Three ratio/price pairs are 0.85/$80,000,0.95/$200,000 and 1.10/$400,000. Define PRD as mean ratio divided by sale-price-weighted mean ratio. Which rounded PRD and observed sample tendency follow?
Correct Answer
B) About 0.94; higher-price sales have higher observed ratios
Mean ratio=(0.85+0.95+1.10)/3=0.9667. Price-weighted mean=(68000+190000+440000)/(80000+200000+400000)=698000/680000=1.0265. PRD=0.9417, rounded 0.94. Higher-price sales here have higher observed ratios, a progressive sample pattern. This is not proof of population inequity or a binding compliance decision.
Why This Is the Correct Answer
Option B is correct because the mean ratio is (0.85 + 0.95 + 1.10) ÷ 3 = 0.9667, and the price-weighted mean is (68,000 + 190,000 + 440,000) ÷ (80,000 + 200,000 + 400,000) = 698,000 ÷ 680,000 = 1.0265. PRD = 0.9667 ÷ 1.0265 = 0.9417, about 0.94. In this sample the highest-priced sale has the highest ratio, a progressive pattern, though this alone does not prove population inequity.
Why the Other Options Are Wrong
Option A: About 1.03; higher-price sales have lower observed ratios
Option A, about 1.03 with higher-price sales having lower ratios, uses roughly the weighted mean ratio (1.0265) as the PRD and states the opposite tendency. The PRD is 0.9667 ÷ 1.0265 ≈ 0.94, and here higher-price sales have higher ratios.
Option C: Exactly 1.00; each property has the same ratio
Option C, exactly 1.00 because each property has the same ratio, contradicts the data, since the ratios are 0.85, 0.95 and 1.10. The PRD is 0.9667 ÷ 1.0265 ≈ 0.94.
Option D: Exactly 1.10; the maximum single ratio is the PRD
Option D, exactly 1.10 because the maximum ratio is the PRD, misdefines PRD. PRD is the mean ratio divided by the weighted mean ratio, 0.9667 ÷ 1.0265 ≈ 0.94, not the largest single ratio.
Below one, big ones high
A PRD under 1.00 means the weighted mean is pulled up, so the expensive sales carry the higher ratios.
How to use: Use this to interpret the direction of a computed PRD.
Exam Tip
Compute the weighted mean by summing assessed values (ratio times price) and dividing by total price, then divide the simple mean by it.
Common Mistakes to Avoid
- -Inverting the PRD by dividing the weighted mean by the mean
- -Reading a PRD below 1.00 as regressive
- -Using the simple sum of prices without weighting ratios by price
Concept Deep Dive
Analysis
This question tests computing the price-related differential (PRD) and interpreting its direction. PRD is the mean ratio divided by the sale-price-weighted mean ratio. A PRD below 1.00 indicates higher-priced sales have higher ratios in the sample, a progressive pattern.
Background Knowledge
The PRD compares the simple mean ratio with the sale-price-weighted mean ratio to detect vertical inequity related to price. A PRD above 1.00 suggests lower-priced properties have relatively higher ratios (regressivity), while a PRD below 1.00 suggests higher-priced properties have relatively higher ratios (progressivity). The weighted mean and PRD are sensitive to high-priced sales.
Real-World Application
An assessor's ratio study shows a PRD of about 0.94, with high-value homes assessed at higher ratios than lower-value homes. Analysts treat this as a signal to investigate the valuation model for higher-priced properties, while recognizing the sample alone is not a binding compliance finding.
More Statistics Questions
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Previous Question
Five sale prices are $200,000, $210,000, $220,000, $230,000 and $240,000. Add one sale at $1,000,000. What are the new mean and median?
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Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.