The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. 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) 10.00%
Ordered median=1; absolute deviations are 0.2, 0.15, 0.1, 0.08, 0.05, 0.02, 0.02, 0.05, 0.08, 0.1, 0.15, 0.2. Their mean=0.1; COD=0.1/1×100=10%. The unscaled decimal fraction is not the percentage answer.
Why This Is the Correct Answer
Option A is correct because the two middle ratios, 0.98 and 1.02, average to a median of 1.00. The absolute deviations total 1.20, so their average is 1.20 / 12 = 0.10. Then 0.10 / 1.00 x 100 = 10.00%.
Why the Other Options Are Wrong
Option B: 0.10%
Option B, 0.10%, is the unscaled average deviation ratio (0.10 / 1.00) shown with a percent sign but not multiplied by 100. The COD is 0.10 x 100 = 10.00%.
Option C: 0.40%
Option C, 0.40%, matches the range of the ratios (1.20 - 0.80 = 0.40) left unscaled, which is not the COD. The COD averages the absolute deviations (1.20 / 12 = 0.10) and scales by 100 to give 10.00%.
Option D: 20.00%
Option D, 20.00%, uses only the largest absolute deviation (0.20) multiplied by 100. The COD averages all twelve deviations: 1.20 / 12 = 0.10, giving 10.00%.
Average all, then times hundred
COD averages every absolute deviation, not just the biggest, and then multiplies by 100.
How to use: Use this to avoid picking answers built on a single deviation or an unscaled decimal.
Exam Tip
With many ratios, total the absolute deviations carefully and divide by the full count of ratios.
Common Mistakes to Avoid
- -Using only the largest deviation
- -Reporting the decimal instead of the percentage
- -Averaging over the wrong number of ratios
Concept Deep Dive
Analysis
This question tests a COD calculation with twelve ratios. You average the sixth and seventh ordered ratios for the median, average the absolute deviations, divide by the median and multiply by 100. The distractors use unscaled decimals or a single extreme deviation.
Background Knowledge
The COD measures the average percentage deviation of ratios from the median ratio. Higher CODs indicate less uniform assessments, and the result must be scaled by 100 to read as a percentage.
Real-World Application
An appraiser reviewing an assessor's study of twelve sales calculates a COD of 10.00%. The appraiser uses that figure to describe the uniformity of assessments in the area and compares it with the study's stated benchmarks.
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For positive assessment-to-sale ratios, define COD as 100 × mean absolute deviation from their median ÷ median. How does it differ from the median ratio?