Sale prices are $380,000, $400,000, $420,000, $440,000, $460,000. Find the arithmetic mean of their absolute deviations from the sample mean. Report to the nearest dollar.
Correct Answer
B) $24,000
The center is $420,000. Absolute deviations are $40,000, $20,000, 0, $20,000, $40,000. Their mean is $24,000. Averaging absolute deviations and taking the median of those deviations are different operations.
Why This Is the Correct Answer
Option B is correct because the mean is $420,000, the absolute deviations are $40,000, $20,000, $0, $20,000 and $40,000, and their sum of $120,000 divided by 5 is $24,000. This averages the absolute deviations, which is a different operation from taking the median of those deviations.
Why the Other Options Are Wrong
Option A: $20,000
Option A, $20,000, is the median of the absolute deviations ($0, $20,000, $20,000, $40,000, $40,000), not their mean. Averaging them gives $120,000 / 5 = $24,000.
Option C: $420,000
Option C, $420,000, is the sample mean of the prices, which is the center used in the calculation, not a measure of deviation. The average absolute deviation from that center is $24,000.
Option D: $80,000
Option D, $80,000, is the range, the largest value minus the smallest ($460,000 - $380,000). The question asks for the mean of the absolute deviations, which is $24,000.
Center, distance, average
Find the center, measure each distance ignoring sign, then average the distances. Do not stop at the center or switch to the median.
How to use: Use these three steps for any average absolute deviation problem.
Exam Tip
Read carefully whether the question asks for the mean or the median of the deviations. Both often appear as answer choices.
Common Mistakes to Avoid
- -Taking the median of the deviations instead of the mean
- -Reporting the mean or range instead of the deviation measure
- -Forgetting to use absolute values so deviations cancel to zero
Concept Deep Dive
Analysis
This calculation tests the average absolute deviation from the mean, a measure of dispersion. Find the sample mean, take each price's absolute distance from it, and average those distances. The distractors substitute a different statistic, such as the median deviation, the mean itself or the range.
Background Knowledge
Measures of scale describe how spread out data are; common ones include the range, variance, standard deviation and average absolute deviation. The average absolute deviation is the mean of the absolute distances of the observations from the center. In appraisal, dispersion measures help gauge how consistent a set of comparable prices is.
Real-World Application
An appraiser examines five recent sales in a subdivision to see how tightly prices cluster. Computing the average absolute deviation from the mean shows the typical distance of a sale from the average price. A small spread supports confidence in the market range, while a large one prompts further analysis.
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.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
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.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
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
Observed sale prices are $80,000, $90,000, $100,000, $110,000. Calculate the range (maximum−minimum), to the nearest dollar.
Next Question
An appraiser arranges twenty sale prices in ascending order: $290,000, $310,000, $330,000, $350,000, $370,000, $390,000, $410,000, $430,000, $450,000, $470,000, $490,000, $510,000, $530,000, $550,000, $570,000, $590,000, $610,000, $630,000, $650,000, $670,000. What is the 10th percentile by the nearest-rank method, which takes the lowest price at or below which at least 10% of the sales fall?