An R-squared of 0.86 in a sales model indicates that:
Correct Answer
D) 86 percent of price variation is explained by the model
Why this is correct: R-squared (R^2) is a statistical measure that represents the proportion of the variance in the dependent variable (e.g., sale price) that is explained by the independent variables in the regression model. An R^2 of 0.86 means 86% of the price variation is explained. Why the other choices are wrong: It does not indicate coefficient accuracy to within 14%. It does not guarantee the model will predict any single sale within 86%. It is not a count of significant observations. Exam tip: High R^2 means the model fits the data well, but it doesn't guarantee the right variables are used or that predictions are accurate.
Why This Is the Correct Answer
Eighty-six percent of price variation being explained by the model is the definition of an R-squared of 0.86. The statistic is a proportion of variance, unitless, and always between zero and one for a properly specified model. Reading it correctly keeps the appraiser from overclaiming, since good fit is not the same as good prediction or correct specification. The dollar magnitude of typical error comes from the standard error of the estimate instead.
Why the Other Options Are Wrong
Option A: The coefficients are accurate to within 14 percent
Coefficient accuracy is reported by each coefficient's standard error and the associated t-statistic, one per variable, not by a single overall fit measure. R-squared says nothing about how precisely any individual slope was estimated. A model can have a high R-squared while containing several coefficients too imprecise to use as adjustments.
Option B: The model will predict any sale within 86 percent
Prediction accuracy for individual sales is described by the standard error of the estimate and the resulting prediction intervals, expressed in dollars. An R-squared of 0.86 offers no guarantee about any single prediction, and individual errors can be large even in a well-fitting model. The option converts a variance proportion into an accuracy promise it cannot make.
Option C: 86 of the observations were statistically significant
R-squared is a proportion, not a count, and it has no relationship to how many observations were statistically significant. Significance attaches to coefficients rather than to observations in any case, so the sentence misuses two concepts at once. The number 86 in the option is the decimal read as a count.
R-Squared Explains, SEE Predicts
R-squared answers how much of the variation the model accounts for, as a share. The standard error of the estimate answers how far off a prediction typically lands, in dollars. Different questions, different statistics.
How to use: When a stem quotes a decimal between zero and one, think proportion of variance explained. When it quotes a dollar figure, think prediction error. Never let one answer the other's question.
Exam Tip
Prefer adjusted R-squared when comparing models with different numbers of variables. Plain R-squared can only increase as variables are added, even useless ones.
Common Mistakes to Avoid
- -Reading R-squared as a guarantee of prediction accuracy
- -Quoting plain R-squared when comparing models of differing complexity
- -Treating good fit as evidence that the model is correctly specified
Concept Deep Dive
Analysis
R-squared, the coefficient of determination, reports the proportion of variance in the dependent variable that the model's independent variables collectively explain. An R-squared of 0.86 means 86 percent of the variation in sale price across the sample is accounted for by the variables in the model, leaving 14 percent attributable to factors the model does not capture, whether unmeasured characteristics such as condition and view, transaction-specific circumstances, or random variation. The statistic describes fit on the data the model was estimated from and says nothing about several things appraisers care about. It does not indicate whether the right variables were chosen, since a model can fit well while omitting a variable that matters and correlating with one that does not. It does not measure prediction error in dollars, which is what the standard error of the estimate reports. And it does not establish causation. Adjusted R-squared is generally the better statistic to quote, because plain R-squared can only rise as variables are added, whereas adjusted R-squared penalizes variables that do not earn their place.
Background Knowledge
You need the components of regression output and what each measures: coefficients, their standard errors, t-statistics and p-values, R-squared and adjusted R-squared, the F-statistic, and the standard error of the estimate. You should also know that fit statistics describe performance on the estimation sample and do not establish correct specification or causation.
Real-World Application
An appraiser reports a model with adjusted R-squared of 0.86 and a standard error of $19,000 on $350,000 homes. She notes the model fits well overall but that individual predictions carry meaningful error, uses it as a corroborating check, and relies on her verified grid for the conclusion.
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?
Which measure would best summarize the most common lot size in a subdivision?
Paired sales analysis and regression differ mainly in that regression:
Price per square foot declines as homes get larger. What does this imply for a linear regression of price on area?
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