Sale prices are $390,000, $400,000, $410,000, $420,000, $430,000. Find the arithmetic mean of their absolute deviations from the sample median. Report to the nearest dollar.
Correct Answer
C) $12,000
The center is $410,000. Absolute deviations are $20,000, $10,000, 0, $10,000, $20,000. Their mean is $12,000. Averaging absolute deviations and taking the median of those deviations are different operations.
Why This Is the Correct Answer
Option C is correct because the median is $410,000 and the absolute deviations are $20,000, $10,000, $0, $10,000 and $20,000. Their sum is $60,000, and $60,000 / 5 = $12,000. Averaging the absolute deviations is what the question asks for.
Why the Other Options Are Wrong
Option A: $10,000
Option A, $10,000, is the median of the absolute deviations (sorted $0, $10,000, $10,000, $20,000, $20,000), not their mean. The mean is $60,000 / 5 = $12,000.
Option B: $410,000
Option B, $410,000, is the sample median itself, the center from which deviations are measured, not the average deviation. The mean absolute deviation is $60,000 / 5 = $12,000.
Option D: $40,000
Option D, $40,000, is the range, $430,000 − $390,000, which uses only the extremes. The mean absolute deviation is $60,000 / 5 = $12,000.
Center, distance, average
Find the center, measure each distance as a positive number, then average the distances. Averaging is not the same as taking the middle distance.
How to use: Use this sequence for any average absolute deviation problem.
Exam Tip
Read whether the question wants the mean or the median of the deviations; the two give different answers.
Common Mistakes to Avoid
- -Taking the median of the deviations instead of their mean
- -Reporting the median price as the deviation
- -Using the range as a measure of average deviation
Concept Deep Dive
Analysis
This question tests the mean absolute deviation about the median. The steps are to find the median, take each observation's absolute distance from it, and average those distances. The distractors use the median of the deviations, the median itself, or the range.
Background Knowledge
The average absolute deviation from the median is a measure of spread used in ratio studies; it is the numerator step of the coefficient of dispersion. It differs from the median absolute deviation, which takes the middle deviation instead of the average. Absolute values keep deviations above and below the center from canceling out.
Real-World Application
An appraiser checking how tightly five similar sales cluster computes that they deviate from the median by $12,000 on average. That modest spread supports reconciling the comparables to a narrow value range.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
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An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
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.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. 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.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
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Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
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Previous Question
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.
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