Quartiles of sale-price-to-list-price ratios are Q1 = 0.94 and Q3 = 1.02. What is the quartile coefficient of dispersion?
Correct Answer
D) 0.0408
QCD = (Q3 − Q1) ÷ (Q1 + Q3) = 0.08 ÷ 1.96 = 0.0408. It is a robust alternative to the coefficient of variation.
Why This Is the Correct Answer
Option D is correct because QCD = (Q3 − Q1) / (Q1 + Q3) = (1.02 − 0.94) / (0.94 + 1.02) = 0.08 / 1.96 = 0.0408. Dividing by the sum of the quartiles makes the measure unit-free and comparable across data sets.
Why the Other Options Are Wrong
Option A: 0.08
Option A, 0.08, is only the numerator, Q3 − Q1 = 1.02 − 0.94 = 0.08, which is the interquartile range. QCD divides that by Q1 + Q3 = 1.96, giving 0.0408.
Option B: 0.98
Option B, 0.98, is the midpoint of the quartiles, (0.94 + 1.02) / 2 = 0.98, a measure of location rather than dispersion. QCD is 0.08 / 1.96 = 0.0408.
Option C: 0.0816
Option C, 0.0816, divides the interquartile range by the midpoint of the quartiles: 0.08 / 0.98 = 0.0816. The QCD formula divides by the sum, 0.08 / 1.96 = 0.0408.
Difference over sum
QCD is the quartile difference over the quartile sum: (Q3 − Q1) on top, (Q1 + Q3) on the bottom.
How to use: Use this to set up the QCD fraction without dividing by the midpoint by mistake.
Exam Tip
Dividing by the midpoint instead of the sum doubles the answer; check which denominator the formula uses.
Common Mistakes to Avoid
- -Reporting the interquartile range without dividing
- -Dividing by the midpoint of the quartiles instead of their sum
- -Confusing the quartile midpoint with a dispersion measure
Concept Deep Dive
Analysis
This question tests the quartile coefficient of dispersion (QCD), a robust relative measure of spread. QCD divides the difference between Q3 and Q1 by their sum. Here (1.02 − 0.94) / (0.94 + 1.02) = 0.08 / 1.96 = 0.0408.
Background Knowledge
The quartile coefficient of dispersion is defined as (Q3 − Q1) / (Q1 + Q3). Because it uses quartiles, it is less affected by extreme values than the coefficient of variation. It expresses spread relative to the level of the data, so different markets can be compared.
Real-World Application
An appraiser studying sale-to-list price ratios in two neighborhoods computes QCD for each. A QCD of 0.0408 in one area indicates tightly clustered ratios, suggesting list prices there are a fairly consistent guide to sale prices.
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