While reviewing a data vendor's report on five closed sales in a neighborhood, an appraiser finds four prices: $380,000, $400,000, $450,000 and $550,000. The fifth price is missing, but the report states a median of $400,000 for all five sales. What can the appraiser conclude about the missing price?
Correct Answer
A) It closed at $400,000 or less
With five sales, the median is the third ordered price, so $400,000 must have two prices at or below it. The known prices supply $380,000 below it and $450,000 and $550,000 above it, so the missing sale must be the second low price, at $400,000 or less. A price of $300,000 or of exactly $400,000 would both work, which rules out the narrower options. A price above $400,000 would move the median above $400,000.
Why This Is the Correct Answer
Option A is correct because $400,000 stays the third ordered value whenever the missing price is $400,000 or lower. A price above $400,000 pushes the third value up to that price or to $450,000. The size of the missing price below $400,000 does not change the middle position.
Why the Other Options Are Wrong
Option B: It closed at exactly $400,000
Option B says the sale closed at exactly $400,000. That price would fit, but $390,000 or $300,000 would also leave $400,000 as the third ordered value, so the median does not pin down one figure.
Option C: It closed between $380,000 and $400,000
Option C limits the sale to between $380,000 and $400,000. A price below $380,000, such as $300,000, also keeps $400,000 in the third position, so the lower limit has no support.
Option D: It closed at $400,000 or more
Option D says the sale closed at $400,000 or more. A price such as $420,000 would make the ordered list $380,000, $400,000, $420,000, $450,000, $550,000, and the median would be $420,000.
Median knows position, not distance
A median tells you how many values lie on each side, not how far away they are. Use it to reason about which side a value falls on.
How to use: Use this when a question gives a median and asks what it implies about an unknown observation.
Exam Tip
Test a value on each side of the median. Trying one low price and one high price quickly shows which answer holds.
Common Mistakes to Avoid
- -Assuming the missing value must equal the reported median
- -Treating the lowest known price as a floor for the missing value
- -Averaging known values instead of reasoning from position
Concept Deep Dive
Analysis
This tests what the median means rather than how to find it. The median of five values is the third in order, so exactly two positions sit below it. Trying the missing price on each side of $400,000 shows which prices keep $400,000 in the middle.
Background Knowledge
The median is the middle item of an odd number of values arranged by size. It depends on position, so it does not reveal how far other observations lie from the center, which is why it resists extreme values.
Real-World Application
An appraiser checking a data vendor's neighborhood summary finds that one of five listed sales has no recorded price. The reported median does not reveal the price, but it does show on which side of the median the sale falls. The appraiser still confirms the actual price before using the sale.
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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