The assessment-to-sale ratios are 0.9, 0.9, 0.95, 1, 1, 1.05, 1.05. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.
Correct Answer
A) 5.00%
Ordered median=1; absolute deviations are 0.1, 0.1, 0.05, 0, 0, 0.05, 0.05. Their mean=0.05; COD=0.05/1×100=5%. The unscaled decimal fraction is not the percentage answer.
Why This Is the Correct Answer
Option A is correct because the median of the seven ordered ratios is 1.00, and the absolute deviations are 0.10, 0.10, 0.05, 0, 0, 0.05 and 0.05. Their sum is 0.35 and their mean is 0.35 / 7 = 0.05. COD = 0.05 / 1.00 × 100 = 5.00%.
Why the Other Options Are Wrong
Option B: 0.05%
Option B, 0.05%, takes the unscaled average deviation of 0.05 and labels it a percentage without multiplying by 100. The COD is 0.05 / 1.00 × 100 = 5.00%.
Option C: 0.15%
Option C, 0.15%, matches the range of the ratios (1.05 − 0.90 = 0.15) with a percent sign attached, which is neither the average deviation nor scaled by 100. The COD is 0.05 / 1.00 × 100 = 5.00%.
Option D: 10.00%
Option D, 10.00%, uses the largest single deviation (0.10 / 1.00 × 100) instead of the average deviation. The COD averages all deviations: 0.05 / 1.00 × 100 = 5.00%.
Average gap over median
COD = average absolute gap from the median, divided by the median, times 100. Forgetting the times 100 leaves a decimal, not a percent.
How to use: Use this formula outline to work any COD calculation step by step.
Exam Tip
After dividing by the median, multiply by 100 before reporting a percentage.
Common Mistakes to Avoid
- -Reporting the decimal fraction as a percentage
- -Using the largest deviation instead of the average
- -Measuring deviations from the mean rather than the median
Concept Deep Dive
Analysis
This question tests calculating the coefficient of dispersion from a set of ratios. The steps are to find the median ratio, compute the absolute deviations from it, average them, divide by the median and multiply by 100. The distractors come from skipping the scaling or using the wrong deviation figure.
Background Knowledge
The coefficient of dispersion expresses the average absolute deviation from the median ratio as a percentage of the median. Lower values indicate more uniform assessments. It is one of the main uniformity measures in ratio studies.
Real-World Application
An assessor's analyst computes a COD of 5% for a small group of residential ratios. That figure indicates the assessments are fairly uniform, and she documents the calculation in the ratio study report.
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Sale-price observations are $410,000, $280,000, $295,000, $310,000, $355,000, $300,000, $325,000, $290,000, $500,000. Compute the median; order the observations first and average the two middle values if n is even. Report to the nearest dollar.
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