The first quartile of sale prices is $420,000 and the third quartile is $580,000. Compute the quartile coefficient of dispersion (Q3−Q1)/(Q3+Q1), as a decimal to three places.
Correct Answer
A) 0.160
(580000−420000)/(580000+420000)=0.16. This dimensionless quartile measure is not the assessment-ratio COD about the median and is not the unscaled interquartile range.
Why This Is the Correct Answer
Option A is correct because (Q3 − Q1) / (Q3 + Q1) = ($580,000 − $420,000) / ($580,000 + $420,000) = $160,000 / $1,000,000 = 0.160. The result is dimensionless because the dollar units cancel.
Why the Other Options Are Wrong
Option B: 1.381
Option B, 1.381, is Q3 divided by Q1 ($580,000 / $420,000), not the stated formula. The correct calculation is $160,000 / $1,000,000 = 0.160.
Option C: 0.724
Option C, 0.724, is Q1 divided by Q3 ($420,000 / $580,000). The formula requires the difference over the sum: $160,000 / $1,000,000 = 0.160.
Option D: 0.381
Option D, 0.381, is the interquartile range divided by Q1 ($160,000 / $420,000) instead of by Q3 + Q1. The correct denominator is $1,000,000, giving 0.160.
Gap over total
QCD is the gap between the quartiles divided by the total of the quartiles.
How to use: Use this to set up the quartile coefficient of dispersion correctly.
Exam Tip
Follow the stated formula literally: compute the numerator and the denominator separately, then divide. Answers formed by simple quartile ratios are common traps.
Common Mistakes to Avoid
- -Dividing one quartile by the other
- -Dividing the interquartile range by only one quartile
- -Confusing the QCD with the interquartile range
Concept Deep Dive
Analysis
This question tests the quartile coefficient of dispersion, a relative spread measure. Subtract the first quartile from the third, then divide by their sum. The distractors come from dividing the quartiles in other ways.
Background Knowledge
The quartile coefficient of dispersion is (Q3 − Q1) / (Q3 + Q1), a relative measure of the spread of the middle half of the data. It is dimensionless, so it can compare spread across data sets at different price levels. It differs from the interquartile range, which is the unscaled difference Q3 − Q1, and from the assessment-ratio COD, which is measured about the median ratio.
Real-World Application
An appraiser comparing price spread in two neighborhoods with different price levels computes each area's quartile coefficient of dispersion. Because the measure is unit-free, the areas can be compared directly.
More Statistics Questions
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