Three sales at $500,000, $520,000 and $510,000 receive reliability weights of 0.5, 0.3 and 0.2, respectively. What is the weighted mean price, to the nearest dollar?
Correct Answer
D) $508,000
500000×0.5 + 520000×0.3 + 510000×0.2=508000; total weight=1. Weighted mean=508000/1=508000, rounded to $508,000. These weights are assigned for the calculation and are not universally appropriate appraisal weights.
Why This Is the Correct Answer
Option D is correct because $500,000 × 0.5 + $520,000 × 0.3 + $510,000 × 0.2 = $250,000 + $156,000 + $102,000 = $508,000. The weights total 1, so the weighted mean is $508,000 / 1 = $508,000.
Why the Other Options Are Wrong
Option A: $510,000
Option A, $510,000, is the simple unweighted mean: ($500,000 + $520,000 + $510,000) / 3 = $510,000. Applying the assigned weights gives $508,000.
Option B: $520,000
Option B, $520,000, is just the highest price, which carries the middle weight of 0.3. The weighted mean is $250,000 + $156,000 + $102,000 = $508,000.
Option C: $507,000
Option C, $507,000, sorts the prices low to high and then applies the weights in listed order, pairing 0.3 with $510,000 and 0.2 with $520,000: $250,000 + $153,000 + $104,000 = $507,000. Each weight belongs to its own price, so the weighted mean is $250,000 + $156,000 + $102,000 = $508,000.
Weighted mean stays inside
Multiply, add, divide by total weight. With positive weights, the answer must land between the lowest and highest values.
How to use: Use this range check to eliminate impossible weighted mean answers quickly.
Exam Tip
Keep each weight paired with its own value, and remember that a weighted mean with positive weights falls within the range of the values.
Common Mistakes to Avoid
- -Using the simple average instead of the weighted one
- -Re-sorting the values and pairing them with the wrong weights
- -Forgetting to divide by the total weight when weights do not sum to 1
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 sum is divided by the total of the weights. Because the weights here total 1, the weighted sum is itself the answer.
Background Knowledge
A weighted mean adjusts each value by a factor reflecting its relative importance. It always falls within the range of the values when the weights are positive. In appraisal, weights are a matter of judgment, and weights given in a question are not automatically appropriate for an actual assignment.
Real-World Application
An appraiser practicing reconciliation weights three indications by her judgment of their reliability. The weighted result of $508,000 sits slightly below the simple average because the lowest indication received the most weight.
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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