Sale prices are $190,000, $210,000, $230,000, $250,000, $270,000, $300,000, $360,000. Using the median of each half excluding the overall median, Q1=$210,000 and Q3=$300,000. Compute the interquartile range Q3−Q1, to the nearest dollar.
Correct Answer
B) $90,000
Q3−Q1=300000−210000=90000. The supplied quartiles/convention control; the median and full range are different quantities.
Why This Is the Correct Answer
Option B is correct because the interquartile range is Q3 − Q1 = $300,000 − $210,000 = $90,000. The supplied quartiles and convention control the answer. The IQR measures the spread of the middle half of the data.
Why the Other Options Are Wrong
Option A: $45,000
Option A, $45,000, is half the interquartile range ($90,000 / 2), sometimes called the semi-interquartile range. The IQR itself is $300,000 − $210,000 = $90,000.
Option C: $255,000
Option C, $255,000, is the midpoint of the two quartiles, ($210,000 + $300,000) / 2, which is a location figure, not a spread. The IQR is $300,000 − $210,000 = $90,000.
Option D: $300,000
Option D, $300,000, is simply Q3, the upper quartile, with nothing subtracted. The IQR is $300,000 − $210,000 = $90,000.
Top quarter minus bottom quarter
IQR = Q3 minus Q1, the width of the middle half. It is a distance, so it is never a price level like Q3 alone.
How to use: Use this to separate the IQR from the median, the range and the quartiles themselves.
Exam Tip
When the stem gives Q1 and Q3, the IQR is just their difference; do not recalculate quartiles with a different convention.
Common Mistakes to Avoid
- -Halving the IQR
- -Reporting the upper quartile as the IQR
- -Confusing the IQR with the full range
Concept Deep Dive
Analysis
This question tests the interquartile range, which is the upper quartile minus the lower quartile. The stem supplies both quartiles under a stated convention, so the candidate only needs to subtract. The distractors confuse the IQR with related quantities such as half the IQR, the quartile midpoint or Q3 itself.
Background Knowledge
The interquartile range is the upper quartile minus the lower quartile and covers the middle 50 percent of the data. Because it ignores the extremes, it is less affected by outliers than the full range. Different quartile conventions can give slightly different quartiles, so the stated convention matters.
Real-World Application
An appraiser describing a neighborhood notes that the middle half of sales fell between $210,000 and $300,000, an IQR of $90,000. This gives readers a sense of the typical price spread without letting the single $360,000 sale distort it.
More Statistics Questions
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