Gross rent multipliers are 10, 12, 14. Compute the sample variance Σ(x−mean)²/(n−1); report to two decimals in squared multiplier units.
Correct Answer
B) 4.00
Mean=12; sum of squared deviations=8; divide by 2 to obtain 4. Variance is in squared units; standard deviation would be its square root.
Why This Is the Correct Answer
Option B is correct because the mean of 10, 12 and 14 is 12, the squared deviations are 4, 0 and 4 for a sum of 8, and 8 / (3 − 1) = 4.00. Variance is expressed in squared units.
Why the Other Options Are Wrong
Option A: 2.00
Option A, 2.00, is the sample standard deviation, the square root of 4. The question asks for the variance, which is 8 / 2 = 4.00.
Option C: 2.67
Option C, 2.67, comes from dividing by n instead of n − 1: 8 / 3 = 2.67, the population form. The stated formula divides by n − 1, giving 8 / 2 = 4.00.
Option D: 12.00
Option D, 12.00, is the mean of the multipliers, not a measure of spread. The sample variance is 8 / 2 = 4.00.
Square, sum, n minus one
Sample variance: square each deviation, add them up, divide by n minus 1. Stop before taking the square root.
How to use: Use this to avoid confusing sample variance with standard deviation or population variance.
Exam Tip
Check whether the question wants variance or standard deviation and which divisor it specifies.
Common Mistakes to Avoid
- -Dividing by n instead of n − 1 for a sample
- -Reporting the standard deviation as the variance
- -Forgetting to square the deviations
Concept Deep Dive
Analysis
This question tests computing sample variance with the n − 1 divisor. The steps are to find the mean, sum the squared deviations and divide by n − 1. The distractors use the population divisor, take the square root or report the mean.
Background Knowledge
Variance measures spread as the average squared deviation from the mean, with the sample form dividing by n − 1. Because it is in squared units, the standard deviation, its square root, is often easier to interpret. Both are sensitive to extreme values.
Real-World Application
An appraiser comparing three gross rent multipliers from similar rentals computes their variance and standard deviation to show how consistent the multipliers are before selecting one for the income approach.
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