Assessment-to-sale ratios are 1.08, 0.86, 0.97, 1.14, 0.91, 1.02, 0.94. 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
A) 0.970
Ordered values: 0.86, 0.91, 0.94, 0.97, 1.02, 1.08, 1.14. The middle observation is 0.97. Common errors: taking the middle value of the unordered list gives 1.140; using the mean instead of the median gives 0.989; using the midrange instead of the median gives 1.000.
Why This Is the Correct Answer
Option A is correct because the ordered ratios are 0.86, 0.91, 0.94, 0.97, 1.02, 1.08, 1.14, and with seven observations the middle one is the fourth, 0.970. The median is the middle item of an odd number of values arranged by size. No averaging is needed because n is odd.
Why the Other Options Are Wrong
Option B: 0.989
Option B, 0.989, is the arithmetic mean, 6.92 ÷ 7 ≈ 0.989, not the median. The median is the fourth ordered value, 0.970.
Option C: 1.000
Option C, 1.000, is the midrange, (0.86 + 1.14) ÷ 2 = 1.000, which averages only the lowest and highest values. The median is the fourth ordered value, 0.970.
Option D: 1.140
Option D, 1.140, is the middle value of the unordered list (the fourth listed value is 1.14). The values must be sorted first; the fourth ordered value is 0.970.
Sort before you split
Always line the values up from smallest to largest before picking the middle one.
How to use: Use this every time a median question gives values in random order.
Exam Tip
If a listed answer equals the middle number of the unsorted list, it is likely a trap; sort first.
Common Mistakes to Avoid
- -Taking the middle value without sorting
- -Reporting the mean as the median
- -Averaging the lowest and highest values and calling it the median
Concept Deep Dive
Analysis
This question tests finding the median of an odd number of ratios. The key step is ordering the seven values first, then taking the fourth, which is the middle observation. The distractors come from skipping the sort or using a different measure of central tendency.
Background Knowledge
The median is the middle value of ordered data and is less affected by extreme values than the mean. In ratio studies it is a primary measure of appraisal level. For an even count, the median is the average of the two central values.
Real-World Application
An assessor's office reviews seven assessment-to-sale ratios in a neighborhood. Sorting them and taking the middle ratio gives the median level, which is less influenced by the single highest ratio than the mean would be.
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