A set of adjusted sale prices has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation, and why might it be preferred to the standard deviation?
Correct Answer
A) 8%; it does not depend on the unit of measurement
CV = s ÷ mean = $20,000 ÷ $250,000 = 0.08, or 8%. Because it is a ratio, its value does not depend on the unit of measurement. It is not a robust statistic.
Why This Is the Correct Answer
Option A is correct because CV = $20,000 / $250,000 = 0.08, or 8%. As a ratio of two quantities in the same units, it is independent of the unit of measurement, which is why it is sometimes preferred to the standard deviation for comparing variability.
Why the Other Options Are Wrong
Option B: 8%; it is resistant to extreme values
Option B has the correct figure of 8% but gives the wrong reason, claiming CV is resistant to extreme values. CV is built from the mean and standard deviation, both of which are sensitive to extremes, so it is not a robust statistic.
Option C: 12.5%; it does not depend on the unit of measurement
Option C, 12.5%, inverts the ratio, $250,000 / $20,000 = 12.5, and labels it a percentage. The correct calculation is $20,000 / $250,000 = 0.08, or 8%.
Option D: 80%; it measures the spread around the median
Option D, 80%, misplaces the decimal (0.08 treated as 0.80) and wrongly says CV measures spread around the median. CV relates the standard deviation to the mean: $20,000 / $250,000 = 8%.
Spread Over Center
CV puts the standard deviation over the mean, so the small number goes on top and the units cancel out.
How to use: Use this phrase to avoid inverting the ratio and to remember why CV is unit-free.
Exam Tip
In two-part answer choices, verify both the number and the stated reason; a correct figure with a wrong property is still wrong.
Common Mistakes to Avoid
- -Dividing the mean by the standard deviation
- -Believing CV is resistant to extreme values
- -Misplacing the decimal when converting to a percentage
Concept Deep Dive
Analysis
This question tests both the calculation and the purpose of the coefficient of variation (CV). CV equals the standard deviation divided by the mean, and because it is a ratio it does not depend on the unit of measurement. Each option pairs a number with a property, so both parts must be correct.
Background Knowledge
The coefficient of variation expresses the standard deviation as a percentage of the mean. Because units cancel, it allows comparison of variability between data sets with different scales or units, but like the mean and standard deviation it is sensitive to extreme values.
Real-World Application
An appraiser compares variability in adjusted prices for a starter-home market and a luxury market. Because the CV is unit-free and scaled to each mean, it gives a fairer comparison than the raw standard deviations.
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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