Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. 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=$300,000; Σ(x−mean)²=2000000000. Standard deviation=√[2000000000/5]=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 mean is $300,000 and the deviations are -30,000, -10,000, 0, 10,000 and 30,000. Their squares sum to 2,000,000,000. Dividing by n = 5 gives 400,000,000, and its square root is $20,000.
Why the Other Options Are Wrong
Option A: $22,361
Option A, $22,361, divides the squared deviations by n-1 (2,000,000,000 / 4 = 500,000,000, square root about $22,361), which is the sample formula. The question treats the data as the full population, so divide by 5: the square root of 400,000,000 is $20,000.
Option C: $60,000
Option C, $60,000, is the range ($330,000 - $270,000), not the standard deviation. The population standard deviation is the square root of 2,000,000,000 / 5, which is $20,000.
Option D: $300,000
Option D, $300,000, is the mean of the sale prices, a measure of center rather than spread. The population standard deviation is $20,000.
Population n, sample n minus one
When you have everyone (the population), divide by n; when you have only a sample, divide by n minus one.
How to use: Use this to choose the correct divisor when a standard deviation question specifies population or sample.
Exam Tip
Read the divisor instruction carefully; population and sample answers are often both listed.
Common Mistakes to Avoid
- -Dividing by n-1 when the data are treated as a population
- -Reporting the range instead of the standard deviation
- -Forgetting to take the square root of the variance
Concept Deep Dive
Analysis
This question tests the population standard deviation, which divides the sum of squared deviations by n. You find the mean, square each deviation, sum them, divide by 5 and take the square root. The distractors reflect the sample formula, the range and the mean.
Background Knowledge
The standard deviation measures how far observations typically lie from the mean. Dividing squared deviations by n gives the population version, while dividing by n-1 gives the sample version used to estimate a larger population.
Real-World Application
An appraiser analyzing all five sales in a small new subdivision computes a population standard deviation of $20,000. The figure helps describe how tightly prices cluster around the $300,000 mean when explaining the range of value indications.
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Previous Question
Assessment-to-sale ratios are 0.92, 0.95, 0.97, 0.99, 1.02, 1.05. Compute the median; order the observations first and average the two middle values if n is even. Report as a decimal ratio to three places.
Next Question
The assessment-to-sale ratios are 0.9, 0.94, 0.98, 1.02, 1.06, 1.1. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.