Quick Answer: Adjusted R Squared Result

The main Python report compared five regression models using 649 complete cases and G3 final grade as the outcome. The final Python model used 13 predictor degrees of freedom: nine numeric predictors and four categorical indicators. It produced R² = 0.8534 and Adjusted R² = 0.8504. The adjustment penalty was therefore approximately 0.0030. The model-level test was F(13, 635) = 284.43, with p < .001. Its RMSE was approximately 1.236, and its MAE was approximately 0.778.

What Is Adjusted R Squared?

Adjusted R squared is a version of the coefficient of determination that reduces the reported fit when a regression model uses additional predictor degrees of freedom without enough improvement in explained variation. Ordinary R² compares explained variation with total variation. Adjusted R² keeps that basic idea but also considers sample size and model complexity.

In ordinary least-squares regression with an intercept, R² cannot decrease when predictors are added. Even an irrelevant variable may produce a tiny increase because the fitted model receives another degree of freedom. Adjusted R² asks a stricter question: did the reduction in unexplained variation justify the extra complexity? If the answer is yes, adjusted R² can rise. If the added predictor contributes too little, adjusted R² can fall.

This makes adjusted R² especially useful for comparing nested or competing multiple-regression models that use the same dependent variable and the same observations. It is less useful when the models use different outcomes, different transformations, different missing-data samples or different estimation methods.

R Squared vs Adjusted R Squared

For the main Python final model, R² is 0.8534 and adjusted R² is 0.8504. The difference is only about 0.0030. The model therefore retains most of its apparent explanatory power after the complexity adjustment. This does not mean that all 13 predictor terms are individually significant. It means that the model as a whole still has strong adjusted fit.

The difference should not be interpreted using a universal pass/fail cutoff. A small gap can be reassuring, but the substantive value of the model also depends on study design, measurement quality, theory, residual behavior and performance on new observations. Readers can connect this distinction with the guides on correlation vs regression, variance, effect size and statistical power.

Adjusted R Squared Formula

Substituting the rounded Python values gives approximately:

The formula shows why the penalty grows when p rises relative to n. With a large sample and a modest number of useful predictors, the difference between R² and adjusted R² may be small. With a small sample and many predictors, the adjustment can be substantial. This is why the predictor count must mean degrees of freedom rather than merely the number of named columns. A categorical variable with several levels may require multiple dummy coefficients.

Calculation from SSE and SST

R² can first be obtained from the residual sum of squares and total sum of squares:

After calculating R², use the adjusted formula above. In the separate Excel numeric model, SSE is 1003.4409 and SST is 6763.2666, producing R² = 0.8516. With n = 649 and p = 10, adjusted R² becomes 0.8493.

How to Interpret Adjusted R Squared

An adjusted R² of 0.8504 means that the final Python regression accounts for approximately 85.04% of the sample variation in G3 after applying the adjustment for 13 predictor degrees of freedom. It does not mean that 85.04% of individual grades are predicted exactly. It also does not mean the model is 85.04% accurate, because accuracy is not the definition of adjusted R².

Interpretation should answer four questions. First, what is the outcome? Here it is G3 final grade. Second, what sample and model specification produced the statistic? Here the main Python model uses 649 observations and 13 predictor degrees of freedom. Third, how does adjusted R² compare with ordinary R²? The difference is approximately .0030. Fourth, how does the value compare with plausible alternative models? In the Python sequence, adjusted R² rises from .6824 in Model 1 to .8504 in Model 5.

Dataset and Variables Used

The analysis uses a student-performance dataset with 649 rows and 33 original columns. The outcome is G3, the final grade. The main Python report considers nine numeric predictors—G1, G2, studytime, failures, absences, age, Medu, Fedu and traveltime—and four categorical predictors—school, sex, address and famsize. The categorical variables are represented through indicator coefficients, giving the final model 13 predictor degrees of freedom.

Verified Python Results and Five-Model Comparison

The Python report compares five increasingly complex models. Model 1 uses one prior-grade predictor. Model 2 uses the prior-grade sequence. Model 3 uses five academic predictors. Model 4 adds demographic terms for nine predictor degrees of freedom. Model 5 is the full adjusted model with 13 predictor degrees of freedom.

