The assessment-to-sale ratios are 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15. 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) 8.00%
Ordered median=1; absolute deviations are 0.15, 0.1, 0.08, 0.05, 0.02, 0.02, 0.05, 0.08, 0.1, 0.15. Their mean=0.08; COD=0.08/1×100=8%. The unscaled decimal fraction is not the percentage answer.
Why This Is the Correct Answer
Option A is correct because the median is 1.00, the absolute deviations (0.15, 0.10, 0.08, 0.05, 0.02, 0.02, 0.05, 0.08, 0.10, 0.15) average 0.80 / 10 = 0.08, and COD = 0.08 / 1.00 × 100 = 8.00%. The final multiplication by 100 converts the decimal to a percentage.
Why the Other Options Are Wrong
Option B: 0.08%
Option B, 0.08%, is the unscaled average absolute deviation ratio of 0.08 with a percent sign attached. The formula multiplies by 100, so COD = 0.08 × 100 = 8.00%.
Option C: 0.30%
Option C, 0.30%, matches the range of the ratios, 1.15 − 0.85 = 0.30, which is not part of the COD formula. COD = (0.80 / 10) / 1.00 × 100 = 8.00%.
Option D: 8.89%
Option D, 8.89%, divides the deviation sum of 0.80 by 9 (n − 1, as in a sample variance) instead of all 10 ratios: 0.80 / 9 = 0.0889, times 100 = 8.89%. The COD uses the average absolute deviation, 0.80 / 10 = 0.08, giving 8.00%.
Deviate, average, divide, times 100
COD follows four steps: absolute deviations from the median, their average over all the ratios, divided by the median, times 100.
How to use: Use this sequence to compute COD without dividing by the wrong count or skipping the percentage step.
Exam Tip
Divide the deviation sum by the full number of ratios, not n − 1, and watch for an answer that is your correct decimal shown as a percent.
Common Mistakes to Avoid
- -Forgetting to multiply by 100
- -Dividing the deviation sum by n − 1 instead of the number of ratios
- -Using the range of the ratios instead of the average absolute deviation
Concept Deep Dive
Analysis
This question tests calculating the coefficient of dispersion (COD) from a list of ratios. With 10 ratios, the median is the average of the 5th and 6th values: (0.98 + 1.02) / 2 = 1.00. The absolute deviations from 1.00 sum to 0.80, average 0.80 / 10 = 0.08, and dividing by the median and multiplying by 100 gives 8.00%.
Background Knowledge
The COD measures assessment uniformity: subtract the median from each ratio, take absolute values, average them over the number of ratios, divide by the median and multiply by 100. A lower COD indicates more uniform ratios. For an even number of ratios, the median is the average of the two middle values.
Real-World Application
An assessor reviewing ten sales in a district finds a COD of 8.00%. The low figure indicates assessments are fairly uniform relative to sale prices in that group.
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