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?
Correct Answer
A) $59,304
For a normal distribution the IQR is about 1.349 standard deviations, so NIQ = 0.7413 × $80,000 = $59,304 approximates the standard deviation.
Why This Is the Correct Answer
Option A is correct because the normalized IQR is 0.7413 × $80,000 = $59,304. For a normal distribution the IQR is about 1.349 times the standard deviation, so this product approximates the standard deviation.
Why the Other Options Are Wrong
Option B: $107,920
Option B, $107,920, comes from multiplying by 1.349 instead of 0.7413: $80,000 × 1.349 = $107,920. Since the IQR is about 1.349 standard deviations, the standard deviation is smaller than the IQR: 0.7413 × $80,000 = $59,304.
Option C: $40,000
Option C, $40,000, is simply half the IQR ($80,000 ÷ 2), which is not the stated normalization. The correct estimate is 0.7413 × $80,000 = $59,304.
Option D: $80,000
Option D, $80,000, is the unnormalized IQR itself, not a standard deviation estimate. Applying the factor gives 0.7413 × $80,000 = $59,304.
IQR is wider than SD
For normal data the IQR spans about 1.35 standard deviations, so the standard deviation is about three-quarters of the IQR. The answer must be smaller than the IQR.
How to use: Use this to reject any answer larger than the given IQR.
Exam Tip
Check the direction of the conversion: a standard deviation estimated from the IQR should be smaller than the IQR for normal data.
Common Mistakes to Avoid
- -Multiplying by 1.349 instead of 0.7413
- -Reporting the raw IQR as the standard deviation
- -Halving the IQR as a shortcut
Concept Deep Dive
Analysis
This question tests the normalized interquartile range as a robust estimate of the standard deviation for normally distributed data. Because the IQR of a normal distribution is about 1.349 standard deviations, multiplying the IQR by 0.7413 (approximately 1 ÷ 1.349) converts it to a standard deviation estimate.
Background Knowledge
The normalized interquartile range (NIQ = 0.7413 × IQR) is a resistant estimate of spread that matches the standard deviation when data are normally distributed. It relies on the fact that the IQR of a normal distribution is about 1.349 standard deviations. Because it uses quartiles, it is less affected by extreme sales than the ordinary standard deviation.
Real-World Application
An appraiser has a sale-price data set with a few extreme listings that inflate the standard deviation. Using the normalized IQR gives a spread estimate that better reflects typical sales.
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?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
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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)?