Assessment-to-sale ratios are 1.2, 1.1, 1, 0.9; corresponding sale prices are $100,000, $200,000, $300,000, $400,000 in the same order. Compute PRD as the arithmetic mean ratio divided by the sale-price-weighted mean ratio. Round to two decimals.
Correct Answer
A) 1.05
Arithmetic mean=1.05; weighted mean=Σ(ratio×sale price)/Σ(sale prices)=1; PRD=1.05/1=1.05. A point result is not a statistical test of population regressivity. Common errors: inverting the ratio (weighted mean ÷ arithmetic mean) gives 0.95; reporting the sale-price-weighted mean ratio instead of the PRD gives 1.00; reporting only the excess of the PRD over 1 gives 0.05.
Why This Is the Correct Answer
Option A is correct because the arithmetic mean ratio is (1.2 + 1.1 + 1 + 0.9) / 4 = 1.05. The weighted mean is Σ(ratio × sale price) / Σ(sale prices) = ($120,000 + $220,000 + $300,000 + $360,000) / $1,000,000 = 1.00. PRD = 1.05 / 1.00 = 1.05.
Why the Other Options Are Wrong
Option B: 0.95
Option B, 0.95, comes from inverting the ratio, dividing the weighted mean by the arithmetic mean (1.00 / 1.05 = 0.95). The PRD is the arithmetic mean divided by the weighted mean: 1.05 / 1.00 = 1.05.
Option C: 1.00
Option C, 1.00, is the sale-price-weighted mean ratio, reported instead of the PRD. Dividing the arithmetic mean by it gives 1.05 / 1.00 = 1.05.
Option D: 0.05
Option D, 0.05, reports only the excess of the PRD over 1 (1.05 − 1). The PRD itself is 1.05 / 1.00 = 1.05.
Weight by price, then compare
Multiply each ratio by its sale price, add, and divide by total price to get the weighted mean; then put the simple mean over it.
How to use: Use this to work any PRD problem that gives ratios and sale prices.
Exam Tip
Compute the weighted mean carefully by multiplying each ratio by its own sale price. Then confirm your answer is the quotient, not an input or the inverse.
Common Mistakes to Avoid
- -Inverting the PRD formula
- -Reporting the weighted mean ratio as the PRD
- -Matching ratios to the wrong sale prices when weighting
Concept Deep Dive
Analysis
This question tests the PRD calculation from paired ratios and sale prices. Compute the simple arithmetic mean of the ratios and the sale-price-weighted mean ratio, then divide the first by the second. The distractors come from inverting the division or reporting an input.
Background Knowledge
The PRD is the mean ratio divided by the weighted mean ratio. In this sample the lowest-priced sale has the highest ratio and the highest-priced sale the lowest, so the weighted mean falls below the simple mean and the PRD exceeds 1. A single point result is descriptive and is not a statistical test of regressivity in the population.
Real-World Application
An assessment office finds that its lowest-priced sale was assessed at 120% of price while its highest-priced sale was at 90%. The PRD of 1.05 summarizes this regressive-looking pattern for further study.
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