The interquartile range of a set of sale prices describes which of the following?
Correct Answer
A) The spread of the middle half of the observations
IQR = Q3 − Q1, the 75th-percentile value minus the 25th-percentile value. It summarizes the spread of the central half of the distribution. Maximum minus minimum is the range; averaging absolute distances gives an absolute-deviation measure. A count above the mean measures neither IQR nor range. For small samples, the chosen quartile convention should be stated when computing a number.
Why This Is the Correct Answer
Option A is correct because the IQR equals Q3 minus Q1, the difference between the 75th and 25th percentile values. Those two points bracket the middle 50 percent of the observations, so the IQR summarizes the spread of the middle half. Because it ignores the extreme tails, it is a more stable spread measure than the standard deviation when the data have outliers.
Why the Other Options Are Wrong
Option B: The gap between the highest and lowest observation
Option B, the gap between the highest and lowest observation, describes the range, not the IQR. The range uses the two extremes, while the IQR uses the first and third quartiles.
Option C: The average distance of each value from the mean
Option C, the average distance of each value from the mean, describes an average absolute deviation. That measure uses every observation's distance from the mean, not the difference between two quartiles.
Option D: The proportion of sales above the mean sale price
Option D, the proportion of sales above the mean sale price, is a count-based share, not a measure of spread. It does not equal the IQR or the range.
IQR covers the middle half
Interquartile means between the quartiles: from the one-quarter mark to the three-quarter mark, which is the middle half.
How to use: Use the word inter, meaning between, to recall that the IQR measures the span between Q1 and Q3.
Exam Tip
Separate range (maximum minus minimum) from IQR (Q3 minus Q1). Questions about outlier-resistant spread usually point to the IQR.
Common Mistakes to Avoid
- -Confusing the IQR with the full range
- -Thinking the IQR is an average distance from the mean
- -Forgetting to state the quartile convention for small samples
Concept Deep Dive
Analysis
This question tests the meaning of the interquartile range (IQR). The IQR is the 75th-percentile value minus the 25th-percentile value, so it measures how spread out the central half of the data is. Candidates must distinguish it from the full range and from deviation-based measures.
Background Knowledge
Quartiles divide ordered data into four equal parts: Q1 at the 25th percentile, Q2 (the median) at the 50th and Q3 at the 75th. The IQR, Q3 minus Q1, is a robust spread measure because extreme values in the tails do not affect it. When computing it for small samples, the quartile convention used should be stated.
Real-World Application
An appraiser analyzing condominium sales notices one penthouse sale far above the rest. She reports the IQR of the sale prices so that the spread of typical units is not distorted by that one extreme sale.
More Statistics Questions
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Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
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