Sale-price observations are $410,000, $420,000, $435,000, $450,000. Compute the median; order the observations first and average the two middle values if n is even. Report to the nearest dollar.
Correct Answer
A) $427,500
Ordered values: $410,000, $420,000, $435,000, $450,000. The average of the two middle observations is 427500. Common errors: computing the mean instead of the median gives $428,750; reporting the lowest observation gives $410,000; reporting the highest observation gives $450,000.
Why This Is the Correct Answer
Option A is correct because the ordered values are $410,000, $420,000, $435,000 and $450,000, and the average of the two middle values is ($420,000 + $435,000) / 2 = $427,500. For an even number of items, the median is the arithmetic average of the two central items.
Why the Other Options Are Wrong
Option B: $428,750
Option B, $428,750, is the mean rather than the median: ($410,000 + $420,000 + $435,000 + $450,000) / 4 = $1,715,000 / 4 = $428,750. The median averages only the two middle values, giving $427,500.
Option C: $410,000
Option C, $410,000, is simply the lowest observation. The median averages the two middle values, $420,000 and $435,000, to get $427,500.
Option D: $450,000
Option D, $450,000, is simply the highest observation. The median is the average of the two central values, ($420,000 + $435,000) / 2 = $427,500.
Line up, find middle
Line the values up in order and walk inward from both ends. If two values are left in the middle, average them.
How to use: Use this whenever a median question has an even count of observations.
Exam Tip
Median and mean distractors often sit close together. Confirm which statistic is asked for and, for an even count, average only the two middle ordered values.
Common Mistakes to Avoid
- -Reporting the arithmetic mean instead of the median
- -Failing to order the values before picking the middle
- -Picking one of the two middle values instead of averaging them
Concept Deep Dive
Analysis
This question tests the median for an even number of observations. Order the four prices, identify the two central items ($420,000 and $435,000), and average them. The result, $427,500, differs from the arithmetic mean of $428,750, which is the main trap.
Background Knowledge
The median is the middle value of an ordered array: the middle item when n is odd, and the average of the two central items when n is even. It is less sensitive to extreme values than the mean. As a descriptive statistic, it does not by itself determine a final value opinion.
Real-World Application
An appraiser summarizing four recent sales in a small subdivision reports the median price as part of the market analysis. Ordering the sales and averaging the two middle prices gives a central figure that is not pulled by the highest sale. The appraiser still analyzes each comparable rather than adopting the median as the value opinion.
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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