A set of comparable sales has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation?
Correct Answer
C) 8.0 percent
Why this is correct: The original explanation gives the formula: Coefficient of Variation (CV) = (Standard Deviation / Mean). Here, CV = $20,000 / $250,000 = 0.08 = 8%. The governing concept is that CV standardizes dispersion relative to the mean, allowing comparison between datasets with different average values. Why the other choices are wrong: 12.5 percent would result from incorrectly dividing the mean by the standard deviation ($250k/$20k). 20.0 percent is simply the standard deviation expressed as a percentage of 100,000, not the mean. 2.5 percent might come from dividing $20,000 by $800,000 or another miscalculation. Exam tip: CV = (Standard Deviation / Mean). It's a percentage, so multiply the decimal result by 100.
Why This Is the Correct Answer
Option C is correct because the coefficient of variation is the standard deviation divided by the mean, so $20,000 divided by $250,000 equals 0.08, or 8.0 percent. Converting the decimal to a percentage requires shifting two places, which is where several distractors are aimed. The result is unitless by construction, which is exactly what makes it comparable across data sets of different price levels. An 8 percent coefficient indicates a reasonably tight group of comparables.
Why the Other Options Are Wrong
Option A: 12.5 percent
12.5 percent comes from inverting the ratio and dividing the mean by the standard deviation, then misreading the resulting 12.5 as a percentage. That quotient is not a measure of relative dispersion at all, and it moves in the wrong direction, growing larger as the data becomes more consistent. The coefficient always places the standard deviation on top.
Option B: 20.0 percent
20.0 percent lifts the leading digits from the $20,000 standard deviation without performing the division. Standard deviation is expressed in dollars and cannot be converted to a percentage by dropping the zeros; it only becomes a rate when divided by the mean. This choice tests whether the candidate actually computes rather than pattern-matches.
Option D: 2.5 percent
2.5 percent appears to come from the mean's leading digits or from a division with the decimal misplaced. The correct quotient is 0.08, which is 8.0 percent, not 2.5 percent. A quick estimate helps here, since $20,000 is a bit less than a tenth of $250,000, so the answer must be a little under 10 percent.
Spread over center
Coefficient of variation puts the spread on top and the center underneath: standard deviation divided by mean. Small answer means tight data, and the answer is always a percentage because the dollars cancel.
How to use: Identify which number is the spread and which is the center, divide in that order, then move the decimal two places. Estimating the fraction first will eliminate any answer of the wrong magnitude.
Exam Tip
Dollars over dollars leaves no units, which is the tell that the result belongs as a percentage. If a choice is expressed in dollars, it is not a coefficient of variation.
Common Mistakes to Avoid
- -Inverting the ratio and dividing the mean by the standard deviation
- -Reporting the coefficient in dollars rather than as a percentage
- -Misplacing the decimal when converting the quotient
- -Comparing raw standard deviations across data sets with very different means
Concept Deep Dive
Analysis
This tests the coefficient of variation, a normalized measure of dispersion that lets an appraiser compare the tightness of data sets expressed in different magnitudes. The standard deviation alone answers how spread out the data is in dollars, but $20,000 of spread means something very different around a $250,000 mean than around a $2,500,000 mean. Dividing the standard deviation by the mean strips out the scale and produces a unitless ratio, usually reported as a percentage, so a set of $250,000 homes and a set of $2,500,000 homes can be compared for relative consistency. Here $20,000 divided by $250,000 is 0.08, or 8 percent. Appraisers use this in comparable selection and in mass appraisal quality control, where a lower coefficient signals a more homogeneous group and therefore a more reliable value indication, and assessment work uses a closely related measure to test uniformity.
Background Knowledge
You need the basic descriptive statistics used in valuation, including mean, median, mode, range, standard deviation, and coefficient of variation, and to know which measure answers which question. You should also understand that the coefficient of variation is unitless and therefore comparable across data sets, and that mass appraisal and assessment ratio studies use related measures to test uniformity.
Real-World Application
Choosing between two sets of comparables for a suburban home, you find one set has a coefficient of variation of 8 percent and the other 19 percent. The tighter set indicates more homogeneous evidence, so you rely on it and explain in the reconciliation why the more dispersed group received less weight.
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