The assessment-to-sale ratios are 0.88, 0.92, 0.96, 0.98, 1.02, 1.04, 1.08, 1.12. 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) 6.50%
Ordered median=1; absolute deviations are 0.12, 0.08, 0.04, 0.02, 0.02, 0.04, 0.08, 0.12. Their mean=0.065; COD=0.065/1×100=6.5%. The unscaled decimal fraction is not the percentage answer.
Why This Is the Correct Answer
Option A is correct because the median of the ordered ratios is 1.00 and the absolute deviations from it are 0.12, 0.08, 0.04, 0.02, 0.02, 0.04, 0.08 and 0.12. These sum to 0.52, and 0.52 / 8 = 0.065 is the average absolute deviation. Dividing by the median of 1.00 and multiplying by 100 gives a COD of 6.50%.
Why the Other Options Are Wrong
Option B: 0.07%
Option B, 0.07%, comes from taking the unscaled average absolute deviation ratio of 0.065, rounding it to 0.07 and labeling it a percentage without multiplying by 100. The correct result is 0.065 / 1.00 x 100 = 6.50%.
Option C: 0.24%
Option C, 0.24%, matches the range of the ratios (1.12 - 0.88 = 0.24) written as a percent, which is a different statistic from the COD. The COD uses the average absolute deviation from the median: 0.52 / 8 = 0.065, then 0.065 / 1.00 x 100 = 6.50%.
Option D: 13.00%
Option D, 13.00%, results from dividing the total absolute deviation by 4 instead of by all 8 ratios (0.52 / 4 = 0.13, x 100 = 13.00%). The average must use all eight deviations: 0.52 / 8 = 0.065, giving a COD of 6.50%.
Median, Deviate, Average, Divide, Hundred
Remember the COD as five steps in order: find the median, take absolute deviations, average them, divide by the median, then multiply by 100.
How to use: Use this step sequence whenever a ratio-study question asks for a COD so you do not skip the averaging or the percentage conversion.
Exam Tip
Check whether the answer choices mix decimals and percentages; a figure 100 times too small often signals a missed conversion.
Common Mistakes to Avoid
- -Reporting the decimal 0.065 as if it were already a percentage
- -Dividing the sum of deviations by half the number of ratios
- -Measuring deviations from the mean instead of the median
Concept Deep Dive
Analysis
This question tests the full coefficient of dispersion (COD) procedure: find the median, take absolute deviations from it, average them, divide by the median and convert to a percentage. With eight ratios the median is the average of the fourth and fifth values, (0.98 + 1.02) / 2 = 1.00. The final multiplication by 100 is the step most often skipped or mislabeled.
Background Knowledge
The coefficient of dispersion is the average absolute deviation of the ratios from the median, expressed as a percentage of the median. It is the most widely used measure of assessment uniformity in ratio studies; a lower COD indicates ratios clustered more tightly around the median.
Real-World Application
An appraiser reviewing an assessor's ratio study for a neighborhood recomputes the COD from the sale ratios to judge whether assessments are uniform. Finding a COD of about 6.5% tells the appraiser the ratios cluster fairly tightly around the median.
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