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.
Correct Answer
B) 0.980
Ordered values: 0.92, 0.95, 0.97, 0.99, 1.02, 1.05. The average of the two middle observations is 0.98. Common errors: using the mean instead of the median gives 0.983; taking only the lower middle value gives 0.970; averaging the lowest and highest values (the midrange) gives 0.985.
Why This Is the Correct Answer
Option B is correct because the six ordered ratios are 0.92, 0.95, 0.97, 0.99, 1.02 and 1.05. The two middle values are 0.97 and 0.99. Their average, (0.97 + 0.99) / 2, is 0.980.
Why the Other Options Are Wrong
Option A: 0.970
Option A, 0.970, takes only the lower middle value. With an even count the median averages both middle values: (0.97 + 0.99) / 2 = 0.980.
Option C: 0.983
Option C, 0.983, is the arithmetic mean, 5.90 / 6, not the median. The median averages the two middle ordered values: (0.97 + 0.99) / 2 = 0.980.
Option D: 0.985
Option D, 0.985, is the midrange, (0.92 + 1.05) / 2, which averages the lowest and highest ratios. The median averages the two middle ordered values: (0.97 + 0.99) / 2 = 0.980.
Even count, split the pair
With an even number of values, the middle is shared by two values, so you split the difference between them.
How to use: Use this whenever you must find a median and the count of observations is even.
Exam Tip
Before computing a median, count the observations and order them; an even count means averaging two values.
Common Mistakes to Avoid
- -Using only one of the two middle values
- -Computing the mean or midrange when the median is asked
- -Forgetting to order the values first
Concept Deep Dive
Analysis
This question tests the median for an even number of observations. With six ordered ratios, there is no single middle value, so you average the third and fourth values. The distractors use one middle value only, compute the mean, or average the two extreme values instead.
Background Knowledge
The median is the middle item in an odd-numbered ordered set and the average of the two central items in an even-numbered set. In ratio studies it is a primary measure of appraisal level because it is less affected by extreme ratios.
Real-World Application
An appraiser reviewing six assessment-to-sale ratios for a condominium project orders them and averages the two middle ratios to find a median of 0.980. That figure summarizes the assessment level for the sample without being distorted by the highest or lowest ratio.
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If an already-high extreme observation increases further without changing the central ordered observations, why does the median change less than the mean?
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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.