The positive observations are 10, 15, 22.5. Compute their simple geometric mean (product of all observations)^(1/n). Report as a dimensionless ratio to three decimals.
Correct Answer
A) 15.000
10×15×22.5 raised to the power 1/3 = 15. The units remain a dimensionless ratio; this differs from their arithmetic average.
Why This Is the Correct Answer
Option A is correct because the product is 10 × 15 × 22.5 = 3,375, and the cube root of 3,375 is 15 (since 15 × 15 × 15 = 3,375). With three observations, the geometric mean is that cube root, 15.000.
Why the Other Options Are Wrong
Option B: 15.833
Option B, 15.833, is the arithmetic mean, (10 + 15 + 22.5) / 3 = 47.5 / 3. The geometric mean multiplies instead: the cube root of 3,375 is 15.000.
Option C: 47.500
Option C, 47.500, is the sum of the three observations, not any kind of mean. The geometric mean is the cube root of their product, 3,375, which is 15.000.
Option D: 10.000
Option D, 10.000, is simply the smallest observation. The geometric mean of 10, 15 and 22.5 is the cube root of 3,375, which is 15.000.
Geometric multiplies, arithmetic adds
Arithmetic mean: add, then divide by n. Geometric mean: multiply, then take the nth root.
How to use: Use this to choose the right operation when a question names the geometric mean.
Exam Tip
If an answer equals the simple average, it is the arithmetic mean, not the geometric mean. For positive data that are not all equal, the geometric mean is lower than the arithmetic mean.
Common Mistakes to Avoid
- -Computing the arithmetic mean instead
- -Using a square root when there are three observations
- -Reporting the sum or a single observation as the mean
Concept Deep Dive
Analysis
This question tests the geometric mean. Multiply all the observations together, then take the nth root, where n is the number of observations. The result differs from the arithmetic mean, which adds the values and divides.
Background Knowledge
The geometric mean is the nth root of the product of n positive values. It is useful for averaging ratios or growth factors, and it does not exceed the arithmetic mean of the same positive data. It requires all values to be positive.
Real-World Application
An analyst studying price relatives from several periods averages the growth factors with a geometric mean, because multiplying factors reflects how changes compound over time better than simply adding them.
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
Previous Question
Assessment-to-sale ratios are 0.88, 0.93, 0.96, 0.99, 1.02, 1.07, 1.11. Compute the arithmetic mean. Report as a decimal ratio to three places.
Next Question
Ordered sale prices are $300,000, $320,000, $340,000, $360,000, $380,000, $400,000, $420,000, $440,000, $460,000, $480,000, $500,000, $520,000, $540,000, $560,000, $580,000, $600,000, $620,000, $640,000, $660,000, $680,000. Using the nearest-rank method, which takes the lowest price at or below which at least 5% of the sales fall, what is the 5th percentile?