Model 5 has the highest adjusted R² and the lowest RMSE and MAE in this sequence. Model 3 has the lowest AIC, while Model 2 has the lowest BIC. This is a useful demonstration that model-selection criteria can disagree because they penalize complexity differently. Adjusted R² chooses Model 5; AIC gives more weight to fit and chooses Model 3; BIC applies a stronger complexity penalty and chooses Model 2.

Key Final-Model Coefficients

The remaining terms have smaller absolute t values and are not statistically significant at .05 in the full model. Their presence can still change other coefficients, contribute jointly or serve a theory-driven adjustment purpose. A nonsignificant individual coefficient does not automatically require deletion, and a high adjusted R² does not make every coefficient meaningful.

Open the complete Adjusted R Squared Python report PDF

SPSS Output Interpretation for Adjusted R Squared

The supplied SPSS PDF verifies the broader 13-predictor specification. The final Model Summary reports R = .924, R² = .853, Adjusted R² = .850 and Std. Error of the Estimate = 1.249. The ANOVA table reports F(13, 635) = 284.425, p < .001. These rounded SPSS results agree with the Python report.

The Python RMSE of approximately 1.2359 is slightly lower than the SPSS standard error of approximately 1.249 because the two summaries use different denominators. RMSE commonly uses n, while the regression standard error uses residual degrees of freedom. This is a calculation convention, not a substantive disagreement.

The SPSS residual statistics show residuals from approximately -9.059 to 5.180, with standardized residuals from approximately -7.250 to 4.146. The residual histogram and normality output therefore deserve attention. A high adjusted R² does not remove the need to inspect extreme cases, distribution shape and influence.

Open the complete Adjusted R Squared SPSS output PDF

Excel Worked Results for Adjusted R Squared

The downloadable Excel workbook provides a transparent calculation using 10 numeric predictors: G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime and health. It contains a Summary sheet, Coefficients sheet, Model Comparison sheet, Worked Data sheet and Formula Guide sheet.

The Excel Model Comparison sheet uses a different sequence from the full Python/SPSS sequence. In that workbook, adjusted R² values are .6824, .8473, .8494, .8492 and .8493. The highest is Model 3 at .8494014, narrowly above the 10-predictor Model 5 value of .8493082. The correct workbook-specific conclusion is therefore that Model 3 gives the best adjusted fit among the numeric workbook models.

Excel Prediction Equation

The 10-predictor workbook uses the following fitted coefficients:

This equation is useful for a worked Excel demonstration, but it should not be mixed with the 13-predictor Python/SPSS coefficient table. The models are closely related but not identical.

Python Chart-by-Chart Interpretation with Real Model Values

The seven Python figures use the same 649 complete observations and the five-model sequence reported in the Python PDF. Every interpretation below names the relevant model, variable or diagnostic and integrates the real numerical values shown by the analysis rather than giving only a general visual description.

Chart 1: Adjusted R Squared Model Comparison

The first bar is much lower than the remaining bars because Model 1 uses only one prior-grade predictor and has adjusted R² of approximately .6824. Adding the second prior-grade measure raises adjusted R² to approximately .8473. Models 3, 4 and 5 cluster close to .85, showing diminishing improvements after the strong G1/G2 information is included.

The highest bar is Model 5 at approximately .8504. The visual difference between Models 3 and 5 is small, but Model 5 is still the adjusted-R² winner in the Python report. This corrects the tempting but inaccurate conclusion that the third bar must be highest simply because the models look nearly tied.

Chart 2: R Squared vs Adjusted R Squared

The two lines remain close because the sample is large relative to the model sizes. The gap is around .0005 for Models 1 and 2, .0012 for Model 3, .0021 for Model 4 and .0030 for Model 5. The widening distance illustrates the increasing complexity charge.

Raw R² rises at every step, from .6829 to .8534. Adjusted R² declines slightly from Model 3 to Model 4, then rises to its maximum in Model 5. The chart therefore shows both important behaviors: R² never punishes added predictors, while adjusted R² can move down or up depending on whether the new terms justify their cost.

Chart 3: Observed vs Predicted Final Model

Most points lie close to the diagonal, especially across the middle and upper grade ranges, which is consistent with the strong R² and adjusted R². The horizontal bands occur because G3 is recorded as an integer grade while predictions are continuous. Points far below the diagonal indicate overprediction; points above it indicate underprediction.

