An appraiser gives reliability weights of 2, 2 and 1 to sale prices of $580,000, $560,000 and $540,000, respectively. What is the weighted mean price, to the nearest dollar?
Correct Answer
A) $564,000
580000×2 + 560000×2 + 540000×1=2820000; total weight=5. Weighted mean=2820000/5=564000, rounded to $564,000. These weights are assigned for the calculation and are not universally appropriate appraisal weights.
Why This Is the Correct Answer
Option A is correct because $580,000 × 2 + $560,000 × 2 + $540,000 × 1 = $2,820,000, and the weights total 2 + 2 + 1 = 5. Dividing gives $2,820,000 / 5 = $564,000. The higher-weighted prices pull the result above the simple average of $560,000.
Why the Other Options Are Wrong
Option B: $560,000
Option B, $560,000, is the unweighted mean: ($580,000 + $560,000 + $540,000) / 3 = $560,000. It ignores the assigned weights; the weighted mean is $2,820,000 / 5 = $564,000.
Option C: $2,820,000
Option C, $2,820,000, is only the weighted sum of price × weight. It still must be divided by the total weight of 5 to give $564,000.
Option D: $940,000
Option D, $940,000, divides the weighted sum by the number of prices instead of the total weight: $2,820,000 / 3 = $940,000. That result is above the highest price, which a weighted mean with positive weights cannot be; the correct result is $2,820,000 / 5 = $564,000.
Multiply, add, divide by weights
For a weighted mean, multiply each value by its weight, add the products, and divide by the total of the weights, not by the count.
How to use: Use this three-step sequence whenever a question gives weights for a set of values.
Exam Tip
Check that your answer lies between the lowest and highest values; a result outside that range signals an error.
Common Mistakes to Avoid
- -Dividing by the number of values instead of the sum of the weights
- -Reporting the weighted sum without dividing
- -Ignoring the weights and taking a simple average
Concept Deep Dive
Analysis
This question tests computing a weighted mean with assigned weights. Each price is multiplied by its weight, the products are summed, and the total is divided by the sum of the weights, not the number of prices. Here ($580,000 × 2 + $560,000 × 2 + $540,000 × 1) / 5 = $2,820,000 / 5 = $564,000.
Background Knowledge
A weighted average adjusts each value by a factor reflecting its relative importance. The weighted mean falls between the lowest and highest values when the weights are positive. Here the weights are given; whether such weights are appropriate in a real assignment is a separate judgment.
Real-World Application
An appraiser reconciling three adjusted comparable prices assigns more weight to the two most similar sales. Using weights of 2, 2 and 1 gives a weighted indication of $564,000. The appraiser explains the reasoning for the weights in the report.
More Statistics Questions
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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.
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