Ratios are 0.88,0.92,0.96,1.04,1.08 and 1.12. Use the average of the two middle ordered values as median. Define COD=100×mean absolute deviation from that median÷median. What COD results?
Correct Answer
C) 8.0%
The ordered middle two ratios are 0.96 and 1.04, so median=1.00. Absolute deviations from the median are 0.12, 0.08, 0.04, 0.04, 0.08 and 0.12, totaling 0.48; their mean=0.48/6=0.08. With the stated formula, COD=100×0.08/1.00=8.0%. Reporting the unscaled 0.08 as a percentage, or dividing the total by 10 or by 4, does not follow the stated formula.
Why This Is the Correct Answer
Option C is correct because the middle ratios are 0.96 and 1.04, so the median is 1.00. The absolute deviations 0.12, 0.08, 0.04, 0.04, 0.08 and 0.12 total 0.48, and their mean is 0.48 / 6 = 0.08. COD = 100 × 0.08 / 1.00 = 8.0%.
Why the Other Options Are Wrong
Option A: 0.08%
Option A, 0.08%, reports the unscaled mean absolute deviation of 0.08 as a percentage without multiplying by 100. The stated formula gives 100 × 0.08 / 1.00 = 8.0%.
Option B: 4.8%
Option B, 4.8%, comes from dividing the 0.48 total by 10 instead of by the 6 ratios: 100 × 0.048 = 4.8%. The correct mean deviation is 0.48 / 6 = 0.08, giving 8.0%.
Option D: 12.0%
Option D, 12.0%, comes from dividing the 0.48 total by 4 instead of 6: 100 × 0.12 = 12.0%. Using all six ratios gives 0.48 / 6 = 0.08 and a COD of 8.0%.
Median, deviate, average, scale
Four moves for COD: find the median, take absolute deviations, average them over all n ratios, then divide by the median and times 100.
How to use: Use these four moves in order to avoid scaling and divisor errors on COD problems.
Exam Tip
Divide the total absolute deviation by the number of ratios, then convert to a percentage. Several distractors come from using the wrong divisor or skipping the times-100 step.
Common Mistakes to Avoid
- -Forgetting to multiply by 100 when expressing COD as a percentage
- -Dividing total absolute deviation by the wrong count
- -Measuring deviations from the mean when the formula calls for the median
Concept Deep Dive
Analysis
This question tests a step-by-step coefficient of dispersion calculation. Find the median of six ratios by averaging the two middle values, compute each absolute deviation from it, average those deviations, divide by the median and multiply by 100. Each distractor reflects a specific slip in scaling or in the divisor.
Background Knowledge
The coefficient of dispersion is the average absolute deviation from the median, expressed as a percentage of the median. When the number of ratios is even, the median is the average of the two middle ratios. A lower COD indicates more uniform assessment.
Real-World Application
An assessor checking a small sample of six sales computes a COD of 8.0% to describe assessment uniformity. Working each step on paper shows the median, deviations and average clearly. The assessor documents the formula used so reviewers can reproduce the figure.
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