A distribution in which the mean exceeds the median is described as:
Correct Answer
C) Positively skewed, with a tail toward higher values
Why this is correct: The correct answer is "Positively skewed, with a tail toward higher values." In a positively skewed (right-skewed) distribution, a long tail extends toward higher values. The mean is sensitive to these extreme high values and is pulled in the direction of the tail, making it greater than the median. This is common in housing price data, where a few very high-priced homes raise the average price above the more typical median price. Why the other choices are wrong: "Perfectly symmetrical around its center" is wrong because in a symmetrical distribution, the mean and median are equal. "Negatively skewed, with a tail toward lower values" is wrong because in a negatively skewed (left-skewed) distribution, the mean is less than the median, pulled down by low outliers. "Bimodal with two distinct peaks" is wrong because bimodality describes the shape's peaks, not the relationship between the mean and median; a bimodal distribution can have its mean either above or below its median. Exam tip: Remember the phrase "mean follows the tail." If the mean > median, the tail (and skew) is positive/right.
Why This Is the Correct Answer
Option C is correct because a mean above the median is the diagnostic signature of positive, or right, skew, and positive skew means the tail extends toward higher values. The wording matches the mechanism: the extreme high observations in the tail are what drag the mean above the positional midpoint. The option also correctly pairs the direction of the skew with the direction of the tail, which is where candidates most often get turned around. No other choice describes a distribution in which the two measures would differ in this direction.
Why the Other Options Are Wrong
Option A: Perfectly symmetrical around its center
In a perfectly symmetrical distribution the mean, median, and mode coincide, so this option describes the one case in which the stated inequality cannot occur. Symmetry means the observations balance identically on both sides of center, leaving nothing to pull the mean off the median. The stem's premise rules this out immediately.
Option B: Negatively skewed, with a tail toward lower values
Negative skew produces the opposite inequality: a tail toward low values drags the mean below the median. The naming convention is the trap, since 'negative' feels like it should describe smaller-than-typical outcomes on the high side to some candidates. Fix the convention by remembering that the sign of the skew names the direction of the tail, and the mean follows the tail.
Option D: Bimodal with two distinct peaks
Bimodality describes a distribution with two peaks, which is a statement about modes rather than about the relationship between the mean and the median. A bimodal distribution can be perfectly symmetric with mean equal to median, or skewed in either direction, so the label does not determine the inequality. Bimodal data in appraisal usually signals that two different submarkets or property types have been pooled and should be analyzed separately.
Mean Follows the Tail
The mean follows the tail like a dog on a leash. A long tail to the right tugs the mean above the median, which is positive skew. A long tail to the left tugs it below, which is negative skew. The median never moves, because it only counts positions, not sizes.
How to use: Whenever a question gives you either the tail direction or the mean-median inequality, use the rule to derive the other. Mean above median means right tail and positive skew; mean below median means left tail and negative skew. Then eliminate any option describing symmetry, since symmetry forces the two measures to be equal.
Exam Tip
Sketch a quick lopsided hump on your scratch paper with the tail on the correct side and mark where the mean and median fall; the picture prevents the sign-convention error that costs most of the points on this topic.
Common Mistakes to Avoid
- -Reversing the skew convention by naming the skew for the side where the bulk of data sits rather than the tail
- -Using an unadjusted mean in a market with a few extreme sales instead of the median or a segmented analysis
- -Treating a bimodal shape as a skew problem rather than as evidence that two submarkets were combined
Concept Deep Dive
Analysis
This question tests how the relationship between measures of central tendency reveals the shape of a distribution. The median is a positional statistic, the value at the fiftieth percentile, and it is unaffected by how extreme the values above and below it happen to be. The mean is a balance point computed from every observation, so it moves toward whichever side carries unusually distant values. When a distribution has a long right tail, a handful of very large observations pull the mean upward while the median stays put, producing mean greater than median. When the tail runs left, the reverse happens. Residential sale price data is the standard example of right skew, because prices are bounded below by zero but have no practical upper bound, which is exactly why market reports lead with median price rather than average price.
Background Knowledge
You need to know the definitions of mean, median, and mode and which of them are sensitive to outliers. You should also know that in a unimodal skewed distribution the mode sits at the peak, the median falls between the mode and the mean, and the mean sits nearest the tail, and that skew is named for the direction the tail points.
Real-World Application
Reviewing sixty sales in a neighborhood, an appraiser finds a mean of $612,000 and a median of $548,000, traces the gap to four estate homes on acreage, and concludes those sales belong to a different market segment. The report relies on the median and on segment-matched comparables rather than on the pooled average.
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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