A regression model reports an R-squared of 0.82. What does this indicate?
Correct Answer
D) The model explains 82 percent of price variation
Why this is correct: The original explanation states R-squared is "the share of variation in the dependent variable accounted for by the model." An R-squared of 0.82 means 82% of the variation in sale prices in this sample is explained by the independent variables in the model. Why the other choices are wrong: Eighty-two percent of sales fall within the range describes a confidence interval, not R-squared. Each variable contributes 82 percent of the value misinterprets R-squared as an allocation to individual variables. The model will be accurate 82 percent of the time incorrectly implies a prediction accuracy rate. Exam tip: R-squared measures explanatory power within the sample data, not predictive accuracy for new properties.
Why This Is the Correct Answer
Option D states the definition correctly: the model explains 82 percent of the variation in price. The key word is variation, since R-squared is computed from sums of squared deviations around the mean rather than from counts of observations or from prediction errors on new data. Framing it as explained variation also keeps the interpretation tied to the sample the model was estimated on. That is exactly what the statistic reports and nothing more.
Why the Other Options Are Wrong
Option A: Eighty-two percent of sales fall within the range
Describing a share of sales falling within a range is the language of a prediction interval or a confidence interval, which quantifies uncertainty around an estimate in the units of the variable. R-squared is a unitless proportion of variance and defines no range at all. The two often appear in the same regression output, which is what makes the confusion easy.
Option B: Each variable contributes 82 percent of the value
R-squared applies to the model as a whole and cannot be allocated to individual variables, particularly when the independent variables are correlated with each other, as living area and bedroom count almost always are. The contribution of a single variable is examined through its coefficient, standard error, and significance test, not through the overall fit statistic. Reading it as a per-variable figure misunderstands the arithmetic entirely.
Option C: The model will be accurate 82 percent of the time
Accuracy on new observations is a different property from in-sample fit, and a model can post a high R-squared while predicting poorly out of sample, which is the classic symptom of overfitting. Predictive performance is assessed with holdout testing or with error measures such as the coefficient of dispersion in mass appraisal. Treating fit as an accuracy rate would let an overfitted model appear trustworthy.
Squared Means Spread
The squared in R-squared points at spread. It answers how much of the scatter in prices the model can account for, not how many sales it got right and not how much any one variable did. Eighty-two percent of the scatter explained, eighteen percent still unaccounted for.
How to use: When an option describes counts of properties, a range, or accuracy on future sales, it is describing something other than R-squared. Keep the option that speaks about variation or variance in the dependent variable. Then remember to ask separately whether the individual coefficients make appraisal sense before relying on the model.
Exam Tip
Adding variables can only raise R-squared, never lower it, so a rising R-squared is not by itself evidence that a model improved; look at adjusted R-squared instead.
Common Mistakes to Avoid
- -Treating a high R-squared as proof the model predicts new sales accurately
- -Adding variables to raise R-squared without checking adjusted R-squared or significance
- -Relying on a model whose coefficients carry signs or magnitudes no appraiser would defend
Concept Deep Dive
Analysis
This question tests what the coefficient of determination measures. In a regression predicting sale price from characteristics such as living area, age, lot size, and quality, the dependent variable varies across the sample, and R-squared reports the proportion of that variation the model accounts for. An R-squared of 0.82 means 82 percent of the observed variation in price is explained by the independent variables together, leaving 18 percent to factors the model omits plus random noise. Three limits matter for appraisers. R-squared describes fit within the sample used to build the model, not accuracy on properties outside it. It always rises when variables are added, even meaningless ones, which is why adjusted R-squared exists to penalize additional terms. And a high R-squared says nothing about whether individual coefficients are reasonable, correctly signed, or statistically significant, so a model can fit well overall while producing an adjustment that no appraiser would defend.
Background Knowledge
You need to know the structure of a regression model, including dependent and independent variables, coefficients, and residuals, and that R-squared is the share of variance in the dependent variable explained by the model. You should also know why adjusted R-squared is preferred when comparing models with different numbers of variables, and that individual coefficients require their own significance testing regardless of overall fit.
Real-World Application
Building a regression on 240 sales to support adjustments, an appraiser obtains an R-squared of 0.82 but finds the garage coefficient is negative and not significant. Rather than trusting the overall fit, the appraiser drops the variable, re-estimates, and supports the garage adjustment with paired sales instead.
More Statistics Questions
A set of comparable sales has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation?
A property sold for $400,000 and resold three years later for $463,050 with no physical change. What compound annual rate does this indicate?
A histogram of neighborhood sale prices shows two distinct peaks. What does this most likely mean?
What does it mean to validate a regression model?
In a market study, what does a frequency distribution of sale prices show?
An appraiser includes months elapsed since each sale as a variable in a price model. What is this intended to capture?
An appraiser presents a statistical analysis in a report. What must accompany it for the reader to weigh it?
An R-squared of 0.86 in a sales model indicates that:
Which measure would best summarize the most common lot size in a subdivision?
Paired sales analysis and regression differ mainly in that regression:
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