Several observations with very low observed G3 sit well below their predicted values. These cases help explain the large negative residual tail. The chart supports strong overall agreement, but it also shows why a single fit statistic should not replace case-level diagnostic review.

Chart 4: Residuals vs Fitted Values

A residuals-versus-fitted plot should show errors scattered around zero without a strong curve or funnel. The supplied image includes positive and negative residuals around the zero line, but it also shows several large negative values. Those points require investigation because they may represent unusual outcomes, influential observations or model misspecification.

Chart 5: Standardized Residual Distribution

The histogram is concentrated near zero, but the left tail extends much farther than the right tail. SPSS reports standardized residuals from about -7.25 to 4.15, and its normality tests are significant. With n = 649, formal normality tests are sensitive, but the extreme left-tail cases are still practically important.

The appropriate conclusion is not that the model is unusable. Instead, report that the central residual mass is near zero while a small number of extreme observations create skewness and heavy tails. Use a Q-Q plot, P-P plot and outlier detection to identify the source of the departures.

Chart 6: Adjusted R Squared Complexity Penalty

The penalty is smallest for the two-predictor model and grows as the model expands to 5, 9 and 13 predictor degrees of freedom. At 13 df, the gap is about .0030. The line is not a measure of overfitting by itself; it is the direct mathematical consequence of the adjusted R² formula combined with the observed R² improvements.

The final point remains small because the sample contains 649 observations. In a much smaller sample, the same 13-df model would receive a larger adjustment. This is why adjusted R² must always be interpreted with both n and p.

Chart 7: Top Coefficient t Values

G2 dominates the chart with t approximately 25.76. G1 is a distant second at about 3.62. Failures and the male indicator have negative t values near -2.43 and -2.05, while traveltime is positive near 2.03. Most other predictors have absolute t values below 2.

The chart distinguishes overall model fit from individual evidence. The model can have adjusted R² above .85 even though many covariates are not individually significant, because the strong prior-grade variables explain most of the variation. Coefficient significance, effect size, multicollinearity and theoretical purpose should be assessed separately.

R Charts, Tables and Independent Validation

The R workflow fits the same five regression specifications with lm(), uses G3 as the outcome, and evaluates the models with R², adjusted R², RMSE, MAE, AIC and BIC. It reproduces the Python ranking: Model 5 has the highest adjusted R² and lowest in-sample RMSE and MAE, Model 3 has the lowest AIC, and Model 2 has the lowest BIC. The R report and chart discussion therefore provide software-to-software validation rather than a different statistical question.

Open the complete Adjusted R Squared R report PDF

R Model-Comparison Results

The consolidated R table prevents the chart section from becoming repetitive. Adjusted R² increases sharply from .6824 in Model 1 to .8473 in Model 2, reaches .8494 in Model 3, falls slightly to .8490 in Model 4, and then reaches the maximum of .8504 in Model 5. The final choice depends on the stated criterion: Model 5 for adjusted explained variance and in-sample error, Model 3 for AIC, or Model 2 for BIC.

Selected R Final-Model Coefficients

R Chart 1: Adjusted R Squared Model Comparison

The R validation chart begins with Model 1 adjusted R² = .6824, then rises to .8473 for Model 2. The large increase of approximately .1649 shows that adding the second prior-grade variable contributes far more explanatory information than the later model expansions. Model 3 reaches .8494, Model 4 records .8490, and Model 5 reaches the highest value of .8504.

The apparent near-tie among Models 3, 4 and 5 should be interpreted numerically. Model 4 is about .0004 lower than Model 3 because its additional terms do not fully overcome the larger predictor penalty. Model 5 then improves adjusted R² by about .0014 over Model 4 and by about .0010 over Model 3. R therefore confirms Model 5 as the adjusted-R² winner, although the practical gain beyond Model 3 is small.

R Chart 2: R Squared vs Adjusted R Squared

Ordinary R² rises monotonically from .6829 to .8478, .8506, .8511 and .8534. Adjusted R² follows the values .6824, .8473, .8494, .8490 and .8504. The two series are close because the sample size is 649, but the separation increases as the model consumes more predictor degrees of freedom.

