For suitable ratio-scale data with a meaningful positive mean away from zero, why may the coefficient of variation be useful?
Correct Answer
D) Expresses dispersion relative to the mean, allowing comparison
CV=SD/mean expresses relative dispersion; multiply by 100 if reporting percent. It can help compare variability across suitable ratio-scale variables with a meaningful zero. A near-zero mean makes the ratio unstable, and interval-scale conversions can change its meaning. CV is not a median finder, correlation coefficient or confidence interval for the mean.
Why This Is the Correct Answer
Option D is correct because CV is the standard deviation expressed relative to the mean (often as a percentage of the mean). Because it removes the effect of the overall level, it allows variability to be compared across suitable ratio-scale data sets, such as a lower-priced and a higher-priced market segment.
Why the Other Options Are Wrong
Option A: Identifies which observation is the median
Option A says CV identifies which observation is the median. The median is found by ordering the data and locating the middle value; CV is a dispersion measure and does not locate any particular observation.
Option B: Converts all of the observed prices into a single index number
Option B says CV converts all of the observed prices into a single index number. CV is a single summary of relative spread, not a price index, and it does not transform the prices themselves.
Option C: Measures the correlation between two variables
Option C says CV measures the correlation between two variables. Correlation describes how two variables move together; CV describes the spread of one variable relative to its own mean.
CV: spread per dollar of mean
Read CV as 'how big is the spread compared with the average': standard deviation on top, mean on the bottom.
How to use: Use this when a question asks which statistic compares variability between groups with different average prices.
Exam Tip
If a question asks how to compare dispersion between data sets with very different averages, look for the measure that divides the standard deviation by the mean.
Common Mistakes to Avoid
- -Confusing CV with a correlation coefficient
- -Using CV on data whose mean is near zero
- -Thinking CV identifies a central value such as the median
Concept Deep Dive
Analysis
This question tests what the coefficient of variation (CV) measures. CV divides the standard deviation by the mean, so it expresses spread relative to the average. That relative scale is what lets analysts compare variability across data sets with different price levels, provided the data are on a ratio scale with a meaningful positive mean.
Background Knowledge
The coefficient of variation equals the standard deviation divided by the mean, and it is multiplied by 100 when reported as a percent. It is useful for comparing relative dispersion when means differ. When the mean is near zero, the ratio becomes unstable and loses meaning.
Real-World Application
An appraiser wants to know whether prices vary more in an entry-level condo segment or a luxury single-family segment. Raw standard deviations are not comparable because the price levels differ, so the appraiser compares each segment's CV instead.
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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