Three sales have assessed values of $180,000, $285,000 and $550,000 and sale prices of $200,000, $300,000 and $500,000. What is the weighted mean ratio?
Correct Answer
A) 1.015
The weighted mean is total assessed value divided by total sale price: $1,015,000 / $1,000,000 = 1.015. The mean of the individual ratios (0.90, 0.95, 1.10) is 0.983, the median is 0.950, and 1,000,000 / 1,015,000 = 0.985 inverts the ratio. Because the highest-value sale has the highest ratio, dollar weighting pulls the weighted mean above the mean.
Why This Is the Correct Answer
Option A is correct because the weighted mean ratio is the total assessed value divided by the total sale price: $1,015,000 / $1,000,000 = 1.015. This is the same as the average assessed value divided by the average sale price. Because the highest-priced sale has the highest ratio (1.10), dollar weighting pulls the result above the simple mean.
Why the Other Options Are Wrong
Option B: 0.983
Option B, 0.983, is the simple mean of the individual ratios: (0.90 + 0.95 + 1.10) / 3 = 0.983. That gives each sale equal weight, whereas the weighted mean is $1,015,000 / $1,000,000 = 1.015.
Option C: 0.950
Option C, 0.950, is the median of the individual ratios 0.90, 0.95 and 1.10. The question asks for the weighted mean, which is $1,015,000 / $1,000,000 = 1.015.
Option D: 0.985
Option D, 0.985, inverts the ratio by dividing total sale price by total assessed value: $1,000,000 / $1,015,000 = 0.985. The assessment ratio places assessed value in the numerator, giving 1.015.
Weighted mean: sum, then divide
For the weighted mean, add up all assessed values and all sale prices first, then divide once. For the simple mean, divide each pair first, then average.
How to use: Use this to choose the order of operations when a ratio question asks for a weighted mean versus a mean.
Exam Tip
Compute the individual ratios as a check: the mean, median and inverted ratio often appear as distractors next to the weighted mean.
Common Mistakes to Avoid
- -Averaging the individual ratios when the question asks for the weighted mean
- -Dividing sale price by assessed value and inverting the ratio
- -Reporting the median ratio instead of the weighted mean
Concept Deep Dive
Analysis
This question tests the difference between the weighted mean ratio and other ratio-study averages. The weighted mean divides total assessed value by total sale price, so higher-priced sales carry more weight. Summing $180,000 + $285,000 + $550,000 = $1,015,000 and $200,000 + $300,000 + $500,000 = $1,000,000 gives 1.015.
Background Knowledge
Ratio studies use several measures of central tendency: the median, the mean and the weighted mean. The weighted mean is total assessed value divided by total sale price, so it weights each ratio by its sale price. Comparing the mean and weighted mean helps reveal whether higher- or lower-value properties are assessed differently.
Real-World Application
An assessor reviewing three sales finds individual ratios of 0.90, 0.95 and 1.10. The weighted mean of 1.015 sits above the simple mean of 0.983 because the $500,000 sale is assessed relatively high. This difference prompts a closer look at how higher-value properties are assessed.
More Statistics Questions
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