Sale prices are $200,000, $220,000, $220,000, $220,000, $230,000, $230,000, $250,000, $270,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
Correct Answer
B) $20,000
Mean=$230,000; Σ(x−mean)²=3200000000. Standard deviation=√[3200000000/8]=20000, rounded to $20,000. Variance and the data range are different quantities.
Why This Is the Correct Answer
Option B is correct because the prices total $1,840,000, so the mean is $230,000. The squared deviations sum to 3,200,000,000, and dividing by n = 8 gives 400,000,000. The square root is $20,000.
Why the Other Options Are Wrong
Option A: $21,381
Option A, $21,381, divides the squared deviations by n-1 (3,200,000,000 / 7, about 457,142,857, square root about $21,381), which is the sample formula. Treating the data as a population means dividing by 8, giving $20,000.
Option C: $70,000
Option C, $70,000, is the range ($270,000 - $200,000), not the standard deviation. The population standard deviation is the square root of 3,200,000,000 / 8, which is $20,000.
Option D: $230,000
Option D, $230,000, is the mean sale price, a measure of center rather than spread. The population standard deviation is $20,000.
Whole group, divide by n
If the data are the whole group, divide the squared deviations by n before taking the square root.
How to use: Use this when a standard deviation question tells you to treat the data as the complete population.
Exam Tip
Expect both the population and sample standard deviations to appear as choices; follow the stated divisor.
Common Mistakes to Avoid
- -Using n-1 when told to treat data as a population
- -Confusing the range with the standard deviation
- -Skipping repeated values when summing squared deviations
Concept Deep Dive
Analysis
This question tests the population standard deviation with repeated values. You find the mean, square each deviation, sum them, divide by n = 8 and take the square root. Distractors reflect the sample formula, the range and the mean.
Background Knowledge
The population standard deviation divides the sum of squared deviations by the number of observations, while the sample version divides by n-1. Both measure typical distance from the mean, unlike the range, which uses only the extremes.
Real-World Application
An appraiser analyzing all eight resales in a townhouse complex computes a population standard deviation of $20,000 around a $230,000 mean. The figure helps the appraiser describe how consistent pricing is within the complex.
More Statistics Questions
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