Sale-price observations are $250,000, $260,000, $255,000, $270,000, $255,000. Compute the median; order the observations first and average the two middle values if n is even. Report to the nearest dollar.
Correct Answer
B) $255,000
Ordered values: $250,000, $255,000, $255,000, $260,000, $270,000. The middle observation is 255000. Common errors: computing the mean instead of the median gives $258,000; reporting the lowest observation gives $250,000; reporting the highest observation gives $270,000.
Why This Is the Correct Answer
Option B is correct because ordering the prices gives $250,000, $255,000, $255,000, $260,000, $270,000. With five observations, the middle (third) value is $255,000. Half the data lie at or below this point and half at or above it.
Why the Other Options Are Wrong
Option A: $258,000
Option A, $258,000, is the mean rather than the median: ($250,000 + $260,000 + $255,000 + $270,000 + $255,000) ÷ 5 = $1,290,000 ÷ 5 = $258,000. The median is the middle ordered value, $255,000.
Option C: $250,000
Option C, $250,000, simply reports the lowest observation. The median is the third of the five ordered values, $255,000.
Option D: $270,000
Option D, $270,000, simply reports the highest observation. The median is the third of the five ordered values, $255,000.
Median is the highway median
Like the median strip down the middle of a highway, the statistical median sits in the middle of the lined-up values.
How to use: Use this picture to remember to line values up in order before picking the middle one.
Exam Tip
Sort the values before locating the median, and count positions carefully when values repeat. If an option equals the arithmetic average, it is likely the mean distractor.
Common Mistakes to Avoid
- -Computing the mean instead of the median
- -Picking the middle value of the unsorted list
- -Reporting the lowest or highest observation as the center
Concept Deep Dive
Analysis
This question tests the basic procedure for finding a median. The observations must first be put in order, and with an odd count of five the median is the third ordered value. The repeated $255,000 value matters because it occupies both the second and third positions.
Background Knowledge
The median is a measure of central tendency that splits an ordered data set in half. It is less affected by extreme values than the mean, which makes it useful for skewed price data. For an odd number of observations it is the single middle value.
Real-World Application
An appraiser summarizing five recent sales in a subdivision reports both the mean and the median. The median of $255,000 gives a typical price that is not pulled upward by the single $270,000 sale.
More Statistics Questions
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For an arithmetic schedule only, allocate historical cost $2,500,000 over 40 years with zero residual. After 8 years, what unindexed scheduled amount remains? This is not an estimate of current market value.
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Sale-price observations are $185,000, $192,000, $188,000, $195,000, $190,000. Compute the median; order the observations first and average the two middle values if n is even. Report to the nearest dollar.