The exact R²-minus-adjusted-R² penalties are approximately .0005, .0005, .0012, .0021 and .0030. The small fall in adjusted R² from Model 3 to Model 4 occurs even though raw R² increases from .8506 to .8511. This is the central reason for reporting adjusted R²: the additional Model 4 terms improve raw fit, but not enough to justify their full complexity cost.

R Chart 3: Observed vs Predicted G3 for the Final Model

The final R model uses 649 observations and 13 predictor degrees of freedom. Its R² = .8534, adjusted R² = .8504, RMSE = 1.2359 and MAE = .7777. Most fitted points lie near the 45-degree agreement line, especially through the central grade range, which is consistent with the model explaining more than 85% of sample variance.

The horizontal rows of points occur because the observed outcome G3 is recorded as an integer while fitted values are continuous. A small number of very low observed grades fall well below their fitted values, indicating overprediction and producing large negative residuals. The chart therefore supports strong aggregate fit but also identifies observations that should be investigated through studentized residuals, leverage and Cook’s distance.

R Chart 4: Residuals vs Fitted Values

The residuals should fluctuate around zero without a systematic curve or progressively widening funnel. The output reports raw residuals from approximately −9.059 to 5.180. Most cases remain much closer to zero, but the long negative side shows that several low observed grades were predicted too highly by the final model.

Those extreme negative cases matter even though adjusted R² is .8504. Adjusted R² summarizes explained variation and does not test linearity, constant variance or influence. The R workflow should therefore pair this plot with formal or graphical checks for heteroskedasticity, nonlinear form and influential observations.

R Chart 5: Standardized Residual Distribution

The standardized residuals range from approximately −7.250 to 4.146. The main body of the distribution is centered near zero, but the negative tail is longer and contains more extreme values than the positive tail. A perfectly normal residual distribution would be more symmetric and would rarely contain standardized residuals with absolute values this large.

The R interpretation should therefore separate central fit from tail behavior. Model 5 explains a large share of the variance, but a small group of cases departs strongly from the fitted relationship. A Q-Q plot, studentized residual review and influence analysis are necessary before claiming that the regression assumptions are fully satisfactory.

R Chart 6: Adjusted R Squared Complexity Penalty

The penalty is .000490 with one predictor degree of freedom, .000471 with two, .001162 with five, .0021 with nine, and .0030 with 13. The slight reduction from Model 1 to Model 2 occurs because the second predictor adds a very large amount of fit, while the larger later models receive progressively stronger adjustments.

The final .0030 penalty is still modest because n = 649 is large relative to 13 predictor degrees of freedom. This does not prove that the model is immune to overfitting. It means only that, under the adjusted-R² formula and this sample size, Model 5 retains most of its raw R² after the complexity correction.

R Chart 7: Top Final-Model Coefficient t Values

G2 dominates with t = 25.7601, followed by G1 with t = 3.6153. The strongest negative displayed terms are failures with t = −2.4308 and the M-coded sex indicator with t = −2.0534. The chart also places traveltime near a positive t value of about 2.03, while most remaining terms have absolute t values below 2.

This chart explains why a model can have adjusted R² = .8504 even when many individual terms are nonsignificant. G1 and especially G2 account for most of the strong fit. The other terms provide smaller incremental adjustments, possible confounding control or group comparisons. Overall adjusted fit, individual coefficient evidence and theoretical inclusion are related but distinct decisions.

Regression Assumptions and Diagnostics

Adjusted R² is not an assumption test. A model can have high adjusted R² and still violate linearity, equal variance, independence, residual distribution or influence conditions. At minimum, examine residual plots, leverage, Cook’s distance and collinearity diagnostics.

The SPSS output shows a maximum Cook’s distance of approximately .140, which is not automatically decisive but is far above the median and deserves case review. The extreme standardized residuals are more immediately visible. Report diagnostic findings rather than hiding them behind the high adjusted R².

Adjusted R Squared, AIC, BIC and Predictive Performance

Adjusted R² is one model-selection criterion, not the only criterion. In the Python table, adjusted R² chooses Model 5, AIC chooses Model 3 and BIC chooses Model 2. This disagreement is informative. Model 5 has the best in-sample adjusted fit and error metrics, but Model 2 or Model 3 may be preferred when parsimony is more important.

