The assessment-to-sale ratios are 0.85, 0.9, 0.95, 1, 1, 1.05, 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
B) 7.50%
Ordered median=1; absolute deviations are 0.15, 0.1, 0.05, 0, 0, 0.05, 0.1, 0.15. Their mean=0.075; COD=0.075/1×100=7.5%. The unscaled decimal fraction is not the percentage answer.
Why This Is the Correct Answer
Option B is correct because the two middle ratios are 1.00 and 1.00, so the median is 1.00. The absolute deviations are 0.15, 0.10, 0.05, 0, 0, 0.05, 0.10 and 0.15, which sum to 0.60 and average 0.60 ÷ 8 = 0.075. COD = 0.075 ÷ 1.00 × 100 = 7.50%.
Why the Other Options Are Wrong
Option A: 0.08%
Option A, 0.08%, takes the unscaled decimal 0.075, rounds it and labels it a percentage without multiplying by 100. The COD is 0.075 ÷ 1.00 × 100 = 7.50%.
Option C: 0.30%
Option C, 0.30%, matches the range of the ratios (1.15 − 0.85 = 0.30), which is not the average absolute deviation and is not scaled to a percentage. The COD is 0.075 ÷ 1.00 × 100 = 7.50%.
Option D: 15.00%
Option D, 15.00%, is double the correct figure; it matches using the largest single deviation (0.15 × 100) instead of the average deviation. The COD uses the average absolute deviation: 0.60 ÷ 8 = 0.075, so 0.075 ÷ 1.00 × 100 = 7.50%.
Median, deviate, average, divide, times 100
COD follows five steps: find the median, take absolute deviations, average them, divide by the median, then multiply by 100. Skipping the last step leaves a decimal, not a percent.
How to use: Use this sequence to avoid stopping at the unscaled decimal.
Exam Tip
Check whether your answer has been multiplied by 100 when the stem asks for a percentage. Divide the sum of deviations by the count of all ratios, including those with zero deviation.
Common Mistakes to Avoid
- -Reporting the unscaled decimal as a percentage
- -Using the largest deviation instead of the average deviation
- -Leaving zero deviations out of the count
Concept Deep Dive
Analysis
This question tests the coefficient of dispersion (COD) calculation from raw ratios. Find the median of the eight ordered ratios by averaging the two middle values, compute each ratio's absolute deviation from the median, average those deviations, divide by the median and multiply by 100.
Background Knowledge
The coefficient of dispersion measures assessment uniformity as the average absolute deviation of ratios from the median, expressed as a percentage of the median. 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's office reviews eight sales in a neighborhood to test assessment uniformity. A COD of 7.50% indicates ratios cluster fairly tightly around the median.
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Seven ordered comparable sale prices are $200,000, $210,000, $220,000, $230,000, $240,000, $250,000, and $380,000. Using the median of each half (excluding the overall median) for the quartiles, what are the range and the interquartile range (IQR = Q3 − Q1)?
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