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eduKateSG · Why Mathematics? · R-Squared: Explained Variation and Its Limits
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 5–9
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 10–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–31
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a regression model needs a clear measure of how much sample variation it accounts for? R-squared, the coefficient of determination, compares the model's regression sum of squares with the total sum of squares, or equivalently subtracts the residual fraction from one.
R-squared is useful descriptive mathematics, but it is easy to overclaim. A high value does not prove causation, correct functional form, useful prediction or good behaviour outside the observed data.
Did You Know? A model can have an impressive R-squared and still miss a curve, contain influential cases or make poor forecasts at new predictor values.
Full chapter index
Chapters 1–4 · Understand the question
Chapters 5–9 · Reproduce the example
Chapters 10–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–31 · Study and verify
Responses vary around their sample mean. Regression divides that total variation into a part represented by fitted values and a residual part left unexplained by the fitted model.
Penn State defines R-squared as SSR/SSTO and, because SSTO=SSR+SSE, also as 1-SSE/SSTO. Read Penn State STAT 501.
SSTO sums squared distances from observations to the response mean. SSR sums squared distances from fitted values to that mean. SSE sums squared distances from observations to fitted values.
| SSTO | SSE | SSR |
|---|---|---|
| 200 | 50 | 150 |
Suppose SSTO=200 and SSE=50. Then SSR=200-50=150 and R-squared=150/200=1-50/200=0.75.
| SSR/SSTO | 1−SSE/SSTO |
|---|---|
| 0.75 | 0.75 |
An R-squared of 0.75 means the fitted regression accounts for 75% of the sample response variation about its mean under the model and data used. It does not mean 75% of individual outcomes are predicted correctly.
Penn State warns that explained variation can be mistaken for causation. A predictor may be associated with the response because of common causes, selection, timing or measurement patterns.
For ordinary least squares with an intercept, centred SSTO=SSR+SSE and R-squared lies from zero to one. Without an intercept or with nonstandard definitions, software may use uncentred quantities and values can behave differently.
Sums of squares carry squared response units, but their ratio is unitless. R-squared does not show whether typical errors are small in the practical units that matter.
Residual standard error returns to response units and describes typical scatter around the model under its assumptions. Report it alongside R-squared.
In ordinary least squares, a larger model includes the smaller model as a special case. Training SSE cannot increase, so R-squared cannot fall even when an added term is unhelpful.
Adjusted R-squared adds a degrees-of-freedom penalty. It can fall when the gain in fit is too small, but it remains an in-sample comparison.
Prediction depends on residual scale, leverage, data shift and validation. PRESS or a genuine test set answers a more direct unseen-case question.
A narrow range of x can produce a low R-squared even when measurement is precise, while a broad range can raise it. Compare studies only with attention to design and population.
A straight line can achieve a high R-squared across a curved relation yet leave systematic residuals. Inspect residual and partial-residual plots.
One extreme case can raise or lower R-squared substantially. Check leverage, studentized residuals and Cook’s distance rather than trusting the summary alone.
Calculate the response mean, SSTO, fitted values, residuals, SSE and SSR in labelled columns. Verify both R-squared formulas and the sum-of-squares identity.
Return R-squared with sample size, model formula, intercept convention, SSE, SSTO and validation metrics. Preserve identical rows across candidate comparisons.
R-squared appears in engineering, economics, environmental science, education research and data analytics. The professional skill is knowing what the proportion does and does not say.
Fit a line to one dataset, then add an influential point. Track R-squared and residual plots. Explain why a single summary cannot diagnose the model.
R-squared is not accuracy, probability, effect size in response units or causal evidence. A lower value does not automatically make research useless.
The adjusted R-squared guide explains the parameter penalty. The PRESS guide adds omitted-case prediction.
Students should show SSTO, SSE and the intercept convention. Parents can ask whether practical error and validation support the headline proportion.
- Recalculate when SSE=80.