For prediction, use cross-validation, a holdout test set or predicted R². Adjusted R² is calculated from the training sample and can still look optimistic when the analyst repeatedly searches many models. Model selection should combine theory, pre-specified goals, diagnostic quality and out-of-sample evidence.

SPSS, Python, R and Excel Workflows

Code Blocks for Adjusted R Squared

SPSS Syntax

* Adjusted R Squared final model in SPSS.
* Assumes the variables are already open and categorical dummies are prepared.

SET PRINTBACK=OFF MPRINT=OFF RESULTS=ON DECIMAL=DOT.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=AdjustedRSquaredOutput.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu Fedu traveltime
                school_dummy sex_dummy address_dummy famsize_dummy
  /STATISTICS COEFF OUTS R ANOVA CHANGE COLLIN TOL CI(95)
  /RESIDUALS HISTOGRAM(ZRESID) NORMPROB(ZRESID)
  /SAVE PRED ZPRED RESID ZRESID COOK LEVER.

OUTPUT SAVE OUTFILE='D:\DATA ANALYSIS\H Regression Tests and Models\Adjusted R Squared\SPSS_Output\spv\Adjusted-R-Squared-Output.spv'.
OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE='D:\DATA ANALYSIS\H Regression Tests and Models\Adjusted R Squared\SPSS_Output\pdf\Adjusted-R-Squared-SPSS-Output.pdf'.

Python Code

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf
from sklearn.metrics import mean_absolute_error, mean_squared_error

DATA_FILE = "dataset.csv"
df = pd.read_csv(DATA_FILE)

formula = (
    "G3 ~ G1 + G2 + studytime + failures + absences + age + "
    "Medu + Fedu + traveltime + C(school) + C(sex) + "
    "C(address) + C(famsize)"
)
model = smf.ols(formula, data=df).fit()

pred = model.predict(df)
valid = model.model.data.row_labels
observed = df.loc[valid, "G3"].astype(float)
predicted = pd.Series(pred, index=valid)

rmse = np.sqrt(mean_squared_error(observed, predicted))
mae = mean_absolute_error(observed, predicted)

print("R squared:", model.rsquared)
print("Adjusted R squared:", model.rsquared_adj)
print("F statistic:", model.fvalue)
print("F-test p-value:", model.f_pvalue)
print("AIC:", model.aic, "BIC:", model.bic)
print("RMSE:", rmse, "MAE:", mae)
print(model.summary())

R Code and R Report PDF

Download the Adjusted R Squared R Report PDF

The R report contains the R model summary, adjusted R-squared calculations, coefficient output, model-comparison results and regression diagnostic charts generated by the R workflow.

df <- read.csv("dataset.csv", stringsAsFactors = TRUE)

model5 <- lm(
  G3 ~ G1 + G2 + studytime + failures + absences + age +
    Medu + Fedu + traveltime + school + sex + address + famsize,
  data = df
)

s <- summary(model5)
cat("R squared:", s$r.squared, "
")
cat("Adjusted R squared:", s$adj.r.squared, "
")
cat("AIC:", AIC(model5), "
")
cat("BIC:", BIC(model5), "
")

pred <- fitted(model5)
res <- residuals(model5)
rmse <- sqrt(mean(res^2))
mae <- mean(abs(res))
cat("RMSE:", rmse, "MAE:", mae, "
")

par(mfrow = c(2, 2))
plot(model5)

Excel Formulas

Assume:
Observed G3 range = L2:L650
Predicted G3 range = M2:M650
Residual range = N2:N650
Squared residual range = O2:O650
Squared deviation range = P2:P650

Residual:
=L2-M2

Squared residual:
=N2^2

Squared deviation from outcome mean:
=(L2-AVERAGE($L$2:$L$650))^2

SSE:
=SUM(O2:O650)

SST:
=SUM(P2:P650)

R squared:
=1-(SSE/SST)

Adjusted R squared:
=1-(1-R_squared)*(n-1)/(n-p-1)

RMSE using n:
=SQRT(SSE/n)

Regression standard error using residual df:
=SQRT(SSE/(n-p-1))

APA-Style Reporting for Adjusted R Squared

Full Python/SPSS Model Report

Model Comparison Report

R Validation Report

Excel Worked-Model Report

Common Mistakes When Interpreting Adjusted R Squared

Downloads and Resources

Related Statistical Guides