Five assessment ratios are 0.90, 0.95, 1.00, 1.05 and 1.10. What is the coefficient of dispersion (COD)?
Correct Answer
B) 6.0
The median is 1.00. Absolute deviations from the median are 0.10, 0.05, 0, 0.05 and 0.10, averaging 0.06; divided by the median and multiplied by 100, the COD is 6.0.
Why This Is the Correct Answer
Option B is correct because the median of the five ratios is 1.00 and the absolute deviations from it are 0.10, 0.05, 0, 0.05 and 0.10. Those deviations total 0.30, so the average absolute deviation is 0.30 / 5 = 0.06. Dividing 0.06 by the median of 1.00 and multiplying by 100 gives a COD of 6.0.
Why the Other Options Are Wrong
Option A: 7.9
Option A, 7.9, comes from computing a standard deviation of the ratios (about 0.079) and multiplying it by 100. The COD uses the average absolute deviation from the median, 0.30 / 5 = 0.06, divided by 1.00 and multiplied by 100, which is 6.0.
Option C: 0.06
Option C, 0.06, is the average absolute deviation divided by the median but not multiplied by 100. The COD is expressed as a percentage: 0.06 / 1.00 x 100 = 6.0.
Option D: 10.0
Option D, 10.0, takes only the largest deviation (0.10) and multiplies it by 100. The COD averages all five deviations: 0.30 / 5 = 0.06, then 0.06 / 1.00 x 100 = 6.0.
Median, deviate, average, divide, hundred
Think of COD as five steps in order: find the median, take absolute deviations, average them, divide by the median, multiply by 100.
How to use: Use this sequence as a checklist whenever a COD calculation question appears so you do not drop the final x100 step.
Exam Tip
When a COD answer choice is a small decimal like 0.06, check whether it is simply the unscaled version of another choice; the COD is reported as a percentage.
Common Mistakes to Avoid
- -Using the standard deviation instead of the average absolute deviation
- -Forgetting to multiply by 100 to express the COD as a percentage
- -Measuring deviations from the mean instead of the median
Concept Deep Dive
Analysis
This question tests whether you can compute the coefficient of dispersion step by step. Find the median ratio, take each ratio's absolute deviation from that median, average those deviations, divide by the median, and multiply by 100. Each distractor comes from skipping or replacing one of these steps.
Background Knowledge
The coefficient of dispersion measures the average percentage deviation of the ratios from the median ratio. It is a widely used measure of appraisal uniformity in ratio studies; a lower COD means the ratios cluster more tightly around the median.
Real-World Application
An appraiser reviewing an assessment office's ratio study for a residential neighborhood recalculates the COD from five sales. Finding a COD of 6.0 tells the appraiser that the assessments in this small sample deviate from the median ratio by an average of 6 percent.
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