Sale-price observations are $410,000, $280,000, $295,000, $310,000, $355,000, $300,000, $325,000, $290,000, $500,000. Compute the median; order the observations first and average the two middle values if n is even. Report to the nearest dollar.
Correct Answer
C) $310,000
Ordered values: $280,000, $290,000, $295,000, $300,000, $310,000, $325,000, $355,000, $410,000, $500,000. The middle observation is 310000. Common errors: computing the mean instead of the median gives $340,556; reporting the lowest observation gives $280,000; taking the middle value of the unordered list gives $355,000.
Why This Is the Correct Answer
Option C is correct because, ordered, the nine prices are $280,000, $290,000, $295,000, $300,000, $310,000, $325,000, $355,000, $410,000 and $500,000. The fifth, middle value is $310,000.
Why the Other Options Are Wrong
Option A: $340,556
Option A, $340,556, is the mean rather than the median: the nine prices sum to $3,065,000, and $3,065,000 / 9 = $340,556. The median is the middle ordered value, $310,000.
Option B: $280,000
Option B, $280,000, is the lowest observation, not the middle one. The median is the fifth of nine ordered values, $310,000.
Option D: $355,000
Option D, $355,000, is the middle (fifth) value of the unordered list as given. The data must be ordered first, which makes the median $310,000.
Line up, then middle
The median is like the middle car in a parked line: line the values up by size first, then pick the center.
How to use: Use this to remember that sorting comes before selecting the median.
Exam Tip
Always sort the data before picking the middle value; test writers often list values out of order.
Common Mistakes to Avoid
- -Picking the middle value without sorting
- -Computing the mean instead of the median
- -Reporting an extreme value
Concept Deep Dive
Analysis
This question tests finding the median of an odd number of observations. The values must be ordered first, and then the middle one is selected. The distractors come from computing the mean, picking the lowest value or taking the middle of the unordered list.
Background Knowledge
The median is the middle value of an ordered data set, or the average of the two central values when the count is even. It is less affected by extreme observations than the mean, which is why the $500,000 sale pulls the mean above the median here. Median prices are commonly used to describe typical market levels.
Real-World Application
An appraiser reports the neighborhood's median sale price as $310,000 after sorting nine recent sales. She notes the mean is higher because of a $500,000 outlier, so the median better represents typical prices.
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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Median prices suggest 9% annual growth while an analysis of resale pairs suggests 4%. Which is a plausible explanation to investigate?
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The assessment-to-sale ratios are 0.9, 0.9, 0.95, 1, 1, 1.05, 1.05. 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.