Five sale prices are $390,000, $410,000, $420,000, $430,000, $520,000. Remove the single smallest and single largest price, then calculate the mean of the remaining three. Report to the nearest dollar.
Correct Answer
D) $420,000
After ordering, retain $410,000, $420,000, $430,000. Their sum is $1,260,000; dividing by 3 gives $420,000. This is trimming, not replacing the extreme observations.
Why This Is the Correct Answer
Option D is correct because removing $390,000 and $520,000 leaves $410,000, $420,000 and $430,000. Their sum is $1,260,000, and $1,260,000 / 3 = $420,000. This is trimming, not replacing the extremes.
Why the Other Options Are Wrong
Option A: $434,000
Option A, $434,000, is the untrimmed mean of all five prices: $2,170,000 / 5 = $434,000. Trimming the extremes gives $1,260,000 / 3 = $420,000.
Option B: $1,260,000
Option B, $1,260,000, is the sum of the three remaining prices without dividing by 3. The trimmed mean is $1,260,000 / 3 = $420,000.
Option C: $412,500
Option C, $412,500, removes only the high $520,000 sale and averages the other four: $1,650,000 / 4 = $412,500. The stated rule also removes the smallest price, $390,000, so the trimmed mean is $1,260,000 / 3 = $420,000.
Trim the ends, average middle
Like trimming bread crusts, cut off the ends and average what remains.
How to use: Use this to separate trimmed means from untrimmed and Winsorized means.
Exam Tip
After trimming, divide by the number of values kept, not the original count.
Common Mistakes to Avoid
- -Averaging all values without trimming
- -Trimming only the high outlier and not the low end
- -Replacing extremes instead of removing them
Concept Deep Dive
Analysis
This question tests calculating a trimmed mean. After ordering, the single smallest and single largest prices are both removed and the remaining three are averaged. The distractors include the untrimmed mean, the unaveraged sum and a one-sided trim.
Background Knowledge
A trimmed mean removes a set number or share of the most extreme observations from each end before averaging. This reduces the influence of outliers such as the $520,000 sale. A Winsorized mean, by contrast, replaces the extremes with the nearest remaining values instead of removing them.
Real-World Application
An appraiser summarizing five sales notices one unusually high price. She reports a trimmed mean of $420,000 alongside the untrimmed mean to show how much the outlier affects the average.
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.
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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.
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