A distribution of sale prices has a long tail toward the high end. What follows for its mean and median?
Correct Answer
C) The mean will lie above the median value
Why this is correct: In a right-skewed distribution (long tail toward high values), the mean is pulled upward by the extreme high values, while the median remains at the 50th percentile. Therefore, the mean will be greater than the median. Why the other choices are wrong: 'The mean will lie below the median value' describes a left-skewed distribution. 'The two measures will be identical in value' occurs only in perfectly symmetrical distributions. 'The median cannot be computed for such data' is false; the median can always be computed for ordered data. Exam tip: For skew, remember 'mean follows the tail'—right tail pulls mean up, left tail pulls mean down.
Why This Is the Correct Answer
Option C follows directly from the mechanism: the high tail pulls the mean up while the median holds at the fiftieth percentile, so the mean lies above the median. The reasoning does not require computation, only an understanding of which statistic is sensitive to magnitude. This is the same relationship as the companion question stated from the other side, which is a useful check that you have the convention right. Any answer placing the mean below or equal to the median would require a different tail or a symmetric shape.
Why the Other Options Are Wrong
Option A: The mean will lie below the median value
A mean below the median is the signature of a left tail, where a small number of unusually low sales drag the average down. Nothing in the stem describes low outliers; it describes a long tail at the high end. Choosing this option means the tail direction was read correctly but the direction of the pull was reversed.
Option B: The two measures will be identical in value
Mean and median coincide only when the distribution is symmetric about its center, which a long one-sided tail rules out by definition. The presence of any meaningful skew guarantees the two measures separate. This choice would be correct only if the stem had described a balanced distribution.
Option D: The median cannot be computed for such data
A median can be computed for any set of ordered values, regardless of shape, and it is in fact the preferred summary precisely when the data are skewed. You sort the observations and take the middle value, or the average of the two middle values when the count is even. Skewness affects interpretation, never computability.
The Heavy End Wins
Picture the distribution as a plank balanced on a fulcrum, with each sale a weight placed at its price. The fulcrum is the mean, and it must slide toward the heavy end where the far-out sales sit. The median is just the middle person in a line, unmoved by how far away the last person stands.
How to use: Translate the stem into a picture before reading the options: tail on the right means the mean slides right, above the median. Then confirm which measure the question is asking you to place. If any option claims a statistic cannot be computed, discard it, because both mean and median exist for any numeric sample.
Exam Tip
Report the median when summarizing skewed market data and say why; describing an average price in a market with a few outsized sales invites a misleading conclusion.
Common Mistakes to Avoid
- -Quoting an average price as evidence of appreciation when a few high sales explain the change
- -Assuming a skewed sample is unusable rather than summarizing it with the median
- -Deleting high sales as outliers without first checking whether they are legitimate arm's-length transactions in the subject's segment
Concept Deep Dive
Analysis
This question runs the same skew relationship in the opposite direction, giving you the tail and asking for the consequence. A long tail toward the high end means most sales cluster in a lower range while a smaller number of transactions extend far above that cluster, which is the ordinary shape of residential price data in a market containing a few luxury or acreage properties. Because the mean incorporates the magnitude of every observation, each of those high sales contributes disproportionately and pulls the average upward. The median only asks how many observations fall on each side, so a sale at $4 million counts exactly as much as a sale at $600,000 in locating the midpoint. The result is mean above median, and the gap between the two is itself a rough gauge of how skewed the data are.
Background Knowledge
You need to know how the median is located in an ordered data set and why it is resistant to outliers, and how the mean is computed and why it is not. You should also know that appraisers prefer the median for summarizing skewed price data and that a large gap between the two measures is a signal to examine the sample for mixed property types or extreme sales.
Real-World Application
Preparing a market conditions analysis, an appraiser notes that average sale price rose eight percent year over year while median price rose two percent, and traces the difference to three new luxury sales rather than to broad appreciation. The time adjustment applied to comparables is based on the median trend and a matched-pair resale analysis.
More Statistics Questions
A set of comparable sales has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation?
A property sold for $400,000 and resold three years later for $463,050 with no physical change. What compound annual rate does this indicate?
A histogram of neighborhood sale prices shows two distinct peaks. What does this most likely mean?
What does it mean to validate a regression model?
In a market study, what does a frequency distribution of sale prices show?
An appraiser includes months elapsed since each sale as a variable in a price model. What is this intended to capture?
An appraiser presents a statistical analysis in a report. What must accompany it for the reader to weigh it?
An R-squared of 0.86 in a sales model indicates that:
Which measure would best summarize the most common lot size in a subdivision?
Paired sales analysis and regression differ mainly in that regression:
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