Ratios are 0.90, 1.00 and 1.10. Define COD = 100 × mean absolute deviation from the median ÷ median, and sample COV = 100 × sample SD ÷ mean, with sample variance dividing squared deviations by n−1. Which pair results, to one decimal percent?
Correct Answer
D) COD 6.7%; COV 10.0%
Median and mean are both 1.00. Mean absolute deviation about the median = (0.10+0+0.10)/3 = 0.0667. COD = 100×0.0667/1 = 6.7% rounded. Sample SD = sqrt((0.01+0+0.01)/2) = 0.10. Sample COV = 100×0.10/1 = 10.0%. Absolute and squared deviations measure spread differently; neither percentage is an automatic statutory compliance threshold.
Why This Is the Correct Answer
Option D is correct because the median and mean are both 1.00, and the absolute deviations from the median are 0.10, 0 and 0.10, averaging 0.20 / 3 = 0.0667, so COD = 100 x 0.0667 / 1.00 = 6.7%. The sample variance is (0.01 + 0 + 0.01) / 2 = 0.01, so the sample SD is 0.10 and COV = 100 x 0.10 / 1.00 = 10.0%.
Why the Other Options Are Wrong
Option A: COD 10.0%; COV 6.7%
Option A, COD 10.0%; COV 6.7%, swaps the two results. The COD is 100 x (0.20 / 3) / 1.00 = 6.7%, and the sample COV is 100 x 0.10 / 1.00 = 10.0%.
Option B: COD 10.0%; COV 10.0%
Option B, COD 10.0%; COV 10.0%, gets the COV right but produces a 10.0% COD by dividing the summed absolute deviations by n-1 (0.20 / 2 = 0.10) instead of by n. The COD defined here averages over all three ratios: 0.20 / 3 = 0.0667, giving 6.7%.
Option C: COD 0%; COV 0%
Option C, COD 0%; COV 0%, results from adding signed deviations (-0.10, 0, +0.10), which cancel to zero. Both measures avoid cancellation by using absolute or squared deviations, giving COD 6.7% and COV 10.0%.
COD absolute, COV squared
COD pairs with the median and absolute deviations; COV pairs with the mean and squared deviations through the standard deviation.
How to use: Use this pairing to set up each formula correctly before calculating when both measures appear in one question.
Exam Tip
Watch the divisor the question specifies: here the COD averages over n while the sample variance divides by n-1.
Common Mistakes to Avoid
- -Swapping the COD and COV results
- -Letting signed deviations cancel to zero
- -Dividing absolute deviations by n-1 when the COD averages over n
Concept Deep Dive
Analysis
This question tests whether you can compute two different dispersion measures from the same data and keep them straight. COD uses absolute deviations from the median, while the sample COV uses squared deviations, an n-1 divisor and the mean. Because the two methods treat deviations differently, they give different percentages here.
Background Knowledge
The COD and the coefficient of variation both express dispersion as a percentage of a central value. COD averages absolute deviations from the median, while COV is the standard deviation, built from squared deviations, expressed as a percentage of the mean; squaring gives larger deviations more weight.
Real-World Application
An appraiser comparing uniformity measures in a small ratio sample computes both COD and COV. Seeing COD at 6.7% and COV at 10.0% from the same three ratios reminds the appraiser that the two statistics are not interchangeable and must be compared only with like measures.
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.
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Land or Site Valuation
4.5% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam
Previous Question
In a market study, what does a frequency distribution of sale prices show?
Next Question
Assessment-to-sale ratios are 1.1, 1, 0.9; corresponding sale prices are $100,000, $200,000, $500,000 in the same order. Compute PRD as the arithmetic mean ratio divided by the sale-price-weighted mean ratio. Round to two decimals.