Ordered sale prices are $105,000, $125,000, $150,000, $175,000, $210,000, $250,000, $290,000, $335,000, $390,000, $490,000, $540,000, $540,000, $590,000, $640,000, $700,000, $780,000, $870,000, $980,000, $1,150,000, $1,400,000. Compute a 40% winsorized mean at each tail: replace the lowest 8 observations with the smallest retained value and the highest 8 with the largest retained value. Average all 20 resulting observations and round to the nearest dollar.
Correct Answer
D) $470,000
Replace lower-tail observations by $390,000 and upper-tail observations by $540,000. The resulting sum is $9,400,000; divide by 20 to obtain $470,000. Winsorization replaces extremes; it does not discard them.
Why This Is the Correct Answer
Option D is correct because after winsorizing, the data contain nine values of $390,000 (8 replaced plus the retained one), one of $490,000 and ten of $540,000 (2 retained plus 8 replaced). The sum is $3,510,000 + $490,000 + $5,400,000 = $9,400,000, and $9,400,000 / 20 = $470,000. Winsorizing keeps all 20 observations but limits the pull of the extremes.
Why the Other Options Are Wrong
Option A: $535,500
Option A, $535,500, is the ordinary mean of all 20 original prices: $10,710,000 / 20 = $535,500. It does not replace the tail values; the winsorized mean is $9,400,000 / 20 = $470,000.
Option B: $515,000
Option B, $515,000, is the median of the 20 prices, the average of the 10th and 11th values: ($490,000 + $540,000) / 2 = $515,000. The question asks for the winsorized mean, $470,000.
Option C: $490,000
Option C, $490,000, is the 40% trimmed mean, which discards the 16 tail observations and averages the remaining four: ($390,000 + $490,000 + $540,000 + $540,000) / 4 = $490,000. Winsorizing replaces those values instead, giving $470,000.
Trim cuts, winsorize clamps
Trimming cuts the tails off and averages fewer values; winsorizing clamps the tail values to the nearest kept value and averages them all.
How to use: Use this to tell trimmed and winsorized mean questions apart.
Exam Tip
Count the observations in your final average: a winsorized mean still divides by the original count, here 20.
Common Mistakes to Avoid
- -Discarding the tail values, which produces a trimmed mean instead
- -Dividing by the number of retained values rather than all 20
- -Replacing the tails with the overall mean instead of the nearest retained value
Concept Deep Dive
Analysis
This question tests the winsorized mean, which replaces extreme values rather than removing them. With 20 ordered prices, the lowest 8 are set to the 9th value ($390,000) and the highest 8 to the 12th value ($540,000). The resulting 20 values sum to $9,400,000, and dividing by 20 gives $470,000.
Background Knowledge
The winsorized mean is similar to the trimmed mean, but instead of dropping the extreme points it sets them to the lowest or highest retained value. Both methods reduce the influence of outliers on the mean. Because the replaced values still count, the winsorized mean can differ noticeably from the trimmed mean.
Real-World Application
An analyst summarizing a broad market with a few very high-priced estates uses a winsorized mean so those sales count but do not dominate. The result is reported alongside the median for context.
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
An appraiser assigns reliability weights of 2, 3, 1 and 2 to sales at $195,000, $240,000, $270,000 and $330,000, respectively. What is the weighted mean price, to the nearest dollar?
Next Question
A sale-price distribution has fourth central moment m4 = 229,635,000 in (thousands of dollars)^4 and second central moment m2 = 8100 in (thousands of dollars)^2. Compute excess kurtosis m4 / m2² − 3. Use moments with the same population (N) divisor, not an unbiased sample variance; report two decimals.