Sale prices are $255,000, $280,000, $300,000, $320,000, $355,000. Using the median of each half including the overall median (Tukey hinges), Q1=$280,000 and Q3=$320,000. Compute the interquartile range Q3−Q1, to the nearest dollar.
Correct Answer
A) $40,000
Q3−Q1=320000−280000=40000. The supplied quartiles/convention control; the median and full range are different quantities.
Why This Is the Correct Answer
Option A is correct because the interquartile range is Q3 − Q1 = $320,000 − $280,000 = $40,000. The stem supplies both quartiles under the Tukey hinge convention, so no further calculation of quartiles is needed.
Why the Other Options Are Wrong
Option B: $20,000
Option B, $20,000, is half of the interquartile range ($40,000 ÷ 2), which is the semi-interquartile range rather than the IQR. The IQR asked for is $320,000 − $280,000 = $40,000.
Option C: $300,000
Option C, $300,000, is the median of the five sale prices, a measure of location rather than spread. The IQR is $320,000 − $280,000 = $40,000.
Option D: $320,000
Option D, $320,000, is the third quartile alone, not the difference between quartiles. Subtracting Q1 gives $320,000 − $280,000 = $40,000.
Top quarter minus bottom quarter
IQR is the upper quartile minus the lower quartile, the width of the middle half of the data. It is a difference, so it is not a single quartile value.
How to use: Use this to reject answer choices that simply repeat one quartile or the median.
Exam Tip
When quartiles are given, use them as stated rather than recalculating under a different convention. Check that your answer is a difference, not a location.
Common Mistakes to Avoid
- -Reporting a quartile or the median instead of the difference
- -Halving the IQR and reporting the semi-interquartile range
- -Recalculating quartiles under a different convention than the one given
Concept Deep Dive
Analysis
This question tests the interquartile range when the quartiles are already supplied under a stated convention. The task is simply to subtract Q1 from Q3 and to avoid confusing the result with the median, a single quartile or half of the IQR.
Background Knowledge
The interquartile range measures the spread of the middle half of a data set as the upper quartile minus the lower quartile. Because it ignores the extreme values, it is less affected by outliers than the full range. Different quartile conventions can produce different quartiles, so the stated convention controls.
Real-World Application
An appraiser summarizing neighborhood sales reports that the middle half of prices spans $280,000 to $320,000. Stating the IQR of $40,000 tells the client how tightly typical sales cluster without being distorted by one unusually high or low sale.
More Statistics Questions
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