Eight sales closed at $200,000, $220,000, $220,000, $220,000, $230,000, $230,000, $250,000 and $270,000. The mean is $230,000. What is the population standard deviation?
Correct Answer
B) $20,000
Why this is correct: The original explanation provides the calculation: squared deviations total 3,200 (in thousands of dollars). Variance = 3,200 / 8 = 400. Standard deviation = √400 = 20, so $20,000. The governing concept is that population standard deviation measures dispersion of all data points around the mean. Step-by-step: 1) Find deviations from mean ($230k): (-30, -10, -10, -10, 0, 0, 20, 40) in $1,000s. 2) Square each: (900, 100, 100, 100, 0, 0, 400, 1600). 3) Sum: 3200. 4) Divide by N (8): 400. 5) Square root: 20 ($20,000). Why the other choices are wrong: $16,000 might result from an incorrect divisor or mis-calculation. $24,000 could be the result of using sample standard deviation formula (dividing by n-1). $40,000 is the range ($270k - $200k = $70k), not the standard deviation. Exam tip: For population standard deviation, divide the sum of squared deviations by N (number of data points), then take the square root.
Why This Is the Correct Answer
The squared deviations total 3,200 in units of thousands squared, and dividing by the population count of 8 gives a variance of 400. The square root of 400 is 20, so the population standard deviation is $20,000. Working in thousands throughout keeps the numbers manageable and avoids the arithmetic errors that come from squaring six-digit figures. A quick reasonableness check supports the result: most of the sales sit within $20,000 of the $230,000 mean, which is what a standard deviation of that size describes.
Why the Other Options Are Wrong
Option A: $16,000
$16,000 does not come out of this data set under either the population or the sample formula. It is in the neighborhood of the mean absolute deviation, which averages the unsigned differences and comes to $15,000 here, and that similarity is what makes it tempting to a candidate who skipped the squaring step. Squaring is what makes standard deviation weight the large outliers more heavily than the small ones, so any method that omits it produces a different and smaller statistic.
Option C: $24,000
$24,000 is offered as though it were the sample standard deviation, but the sample calculation divides 3,200 by 7 rather than 8, giving a variance of about 457 and a standard deviation of roughly $21,400. So this figure is neither the population nor the sample answer for this data. It exists to reward a candidate who correctly senses that the sample figure is larger without checking how much larger.
Option D: $40,000
$40,000 is the largest single deviation in the set, the distance from the $270,000 high sale to the $230,000 mean. A standard deviation is an average-like measure of spread and cannot equal the most extreme deviation unless every point is that far out. Note also that the range of this data is $70,000, from $200,000 to $270,000, so $40,000 is not the range either.
Deviate, Square, Average, Root
Four verbs in order. Deviate from the mean, square each result, average the squares, then root the average. If the stem says population you average over all N; if it says sample you average over one fewer than you have.
How to use: Scale the data into thousands before starting so the squares stay small. Then read the stem for the words population or sample, since that single word chooses your divisor and separates two of the answer choices.
Exam Tip
Before selecting, glance at the data and ask whether your answer looks like a typical distance from the mean. A standard deviation larger than most individual deviations, or as large as the range, is arithmetically impossible.
Common Mistakes to Avoid
- -Using the sample divisor when the stem specifies a population, or the reverse
- -Reporting the variance instead of taking the square root to return to dollars
- -Averaging unsigned deviations without squaring, which produces mean absolute deviation
Concept Deep Dive
Analysis
Standard deviation measures how widely a set of values spreads around its mean, and the calculation is a fixed five-step routine. Take each value's deviation from the mean, square the deviations so that positive and negative differences do not cancel, sum the squares, divide by the count to get variance, then take the square root to return to the original units. With a mean of $230,000, the eight deviations in thousands are minus 30, minus 10, minus 10, minus 10, zero, zero, 20, and 40. Their squares are 900, 100, 100, 100, 0, 0, 400, and 1,600, which sum to 3,200. The stem specifies the population standard deviation, so the divisor is N, which is 8, giving a variance of 400 and a standard deviation of 20, or $20,000. The population versus sample distinction is the deliberate hinge of the question: the sample formula would divide by n minus 1, which is 7 here, and would produce a slightly larger figure.
Background Knowledge
You need the five-step standard deviation routine and the difference between the population divisor N and the sample divisor n minus 1, along with the fact that dividing by the smaller number always yields the larger statistic. You should know that variance is in squared units and that the square root returns the measure to dollars, which is why standard deviation rather than variance is quoted in market analysis. You should also recognize related dispersion measures such as range, mean absolute deviation, and coefficient of variation, since answer sets often mix them in.
Real-World Application
Analyzing a subdivision of similar homes, an appraiser computes a mean sale price and a standard deviation to judge how tightly the market clusters. A small standard deviation supports a narrow reconciled range and a tight adjustment grid, while a wide one signals meaningful differences among the properties that the appraiser must identify and adjust for before relying on the average.
More Statistics Questions
A set of comparable sales has a mean of $250,000 and a standard deviation of $20,000. What is the coefficient of variation?
A property sold for $400,000 and resold three years later for $463,050 with no physical change. What compound annual rate does this indicate?
A histogram of neighborhood sale prices shows two distinct peaks. What does this most likely mean?
What does it mean to validate a regression model?
In a market study, what does a frequency distribution of sale prices show?
An appraiser includes months elapsed since each sale as a variable in a price model. What is this intended to capture?
An appraiser presents a statistical analysis in a report. What must accompany it for the reader to weigh it?
An R-squared of 0.86 in a sales model indicates that:
Which measure would best summarize the most common lot size in a subdivision?
Paired sales analysis and regression differ mainly in that regression:
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