First quartile Q1 is $390,000 and third quartile Q3 is $470,000. Compute the midhinge (Q1+Q3)/2 to the nearest dollar.
Correct Answer
D) $430,000
(390000+470000)/2=430000. The midhinge averages two quartile values, while the interquartile range subtracts them.
Why This Is the Correct Answer
Option D is correct because the midhinge is (Q1 + Q3) / 2 = ($390,000 + $470,000) / 2 = $860,000 / 2 = $430,000. It averages the two quartile values, giving a location measure that is not driven by the extremes.
Why the Other Options Are Wrong
Option A: $80,000
Option A, $80,000, is Q3 minus Q1: $470,000 − $390,000 = $80,000, which is the interquartile range, a measure of spread. The midhinge adds the quartiles and divides by 2, giving $430,000.
Option B: $390,000
Option B, $390,000, is simply Q1 restated. The midhinge averages Q1 and Q3: ($390,000 + $470,000) / 2 = $430,000.
Option C: $470,000
Option C, $470,000, is simply Q3 restated. The midhinge averages Q1 and Q3: ($390,000 + $470,000) / 2 = $430,000.
Hinge adds, IQR subtracts
The midhinge is the midpoint between the two hinges (quartiles), so you add and halve; the interquartile range is the distance between them, so you subtract.
How to use: Use this to avoid mixing up the midhinge with the interquartile range on quartile questions.
Exam Tip
When given Q1 and Q3, check whether the question asks for a location (average them) or a spread (subtract them).
Common Mistakes to Avoid
- -Subtracting the quartiles and reporting the interquartile range
- -Reporting one quartile instead of averaging both
- -Treating the midhinge as a measure of spread
Concept Deep Dive
Analysis
This question tests the midhinge, a robust measure of location built from the quartiles. The midhinge averages the first and third quartiles. With Q1 = $390,000 and Q3 = $470,000, the midhinge is ($390,000 + $470,000) / 2 = $430,000.
Background Knowledge
Quartiles divide ordered data into four parts. The midhinge averages Q1 and Q3 and measures location, while the interquartile range (Q3 − Q1) measures spread. Both rely on the middle half of the data, so they are less sensitive to extreme values than the mean or the full range.
Real-World Application
An appraiser summarizing a market of 40 sales reports Q1 of $390,000 and Q3 of $470,000. The midhinge of $430,000 gives a central value based on the middle half of sales, while the $80,000 interquartile range describes how tightly those sales cluster.
More Statistics Questions
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