Ordered sale prices are $200,000, $220,000, $240,000, $280,000, $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, $580,000, $800,000, $950,000. Compute a 15% winsorized mean at each tail: replace the lowest 3 observations with the smallest retained value and the highest 3 with the largest retained value. Average all 20 resulting observations and round to the nearest dollar.
Correct Answer
D) $418,000
Replace lower-tail observations by $280,000 and upper-tail observations by $550,000. The resulting sum is $8,360,000; divide by 20 to obtain $418,000. Winsorization replaces extremes; it does not discard them.
Why This Is the Correct Answer
Option D is correct because the three lowest values ($200,000, $220,000, $240,000) are replaced by $280,000 and the three highest ($580,000, $800,000, $950,000) by $550,000. The 14 retained values sum to $5,870,000; adding 3 × $280,000 and 3 × $550,000 gives $8,360,000. Dividing by 20 gives $418,000.
Why the Other Options Are Wrong
Option A: $443,000
Option A, $443,000, is the ordinary mean of the original 20 prices: $8,860,000 ÷ 20 = $443,000. Winsorizing replaces the extremes first, giving $8,360,000 ÷ 20 = $418,000.
Option B: $420,000
Option B, $420,000, is the median of the 20 prices, the average of the 10th and 11th values: ($410,000 + $430,000) ÷ 2 = $420,000. The winsorized mean averages all 20 adjusted values, $8,360,000 ÷ 20 = $418,000.
Option C: $419,286
Option C, $419,286, is the 15% trimmed mean, which discards the six extremes and averages the 14 remaining values: $5,870,000 ÷ 14 ≈ $419,286. Winsorizing keeps all 20 positions by replacing the extremes, giving $8,360,000 ÷ 20 = $418,000.
Trim cuts, winsorize clips
Trimming cuts the extreme values off; winsorizing clips them to the edge values but keeps them in the count.
How to use: Use this to decide whether to divide by the full count or the reduced count.
Exam Tip
For a winsorized mean, divide by the original number of observations. Dividing by the reduced count gives the trimmed mean instead.
Common Mistakes to Avoid
- -Discarding the extremes and dividing by the reduced count
- -Reporting the plain mean without adjusting the extremes
- -Replacing extremes with the overall mean rather than the nearest retained value
Concept Deep Dive
Analysis
This question tests the winsorized mean, which limits the influence of extremes by replacing them rather than removing them. The three lowest values are set to the smallest retained value and the three highest to the largest retained value, and then all 20 values are averaged. Candidates must distinguish this from the plain mean, the trimmed mean and the median.
Background Knowledge
Trimmed and winsorized means both reduce the effect of outliers on an average. A trimmed mean drops a set share of the highest and lowest values, while a winsorized mean resets them to the nearest retained values and keeps the full count. Both lie between the median and the ordinary mean in robustness.
Real-World Application
An analyst summarizing 20 neighborhood sales sees two estate sales far above the rest. She reports a winsorized mean so those extremes count only at the level of the highest typical sale, avoiding an average pulled up by the outliers.
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.
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Land or Site Valuation
4.5% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam
Previous Question
When a statistical model and a separate applicable appraisal analysis produce different indications, why investigate their assumptions and evidence together?
Next 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.