Four parcels have assessment-to-sale ratios of 1.15 on $90,000; 1.05 on $150,000; 0.98 on $250,000; and 0.92 on $400,000. Define PRD as the unweighted mean ratio divided by the sale-price-weighted mean ratio. Which rounded PRD and descriptive sample tendency are consistent with these data?
Correct Answer
A) PRD about 1.04; higher observed ratios on lower-price sales
The unweighted mean ratio is 1.025. Sale-price-weighted mean = 874,000/890,000 = 0.982. PRD = 1.025/0.982 = 1.0438, rounded to 1.04. Here lower-price sales have higher observed ratios, consistent with a regressive sample pattern. Four observations and a point estimate alone do not establish a population-wide inequity conclusion or pass/fail finding.
Why This Is the Correct Answer
Option A is correct because the unweighted mean of 1.15, 1.05, 0.98 and 0.92 is 4.10 / 4 = 1.025. The weighted mean is (103,500 + 157,500 + 245,000 + 368,000) / 890,000 = 874,000 / 890,000 = 0.982, so the PRD is 1.025 / 0.982 = 1.04. The lowest-priced sales carry the highest ratios, which is a regressive sample pattern.
Why the Other Options Are Wrong
Option B: PRD about 0.96; lower observed ratios on lower-price sales
Option B, PRD about 0.96 with lower ratios on lower-price sales, inverts the fraction (0.982 / 1.025 = 0.96) and misreads the data. The correct PRD is 1.025 / 0.982 = 1.04, and the $90,000 sale has the highest ratio, 1.15, not the lowest.
Option C: PRD exactly 1.00; no further evidence needed for population equity
Option C, PRD exactly 1.00 with no further evidence needed, does not follow from the facts: the calculation gives 1.025 / 0.982 = 1.04. A point estimate from four sales also cannot by itself establish population equity.
Option D: PRD exactly 1.15, obtained by taking the highest single ratio
Option D, PRD exactly 1.15 from the highest single ratio, simply picks the largest ratio. The PRD divides the unweighted mean (1.025) by the weighted mean (0.982), giving 1.04.
Above one leans regressive
A PRD above 1 means cheaper properties tend to carry higher ratios, which leans regressive.
How to use: Use this to match a calculated PRD with the correct description of the ratio pattern.
Exam Tip
Check the raw data directly: if the lowest price has the highest ratio, any choice saying lower ratios on lower-price sales is wrong.
Common Mistakes to Avoid
- -Inverting the PRD fraction
- -Treating a small-sample PRD as conclusive proof of population inequity
- -Weighting by ratio instead of by sale price
Concept Deep Dive
Analysis
This question tests both the PRD calculation and its interpretation. You compute the unweighted mean ratio, the sale-price-weighted mean ratio, divide, and then read the pattern of ratios against price. A sound answer also avoids overstating what four observations can prove.
Background Knowledge
The PRD compares an unweighted mean ratio with a sale-price-weighted mean ratio. When lower-priced properties have higher ratios, the weighted mean falls below the simple mean and the PRD exceeds 1, a pattern described as regressive; small samples call for caution before drawing population-wide conclusions.
Real-World Application
An appraiser testifying about assessment equity calculates a PRD of 1.04 from four sales in a neighborhood. The appraiser describes a regressive pattern in the sample but explains that a larger sample and further testing would be needed before concluding the whole jurisdiction is inequitable.
More Statistics Questions
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Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
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Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
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