Sale prices are $390,000, $400,000, $410,000, $420,000, $430,000. Find the median of their absolute deviations from the sample median. Report to the nearest dollar.
Correct Answer
D) $10,000
The center is $410,000. Absolute deviations are $20,000, $10,000, 0, $10,000, $20,000. Their median is $10,000. Averaging absolute deviations and taking the median of those deviations are different operations.
Why This Is the Correct Answer
Option D is correct because the sample median is $410,000, and the absolute deviations from it are $20,000, $10,000, $0, $10,000 and $20,000. Ordered, these are $0, $10,000, $10,000, $20,000, $20,000. The middle value is $10,000.
Why the Other Options Are Wrong
Option A: $12,000
Option A, $12,000, is the average absolute deviation rather than the median: ($20,000 + $10,000 + $0 + $10,000 + $20,000) ÷ 5 = $60,000 ÷ 5 = $12,000. The median of those deviations is $10,000.
Option B: $410,000
Option B, $410,000, is the sample median of the prices, which is only the first step. The MAD is the median of the absolute deviations from $410,000, which is $10,000.
Option C: $40,000
Option C, $40,000, is the range of the prices: $430,000 − $390,000 = $40,000. The MAD is the median of the absolute deviations, $10,000.
Median of the distances
MAD is a median twice: first the median of the data, then the median of each value's distance from it.
How to use: Use this to remember that the final step is another median, not an average.
Exam Tip
Watch for the average absolute deviation as a distractor; it averages the distances, while the MAD takes their median.
Common Mistakes to Avoid
- -Averaging the absolute deviations instead of taking their median
- -Reporting the sample median as the answer
- -Confusing the MAD with the range
Concept Deep Dive
Analysis
This question tests the median absolute deviation (MAD). First find the sample median, then compute each observation's absolute distance from it, and finally take the median of those distances. Candidates must not average the distances or stop after finding the center.
Background Knowledge
The median absolute deviation is a robust measure of spread built entirely from medians. Because it uses medians instead of means, a few extreme values have little effect on it. It differs from the average absolute deviation, which averages the distances instead.
Real-World Application
An appraiser checking the spread of five similar condo sales computes a MAD of $10,000. Because the MAD resists outliers, she uses it to flag any later sale that falls far outside the typical range for verification.
More Statistics Questions
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Previous Question
The assessment-to-sale ratios are 0.86, 0.94, 1, 1.06, 1.14. 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.
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