A set of sale prices has a median of $420,000 and an average absolute deviation from the median of $21,000. What is the coefficient of dispersion, as a percentage to two decimals?
Correct Answer
B) 5.00%
21000/420000×100=5%. The ratio must be multiplied by 100 to report a percentage; the deviation is an average absolute deviation, not a median deviation.
Why This Is the Correct Answer
Option B is correct because COD = (21,000 ÷ 420,000) × 100 = 0.05 × 100 = 5.00%. The average absolute deviation is divided by the median, and the ratio is multiplied by 100 to report it as a percentage. This follows the IAAO sequence of computing the average absolute deviation, dividing by the median and multiplying by 100.
Why the Other Options Are Wrong
Option A: 0.05%
Option A, 0.05%, comes from 21,000 ÷ 420,000 = 0.05 and then attaching a percent sign without multiplying by 100. The correct result is 0.05 × 100 = 5.00%.
Option C: 20.00%
Option C, 20.00%, comes from inverting the ratio: 420,000 ÷ 21,000 = 20. COD divides the deviation by the median, so the correct figure is (21,000 ÷ 420,000) × 100 = 5.00%.
Option D: 95.00%
Option D, 95.00%, comes from subtracting the correct 5% from 100% (100 − 5 = 95). COD is the deviation measure itself, (21,000 ÷ 420,000) × 100 = 5.00%, not its complement.
Deviation over median, times 100
Think of COD as the typical miss divided by the middle, then scaled to a percent: small number on top, median on the bottom, times 100.
How to use: Use this whenever a COD question gives you the average absolute deviation and the median directly.
Exam Tip
Check whether your answer is a plausible percentage; if you get a decimal like 0.05, you probably forgot to multiply by 100.
Common Mistakes to Avoid
- -Forgetting to multiply the ratio by 100
- -Dividing the median by the deviation instead of the deviation by the median
- -Reporting 100% minus the COD as if it were the COD
Concept Deep Dive
Analysis
This question tests the coefficient of dispersion (COD) formula when the average absolute deviation and the median are already given. The only steps are to divide the average absolute deviation by the median and then multiply by 100 to express the result as a percentage. Most errors come from skipping the multiplication by 100, inverting the ratio, or subtracting from 100.
Background Knowledge
The coefficient of dispersion measures uniformity by expressing the average absolute deviation of observations from the median as a percentage of the median. A lower COD indicates observations clustered more tightly around the median. It is widely used in ratio studies to evaluate assessment uniformity.
Real-World Application
An appraiser reviewing a neighborhood price study sees an average absolute deviation of $21,000 around a $420,000 median. Computing a COD of 5% tells the appraiser the sales are fairly tightly grouped, which supports using the median as a reasonable central indicator for that segment.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. 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.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
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