- Explain why changing the predictor range can change R-squared.
- Describe a high-R-squared model that still fails a residual check.
### Is 0.8 always good? No. Context, error units and purpose matter.
### Does R-squared prove cause? No.
### Can it compare different responses? Usually not without careful justification.
Suppose two models use the same 40 observations. Model A has SSE 60 and Model B has SSE 55 with two extra parameters. Ordinary R-squared favours B, but the decision should also examine adjusted R-squared, PRESS, residual shape and whether the extra terms have a defensible mechanism. Writing all five items beside the score prevents a descriptive proportion from becoming an automatic model-selection rule.
R-squared from a model of log income describes variation on the log scale, not the original dollar scale. Back-transformed prediction has additional complications. Compare models on different response scales only with a criterion designed for that comparison and a clear practical question.
Aggregating individual observations into group means can change both total and residual variation, often producing a very different R-squared. State the unit of analysis. A strong relationship among school averages, for example, does not automatically describe variation among individual students.
Two unrelated quantities that both trend over time can produce a high R-squared. Plot time, examine stationarity and use a design appropriate to temporal dependence. A fit statistic does not remove the need for a plausible mechanism.
The usual centred R-squared compares the regression with predicting the sample mean. That baseline may not match the real operational alternative, such as yesterday’s value or an existing production model. Report performance against the decision-relevant benchmark as well.
A careful conclusion names the response, sample and model: “The fitted model accounted for 75% of the observed variation in y about its sample mean.” Then add the residual error scale and validation result. This wording keeps the claim descriptive and auditable.
Use the Mathematics Learning Hub to connect ratios, variance and functions. Rebuild an ANOVA table and verify every sum-of-squares identity.
Extended practice laboratory
Practice case 1: Calculate response mean
Setup: Start from the article’s R-squared worked example for case 1. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Calculate SSTO
Setup: Start from the article’s R-squared worked example for case 2. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Calculate fitted values
Setup: Start from the article’s R-squared worked example for case 3. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Calculate residuals
Setup: Start from the article’s R-squared worked example for case 4. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate SSE
Setup: Start from the article’s R-squared worked example for case 5. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Calculate SSR
Setup: Start from the article’s R-squared worked example for case 6. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Verify SS identity
Setup: Start from the article’s R-squared worked example for case 7. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Form SSR over SSTO
Setup: Start from the article’s R-squared worked example for case 8. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Form one minus SSE ratio
Setup: Start from the article’s R-squared worked example for case 9. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Convert to percentage
Setup: Start from the article’s R-squared worked example for case 10. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Increase SSE
Setup: Start from the article’s R-squared worked example for case 11. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Add predictor
Setup: Start from the article’s R-squared worked example for case 12. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Change predictor range
Setup: Start from the article’s R-squared worked example for case 13. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Remove intercept
Setup: Start from the article’s R-squared worked example for case 14. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Compare adjusted R squared
Setup: Start from the article’s R-squared worked example for case 15. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Compare residual error
Setup: Start from the article’s R-squared worked example for case 16. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Compare PRESS
Setup: Start from the article’s R-squared worked example for case 17. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Inspect curvature
Setup: Start from the article’s R-squared worked example for case 18. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Inspect leverage
Setup: Start from the article’s R-squared worked example for case 19. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Inspect Cook distance
Setup: Start from the article’s R-squared worked example for case 20. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Change response units
Setup: Start from the article’s R-squared worked example for case 21. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Check same rows
Setup: Start from the article’s R-squared worked example for case 22. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Reject accuracy claim
Setup: Start from the article’s R-squared worked example for case 23. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Reject causal claim
Setup: Start from the article’s R-squared worked example for case 24. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write an R-squared conclusion
Setup: Start from the article’s R-squared worked example for case 25. Task: Change the named quantity, recompute dependent values and compare with the original.
Check: Check the defining identity, preserve units and state one diagnostic or validation step before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.