An appraiser summarizes 20 sale prices with no ties using a winsorized mean, treating one observation in each tail. Verification then shows that the second-highest sale closed at $840,000, not the $820,000 on file. The highest sale closed at $980,000. How much does the corrected winsorized mean rise?
Correct Answer
D) $2,000
The second-highest sale is the largest retained value, and the highest sale is reset to it, so both averaged values move from $820,000 to $840,000. That adds $40,000 to the total, and $40,000 ÷ 20 = $2,000. Counting one moved value gives $1,000, dividing by the 18 retained values gives $1,111, and assuming the upper tail has no influence gives $0.
Why This Is the Correct Answer
Option D is correct because the second-highest sale and the reset highest sale both move from $820,000 to $840,000. The total of the 20 averaged values rises by $40,000. Dividing by 20 gives a $2,000 rise in the winsorized mean.
Why the Other Options Are Wrong
Option A: $0
Option A shows $0, which would be right if the $980,000 sale had changed, because its reset value would not move. The corrected sale is the largest retained value, so the average rises by 40,000 ÷ 20 = $2,000.
Option B: $1,000
Option B shows $1,000, from 20,000 ÷ 20, which counts one moved value. It misses that the highest sale is reset to the corrected figure, so the correct change is 40,000 ÷ 20 = $2,000.
Option C: $1,111
Option C shows $1,111, from 20,000 ÷ 18, which is how a trimmed mean of the 18 middle values would respond. A winsorized mean keeps all 20 observations, so the correct change is 40,000 ÷ 20 = $2,000.
The boundary value counts twice
With one observation reset in each tail, the largest and smallest retained values each appear twice in the average. Move a boundary value and you move two observations.
How to use: Use this when a data correction falls at or next to the tail of a winsorized set, to count how many averaged values change.
Exam Tip
Identify which observation changed and write out the resulting top values before dividing. A change to the replaced value and a change to the boundary value have different effects.
Common Mistakes to Avoid
- -Assuming a winsorized mean does not respond to changes near the tails
- -Counting the corrected sale but not the observation reset to it
- -Dividing by the retained count as if the measure were trimmed
Concept Deep Dive
Analysis
This tests how a winsorized mean actually responds to data, not how to run the recipe. The highest observation is set to the largest retained value, so that retained value appears twice in the average. Any change to it therefore has double the weight of a change to a value in the middle of the array.
Background Knowledge
A winsorized mean limits the effect of the size of the extreme observations, since each is set to the lowest or highest retained value. The retained boundary values carry extra weight as a result, because they stand in for the replaced observations as well as themselves.
Real-World Application
An appraiser verifying a sale with the closing agent finds that one high-end price was recorded $20,000 too low. Because that sale is the largest retained value in a winsorized summary, the reported average moves by more than a simple 20,000 ÷ 20. The appraiser recalculates the summary instead of estimating the change.
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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