Seven ordered comparable sale prices are $200,000, $210,000, $220,000, $230,000, $240,000, $250,000, and $380,000. Using the median of each half (excluding the overall median) for the quartiles, what are the range and the interquartile range (IQR = Q3 − Q1)?
Correct Answer
A) Range $180,000; IQR $40,000
Range = 380000−200000 = $180,000. Overall median is 230000. Excluding it gives lower-half median Q1=$210,000 and upper-half median Q3=$250,000. IQR=250000−210000=$40,000. The high observation affects the extreme-based range more; its magnitude alone does not prove invalid data. Swapping the measures or treating IQR as the full range does not follow the stated convention.
Why This Is the Correct Answer
Option A is correct because the range is $380,000 − $200,000 = $180,000. The overall median is $230,000; excluding it, the lower half ($200,000, $210,000, $220,000) has median Q1 = $210,000 and the upper half ($240,000, $250,000, $380,000) has median Q3 = $250,000, so the IQR is $250,000 − $210,000 = $40,000.
Why the Other Options Are Wrong
Option B: Range $40,000; IQR $180,000
Option B, range $40,000 and IQR $180,000, swaps the two measures. The range is the full spread, $380,000 − $200,000 = $180,000, and the IQR is $250,000 − $210,000 = $40,000.
Option C: Range $180,000; IQR $180,000
Option C, range $180,000 and IQR $180,000, gets the range right but treats the IQR as the full range. The IQR uses the quartiles, $250,000 − $210,000 = $40,000.
Option D: Range $180,000; IQR $30,000
Option D, range $180,000 and IQR $30,000, gets the range right but includes the overall median of $230,000 in each half. That makes the lower half $200,000 to $230,000 with Q1 = $215,000 and the upper half $230,000 to $380,000 with Q3 = $245,000, so $245,000 − $215,000 = $30,000; the stated convention excludes the median, giving Q1 = $210,000, Q3 = $250,000 and an IQR of $40,000.
Range sees extremes, IQR ignores them
The range stretches to include the outlier; the IQR stays with the middle half. With one high sale, expect a big range and a modest IQR.
How to use: Use this to check that your range is larger than your IQR.
Exam Tip
The IQR cannot exceed the range, so any choice with a larger IQR than range is wrong. Follow the stated quartile convention exactly.
Common Mistakes to Avoid
- -Swapping the range and the IQR
- -Reporting the full range as the IQR
- -Including the overall median in each half when the convention excludes it
Concept Deep Dive
Analysis
This question tests computing both the range and the interquartile range under a stated quartile convention and recognizing how an extreme value affects each. The range uses the two extremes, while the IQR uses the medians of the lower and upper halves after excluding the overall median.
Background Knowledge
The range is the largest value minus the smallest, so one high sale can enlarge it greatly. The interquartile range measures the spread of the middle half and is more resistant to extremes. A high observation affects the range more, but its size alone does not prove the data are invalid.
Real-World Application
An appraiser finds seven comparable sales, one of which sold far above the rest. Reporting both the range and the IQR shows the client that most sales cluster tightly even though one sale widens the overall spread, prompting the appraiser to verify that sale.
More Statistics Questions
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Sale prices have an interquartile range of $80,000. Using the normalized interquartile range (0.7413 × IQ), what is the approximate standard deviation for normally distributed data?
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The assessment-to-sale ratios are 0.85, 0.9, 0.95, 1, 1, 1.05, 1.1, 1.15. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.