Assessment ratios for three parcels are 0.90, 1.00, and 1.10 (sample mean = 1.00). Using the sample formulas (divide squared deviations by n−1), what are the sample variance and the sample standard deviation?
Correct Answer
B) Variance = 0.01; standard deviation = 0.10
Squared deviations sum to 0.02. Sample variance=0.02/(3−1)=0.01 in squared ratio units; sample SD=sqrt(0.01)=0.10 in ratio units. SD is the square root of variance, so interchanging them is incorrect. The observations are not identical and thus do not have zero variance.
Why This Is the Correct Answer
Option B is correct because the deviations from the mean of 1.00 are −0.10, 0 and +0.10, whose squares sum to 0.02. The sample variance is 0.02 ÷ (3 − 1) = 0.01, and the sample standard deviation is √0.01 = 0.10.
Why the Other Options Are Wrong
Option A: Variance = 0.10; standard deviation = 0.01
Option A, variance 0.10 and standard deviation 0.01, swaps the two measures. The variance is 0.02 ÷ 2 = 0.01, and the standard deviation is its square root, 0.10.
Option C: Variance = 0.10; standard deviation = 0.10
Option C, variance 0.10 and standard deviation 0.10, cannot be right because the standard deviation is the square root of the variance, and √0.10 is about 0.32, not 0.10. The correct values are variance 0.01 and standard deviation 0.10.
Option D: Variance = 0; standard deviation = 1.00
Option D, variance 0 and standard deviation 1.00, does not follow from the facts. Zero variance would require identical ratios, and the ratios 0.90, 1.00 and 1.10 differ; the correct values are variance 0.01 and standard deviation 0.10.
Variance squared, SD rooted
Variance lives in squared units; the standard deviation is its square root in the original units. For values below 1, the square root is larger than the variance.
How to use: Use this to check which number is the variance and which is the standard deviation.
Exam Tip
Verify that the standard deviation squared equals the variance; any pair that fails this check is wrong.
Common Mistakes to Avoid
- -Swapping variance and standard deviation
- -Dividing by n instead of n−1 for a sample
- -Assuming small values of variance mean zero variance
Concept Deep Dive
Analysis
This question tests sample variance and standard deviation and the relationship between them. Square the deviations from the mean, sum them, divide by n−1 for the variance, and take the square root for the standard deviation.
Background Knowledge
Sample variance is the sum of squared deviations from the mean divided by n−1, expressed in squared units. The standard deviation is the square root of the variance and is in the original units. Only data with identical values have zero variance.
Real-World Application
An assessor computes the spread of three assessment ratios in a small neighborhood. Reporting a standard deviation of 0.10 in ratio units is easier for reviewers to interpret than the variance of 0.01.
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