Assessment-to-sale ratios are 1.1, 1, 0.9; corresponding sale prices are $100,000, $200,000, $500,000 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)=0.95; PRD=1/0.95=1.0526. 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 of 1.1, 1.0 and 0.9 is 1.00. The weighted mean is (1.1 x 100,000 + 1.0 x 200,000 + 0.9 x 500,000) / 800,000 = 760,000 / 800,000 = 0.95. The PRD is 1.00 / 0.95 = 1.0526, 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, 760,000 / 800,000, reported instead of the PRD. The PRD divides the arithmetic mean by that figure: 1.00 / 0.95 = 1.05.
Option C: 1.00
Option C, 1.00, is the arithmetic mean ratio reported instead of the PRD. That mean must still be divided by the weighted mean of 0.95, giving 1.05.
Option D: 0.05
Option D, 0.05, reports only the excess of the PRD over 1 (1.05 - 1). The question asks for the PRD itself, 1.00 / 0.95 = 1.05.
Plain over weighted
PRD puts the plain (unweighted) mean on top and the price-weighted mean on the bottom.
How to use: Use this phrase to set up the PRD fraction in the right order before dividing.
Exam Tip
Several PRD choices are often intermediate results; finish the division and check that you are reporting the ratio of the two means.
Common Mistakes to Avoid
- -Reporting the weighted mean ratio as the PRD
- -Inverting the fraction and dividing weighted mean by simple mean
- -Weighting by ratios instead of sale prices
Concept Deep Dive
Analysis
This question tests the price-related differential calculation from paired ratio and sale-price data. You must compute the simple arithmetic mean ratio, then the sale-price-weighted mean ratio, and divide the first by the second. Each distractor reports an intermediate figure or a fragment of the result instead of the PRD.
Background Knowledge
The price-related differential compares the simple mean ratio with the sale-price-weighted mean ratio to look for a relationship between assessment level and price. A PRD above 1 occurs when higher-priced sales tend to have lower ratios, pulling the price-weighted mean below the simple mean; this pattern is associated with regressivity, though a single point result is not a statistical test of population regressivity.
Real-World Application
An appraiser reviewing a county ratio study sees that the most expensive sale has the lowest ratio. Computing a PRD of 1.05 from the paired data, the appraiser notes a regressive sample pattern while recognizing that three sales alone cannot establish a population-wide conclusion.
More Statistics Questions
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