Assessment ratios for three parcels are 0.85, 1.00, and 1.15 (sample mean = 1.00). What are the population standard deviation (divide squared deviations by n) and the sample standard deviation (divide by n−1), and which is larger?
Correct Answer
A) Population SD ≈ 0.122; sample SD = 0.15; sample SD is larger
Squared deviations total 0.045. Population-form SD = sqrt(0.045/3) = 0.1225…, approximately 0.122; sample-form SD = sqrt(0.045/2) = 0.150. With the same nonzero squared-deviation sum, dividing by n−1 produces a larger SD than dividing by n. These are decimal ratio units, not percentage changes.
Why This Is the Correct Answer
Option A is correct because the deviations from the mean of 1.00 are −0.15, 0 and 0.15, whose squares total 0.045. Dividing by n = 3 gives 0.015, and the square root is about 0.122 (population SD). Dividing by n−1 = 2 gives 0.0225, and the square root is 0.150 (sample SD), so the sample SD is larger.
Why the Other Options Are Wrong
Option B: Population SD = 0.15; sample SD ≈ 0.122; population SD is larger
Option B swaps the two results, calling the population SD 0.15 and the sample SD about 0.122. Dividing 0.045 by n = 3 and taking the square root gives about 0.122 for the population form, while dividing by n−1 = 2 gives 0.150 for the sample form, so the sample SD is the larger one.
Option C: Both equal 0.15, because n and n−1 give the same result for three observations
Option C claims both equal 0.15 because n and n−1 give the same result for three observations. They do not: 0.045 ÷ 3 = 0.015 and 0.045 ÷ 2 = 0.0225, giving square roots of about 0.122 and 0.150.
Option D: Population SD = 0; sample SD = 1.00, the mean of the ratios
Option D gives a population SD of 0 and a sample SD of 1.00, the mean of the ratios. A standard deviation of 0 would require all three ratios to be identical, which they are not, and the mean of 1.00 is a measure of center, not of spread.
Smaller divisor, bigger answer
The sample formula divides by one less (n−1), and dividing the same positive total by a smaller number gives a bigger result.
How to use: Use this when a question asks which standard deviation form is larger for the same data.
Exam Tip
Compute the sum of squared deviations once, then divide by n and by n−1 separately. Check which form each answer labels before choosing.
Common Mistakes to Avoid
- -Swapping the n and n−1 results between the population and sample forms
- -Assuming the two forms give the same result for small samples
- -Confusing the mean of the ratios with their standard deviation
Concept Deep Dive
Analysis
This question tests the difference between the population and sample forms of the standard deviation. Both use the same sum of squared deviations from the mean; the only difference is the divisor, n or n−1. Because n−1 is smaller than n, the sample form produces a larger result whenever the deviations are not all zero.
Background Knowledge
The standard deviation is the square root of the average squared deviation from the mean. The population form divides by n, while the sample form divides by n−1 to correct for estimating the mean from the same data. For assessment ratios the result is in decimal ratio units, not percentages.
Real-World Application
An assessor reviewing a small neighborhood with three sold parcels computes the spread of their assessment ratios. The software reports the sample standard deviation of 0.150, while a hand calculation using n gives about 0.122, and the assessor recognizes the difference comes from the divisor, not an error.
More Statistics Questions
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.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. 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.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Land or Site Valuation
4.5% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam