Five competitive closed sales have a mean price of $480,000 and a sample standard deviation of $24,000. What is the coefficient of variation (COV) of the sale prices?
Correct Answer
C) 0.050 or 5.0%
COV = sample standard deviation ÷ sample mean = 24,000 ÷ 480,000 = 0.050, or 5.0%. The dollars cancel, so COV has no unit. Common errors: inverting the ratio (mean ÷ standard deviation) gives 20.0; also dividing by the 5 sales gives 0.010; using the standard error (24,000 ÷ √5) instead of the standard deviation gives 0.022. COV describes relative dispersion; it is not an uncertainty interval or a value conclusion.
Why This Is the Correct Answer
Option C is correct because COV = $24,000 ÷ $480,000 = 0.050, or 5.0%. The dollar units cancel, so the COV is a unitless measure of relative dispersion.
Why the Other Options Are Wrong
Option A: 20.0 or 2,000%
Option A, 20.0 or 2,000%, inverts the ratio: $480,000 ÷ $24,000 = 20.0. The COV divides the standard deviation by the mean: $24,000 ÷ $480,000 = 0.050.
Option B: 0.010 or 1.0%
Option B, 0.010 or 1.0%, also divides by the 5 sales: $24,000 ÷ $480,000 ÷ 5 = 0.010. The sample size is already reflected in the standard deviation, so COV = $24,000 ÷ $480,000 = 0.050.
Option D: 0.022 or 2.2%
Option D, 0.022 or 2.2%, uses the standard error instead of the standard deviation: ($24,000 ÷ √5) ÷ $480,000 ≈ 0.022. The COV uses the standard deviation: $24,000 ÷ $480,000 = 0.050.
Spread over center
COV puts the spread (standard deviation) on top and the center (mean) on the bottom. If the answer is bigger than 1 for typical sale data, it is probably flipped.
How to use: Use this to check the order of division in COV problems.
Exam Tip
Confirm that the numerator is the standard deviation, not the standard error, and that you did not divide by the sample size again.
Common Mistakes to Avoid
- -Dividing the mean by the standard deviation
- -Using the standard error instead of the standard deviation
- -Dividing again by the number of sales
Concept Deep Dive
Analysis
This question tests the coefficient of variation as the sample standard deviation divided by the sample mean. Divide in the correct order, keep the standard deviation rather than the standard error, and express the result as a decimal or percentage.
Background Knowledge
The coefficient of variation expresses the standard deviation as a percentage of the mean, allowing comparison of dispersion across data sets with different price levels. It describes relative spread and is not an uncertainty interval or a value conclusion. The standard error measures uncertainty in the mean and is a different statistic.
Real-World Application
An appraiser compares price dispersion in two neighborhoods with very different price levels. Computing the COV for each lets the appraiser see which market has relatively more varied prices.
More Statistics Questions
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Previous Question
The assessment-to-sale ratios are 0.9, 0.95, 1, 1.05, 1.15. 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.
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Sale-price mean is $510,000, mode $420,000 and standard deviation $90,000. Calculate Pearson's first skewness coefficient (mean − mode) / standard deviation, to two decimals.