Assessment-to-sale ratios are 0.85, 0.92, 1, 1.08, 1.15; corresponding sale prices are $500,000, $400,000, $300,000, $200,000, $100,000 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; weighted mean=Σ(ratio×sale price)/Σ(sale prices)=0.9493; PRD=1/0.9493=1.0534. A point result is not a statistical test of population regressivity. Common errors: reporting the sale-price-weighted mean ratio instead of the PRD gives 0.95; reporting the arithmetic 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 (0.85 + 0.92 + 1 + 1.08 + 1.15) / 5 = 1.00. The weighted mean is Σ(ratio × sale price) / Σ(sale prices) = $1,424,000 / $1,500,000 = 0.9493. PRD = 1.00 / 0.9493 = 1.0534, which rounds to 1.05.
Why the Other Options Are Wrong
Option B: 0.95
Option B, 0.95, is the sale-price-weighted mean ratio (0.9493 rounded), reported instead of the PRD. The PRD divides the arithmetic mean by that figure: 1.00 / 0.9493 = 1.05.
Option C: 1.00
Option C, 1.00, is the arithmetic mean ratio, reported instead of the PRD. Dividing it by the weighted mean of 0.9493 gives a PRD of 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 1.00 / 0.9493 = 1.05.
Simple over weighted
PRD puts the simple mean on top and the weighted mean on the bottom; if expensive sales have low ratios, the bottom shrinks and PRD rises above 1.
How to use: Use this to set up any PRD calculation in the right order.
Exam Tip
Compute both means before dividing, and make sure the final answer is the quotient, not one of the inputs or the difference from 1.
Common Mistakes to Avoid
- -Reporting the weighted mean ratio as the PRD
- -Inverting the division
- -Reporting only the amount by which PRD exceeds 1
Concept Deep Dive
Analysis
This question tests the price-related differential (PRD) calculation from paired ratio and price data. You need two figures: the simple arithmetic mean of the ratios and the sale-price-weighted mean ratio. PRD is the first divided by the second.
Background Knowledge
The PRD is calculated by dividing the mean ratio by the weighted mean ratio. Because the weighted mean gives more influence to higher-priced sales, a PRD above 1 indicates that higher-priced properties tend to have lower ratios in the sample. A single point figure is descriptive and is not by itself a statistical test of regressivity in the population.
Real-World Application
An assessment office studies five recent sales and finds the low-priced homes assessed at higher ratios than the high-priced homes. The PRD of about 1.05 summarizes that pattern for further review with a larger sample.
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