Sale prices are $210,000, $245,000, $280,000, $310,000 and $360,000. Their reliability weights are 1, 3, 2, 3 and 1, respectively. What is the weighted mean price, to the nearest dollar?
Correct Answer
A) $279,500
210000×1 + 245000×3 + 280000×2 + 310000×3 + 360000×1=2795000; total weight=10. Weighted mean=2795000/10=279500, rounded to $279,500. 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 the weighted products are $210,000 x 1 + $245,000 x 3 + $280,000 x 2 + $310,000 x 3 + $360,000 x 1 = $2,795,000. The weights total 10, so the weighted mean is $2,795,000 / 10 = $279,500.
Why the Other Options Are Wrong
Option B: $281,000
Option B, $281,000, is the unweighted mean, $1,405,000 / 5 = $281,000, which ignores the assigned weights. Applying them gives $2,795,000 / 10 = $279,500.
Option C: $280,000
Option C, $280,000, is the median of the five prices, the middle value, rather than a weighted average. The weighted mean is $2,795,000 / 10 = $279,500.
Option D: $2,795,000
Option D, $2,795,000, is the sum of the weighted products without dividing by the total weight. Dividing by 10 gives $279,500.
Multiply, Sum, Divide by Weights
Weight each value, add the products, then divide by the total of the weights, not by the count of values.
How to use: Use this three-step phrase whenever a question gives weights for an average.
Exam Tip
Divide by the sum of the weights; dividing by the number of values or skipping the division produces common wrong answers.
Common Mistakes to Avoid
- -Ignoring the weights and averaging the prices directly
- -Dividing by the number of values instead of the total weight
- -Forgetting to divide the sum of products at all
Concept Deep Dive
Analysis
This question tests the weighted mean: multiply each price by its weight, sum the products and divide by the sum of the weights. The distractors come from ignoring the weights, using the median or forgetting the final division.
Background Knowledge
A weighted mean adjusts each value by a factor reflecting its relative importance in the whole. In appraisal, weighting is used when some data deserve more emphasis, but the weights themselves need a supportable rationale.
Real-World Application
An appraiser gives greater weight to two highly comparable sales and less to three that needed larger adjustments. Computing a weighted mean of the adjusted prices helps summarize the indications, with the weighting rationale explained in the report.
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