If an already-high extreme observation increases further without changing the central ordered observations, why does the median change less than the mean?
Correct Answer
A) It is determined by the central ordered observations
A median depends on the central ordered observation or two central observations. Increasing an already-high extreme value leaves those central observations unchanged while changing the mean. Thus it is less sensitive to extreme magnitudes. This does not make it universally preferable, always higher than the mean or dependent on a larger sample size.
Why This Is the Correct Answer
Option A is correct because the median is determined by the central ordered observation, or the average of the two central observations. Increasing an already-high extreme value leaves those central observations unchanged, so the median stays the same. The mean, by contrast, changes because the larger value enters its sum.
Why the Other Options Are Wrong
Option B: It incorporates each observed magnitude into its sum
Option B, incorporating each observed magnitude into its sum, describes the arithmetic mean. That is exactly why the mean is more sensitive to the extreme value, so it explains the opposite behavior.
Option C: It exceeds the arithmetic mean in right-skewed data
Option C, exceeding the arithmetic mean in right-skewed data, is not a reason for stability, and in right-skewed data the mean is typically pulled above the median rather than below it.
Option D: It requires a larger sample size than the mean
Option D, requiring a larger sample size than the mean, is not true; the median can be computed from any sample size, and sample size does not explain its resistance to extreme values.
Middle seat, not total weight
The median is about who sits in the middle seat, while the mean is about the total weight of everyone.
How to use: Use this contrast to explain why changing an end value moves the mean but not the median.
Exam Tip
When an option describes summing every value, it is describing the mean, not the median.
Common Mistakes to Avoid
- -Attributing the mean's summing property to the median
- -Believing the median is always higher than the mean
- -Thinking the median needs a larger sample than the mean
Concept Deep Dive
Analysis
This question tests why the median resists extreme values. The median is fixed by position in the ordered data, while the mean adds every value. If the highest value grows but the middle values stay put, the median does not move while the mean does.
Background Knowledge
The median divides ordered data into two equal halves and is less affected by extreme values than other measures of central tendency. This resistance does not make it universally the best summary; the choice of measure depends on the purpose of the analysis.
Real-World Application
In a neighborhood where one estate sells for far more than any other home, an appraiser notes that the mean sale price rises sharply while the median stays near typical values. The appraiser reports the median as a better picture of a typical sale in that data set.
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.
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Land or Site Valuation
4.5% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam
Previous Question
Four parcels have assessment-to-sale ratios of 1.15 on $90,000; 1.05 on $150,000; 0.98 on $250,000; and 0.92 on $400,000. Define PRD as the unweighted mean ratio divided by the sale-price-weighted mean ratio. Which rounded PRD and descriptive sample tendency are consistent with these data?
Next Question
Assessment-to-sale ratios are 0.92, 0.95, 0.97, 0.99, 1.02, 1.05. Compute the median; order the observations first and average the two middle values if n is even. Report as a decimal ratio to three places.