Price per square foot in a market has a first quartile of $210 and a third quartile of $290. What is the interquartile range, and what kind of measure is it?
Correct Answer
C) $80; a robust measure of scale
IQR = upper quartile − lower quartile = $290 − $210 = $80. The interquartile range is used as a robust measure of scale.
Why This Is the Correct Answer
Option C is correct because the IQR is the upper quartile minus the lower quartile: $290 − $210 = $80. The interquartile range is used as a robust measure of scale, meaning it describes spread while being little affected by extreme observations.
Why the Other Options Are Wrong
Option A: $500; a measure of central tendency
Option A gives $500 and calls it a measure of central tendency. $500 is the sum of the quartiles ($290 + $210), not their difference, and the IQR measures spread, not central tendency.
Option B: $80; a measure of central tendency
Option B gives the right figure, $80, but calls it a measure of central tendency. The IQR measures how spread out the middle half of the data is, so the classification is wrong and the option is not the best answer.
Option D: $250; a robust measure of scale
Option D gives $250, which is the midpoint of the quartiles, ($290 + $210) / 2, not the IQR. The IQR is $290 − $210 = $80.
IQR is the middle-half width
Picture a box covering the middle half of the data; the IQR is how wide that box is, Q3 minus Q1.
How to use: Use this to remember both how to compute the IQR and that it measures spread.
Exam Tip
For two-part answers, check both the number and the classification. A correct figure with the wrong description is still wrong.
Common Mistakes to Avoid
- -Adding the quartiles instead of subtracting them
- -Calling the IQR a measure of central tendency
- -Reporting the midpoint of the quartiles as the IQR
Concept Deep Dive
Analysis
This question tests both the calculation and the classification of the interquartile range (IQR). The IQR is the third quartile minus the first quartile. It describes spread, not a typical value, and it is resistant to extreme values.
Background Knowledge
The interquartile range is the difference between the 75th and 25th percentiles of a variable and covers the middle half of the data. Because it ignores the top and bottom quarters, it is a robust measure of scale. Measures of central tendency, such as the mean and median, describe a typical value instead.
Real-World Application
An appraiser reports that price per square foot in a market area has an IQR of $80, from $210 to $290. Because a few unusual sales do not move this figure much, it gives a stable picture of typical spread.
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Previous Question
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.
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