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?
Correct Answer
B) $90,000
Q3−Q1=495000−405000=90000. The interquartile range is the difference between the quartiles; the median and full range are different quantities.
Why This Is the Correct Answer
Option B is correct because IQR = Q3 − Q1 = $495,000 − $405,000 = $90,000. The interquartile range measures the spread of the middle half of the data. The quartiles are given, so no further calculation from raw data is needed.
Why the Other Options Are Wrong
Option A: $45,000
Option A, $45,000, is half the interquartile range, $90,000 ÷ 2, which is the semi-interquartile range. The IQR itself is $495,000 − $405,000 = $90,000.
Option C: $450,000
Option C, $450,000, is the average of the two quartiles, ($405,000 + $495,000) ÷ 2, a location measure rather than a spread. The IQR is $495,000 − $405,000 = $90,000.
Option D: $495,000
Option D, $495,000, is simply Q3 without subtracting Q1. The IQR is $495,000 − $405,000 = $90,000.
Top quarter minus bottom quarter
The IQR is the distance from Q1 up to Q3, the width of the middle half of the data.
How to use: Use this whenever a question gives Q1 and Q3 and asks for the interquartile range.
Exam Tip
IQR is a subtraction, so it must be smaller than Q3 and cannot be a price level such as a quartile or the quartile average; if your answer equals or sits between the quartiles, you likely averaged or reported a quartile.
Common Mistakes to Avoid
- -Halving the IQR
- -Averaging the quartiles instead of subtracting
- -Confusing the IQR with the full range
Concept Deep Dive
Analysis
This question tests the definition of the interquartile range, which is the third quartile minus the first quartile. With the quartiles given, the calculation is a single subtraction. Distractors come from halving the result, averaging the quartiles, or reporting one quartile.
Background Knowledge
The interquartile range captures the spread of the central 50% of observations and is resistant to extreme values. It differs from the full range, which uses the maximum and minimum, and from the median, which measures location. The IQR is often used to identify potential outliers.
Real-World Application
An appraiser describing a submarket reports that the middle half of sales fell between $405,000 and $495,000, an IQR of $90,000. This tells readers how tightly typical sales cluster without being distorted by a few unusual transactions.
More Statistics Questions
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