a similarly large negative residual.

These observations do not erase strong global fit, but they show that aggregate explained variation can coexist with serious individual errors.

R Squared is arithmetically defined even when residuals are nonnormal or heteroscedastic, but inferential claims and prediction intervals still depend on model conditions.

Residual-versus-fitted and distribution checks must accompany the headline ratio. A large value is not a diagnostic certificate.

Strong overlap among predictors can leave total R Squared high while individual coefficients become unstable or uncertain.

The dominance of prior grades means the full model can predict well even when small additional slopes are sensitive to specification choices.

Adding flexible terms can raise fitted R Squared by absorbing sample noise. The monotonic property makes training fit a weak model-selection criterion by itself.

Use validation, adjusted fit or prespecified nested tests. The intended use and feature availability should guide the final equation.

G2 is measured close to final G3 and accounts for most of the fitted variation. A model intended for early intervention may not have G2 available.

High R Squared does not make a temporally unavailable predictor suitable for deployment. Report the prediction time and feature availability.

Define the outcome scale, model formula, sample and purpose. Reconstruct SST, SSE and SSR; report ordinary and adjusted fit; then add RMSE, MAE, nested changes and diagnostics.

For prediction claims, replace fitted values with protected validation predictions. For explanatory claims, avoid allocating the total ratio causally among correlated predictors.