An appraiser is summarizing ten verified sales of similar homes, which sold for $210,000, $240,000, $250,000, $260,000, $270,000, $280,000, $290,000, $300,000, $400,000 and $640,000. One colleague calculates a trimmed mean and another a winsorized mean, each treating one observation in each tail. What are the correct results?
Correct Answer
C) Trimmed mean $286,250; winsorized mean $293,000
Trimming drops the two extreme sales, so the middle eight prices total $2,290,000 and the trimmed mean is $2,290,000 ÷ 8 = $286,250. Winsorizing keeps ten observations but resets $210,000 to $240,000 and $640,000 to $400,000, so the total is $2,930,000 and the mean is $293,000. Swapping the two labels gives option a, dropping only the high sale gives $2,500,000 ÷ 9 = $277,778, and the plain mean of the original prices gives $314,000.
Why This Is the Correct Answer
Option C is correct because removing $210,000 and $640,000 leaves $2,290,000 across eight sales, or $286,250. Resetting those two sales to $240,000 and $400,000 gives $2,930,000 across ten observations, or $293,000. The winsorized figure is higher here because the high tail was much farther from the center and still counts at $400,000.
Why the Other Options Are Wrong
Option A: Trimmed mean $293,000; winsorized mean $286,250
Option A reverses the labels, showing a trimmed mean of $293,000 and a winsorized mean of $286,250. The $286,250 comes from dropping the two extremes, 2,290,000 ÷ 8, which is trimming; the $293,000 comes from resetting them, 2,930,000 ÷ 10, which is winsorizing.
Option B: Trimmed mean $277,778; winsorized mean $293,000
Option B shows a trimmed mean of $277,778 beside a winsorized mean of $293,000. The winsorized figure is right, but $277,778 drops only the $640,000 sale: 2,500,000 ÷ 9. One observation is treated in each tail, so the $210,000 sale also leaves, and the correct figure is 2,290,000 ÷ 8 = $286,250.
Option D: Trimmed mean $286,250; winsorized mean $314,000
Option D shows a trimmed mean of $286,250 beside a winsorized mean of $314,000. The trimmed figure is right, but $314,000 is the plain mean of the original prices: 3,140,000 ÷ 10. Winsorizing must first reset $210,000 to $240,000 and $640,000 to $400,000, which gives 2,930,000 ÷ 10 = $293,000.
Trim cuts, winsorize swaps
Trimming a hedge cuts pieces away, so the count falls. Winsorizing swaps the extremes for their nearest neighbors, so the count stays the same.
How to use: Use this when an exam names a robust mean without listing its steps, to decide whether the divisor changes.
Exam Tip
Check the divisor first. A trimmed result divides by the retained count, and a winsorized result divides by the original count.
Common Mistakes to Avoid
- -Trimming one tail when the method treats both tails
- -Treating trimmed and winsorized means as the same calculation
- -Reporting the untreated mean after saying the tails were treated
Concept Deep Dive
Analysis
This tests whether you know what each robust average does to the extreme sales, since the steps have to come from memory. A trimmed mean removes the tail observations and divides by the smaller count. A winsorized mean keeps the count and resets each tail observation to the nearest retained value.
Background Knowledge
Trimmed and winsorized means both reduce the pull of extreme observations on an average. They differ in how they treat those observations: trimming removes them, while winsorizing sets them to the lowest or highest retained value and keeps them in the count.
Real-World Application
An appraiser reviewing a market summary sees one reported average of $286,250 and another of $293,000 for the same ten sales. Recomputing both shows the figures are not in conflict, because one analyst trimmed and the other winsorized the same tails. The report should name the method used.
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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