Market data show these sale prices, from lowest to highest: $300,000, $330,000, $360,000, $390,000, $410,000, $440,000, $460,000, $480,000, $500,000, $530,000. Using the nearest-rank method, which takes the lowest price at or below which at least 20% of the sales fall, what is the 20th percentile?
Correct Answer
D) $330,000
r=ceil(0.2×10)=2; observation 2 is $330,000. A different percentile convention can yield a different answer, so the stated nearest-rank method controls.
Why This Is the Correct Answer
Option D is correct because r = ceil(0.20 × 10) = ceil(2) = 2, and the second ordered observation is $330,000. Because 0.20 × 10 is exactly 2, no rounding up is needed, and position 2 is counted from one.
Why the Other Options Are Wrong
Option A: $300,000
Option A is wrong because $300,000 is the first observation, the minimum, at position 1. The nearest-rank method gives rank ceil(0.20 × 10) = 2, which is $330,000.
Option B: $360,000
Option B is wrong because $360,000 is the third observation, which results from adding 1 to the rank (2 + 1 = 3) or counting positions from zero. With one-based positions, rank 2 is $330,000.
Option C: $425,000
Option C is wrong because $425,000 is the median, the average of the fifth and sixth observations ($410,000 + $440,000) / 2, which is the 50th percentile. The 20th percentile by the stated rule is the second observation, $330,000.
Rank, round up, read
Multiply the percentile by the count to get the rank, round up if needed, then read that position counting from one.
How to use: Use this whenever a question specifies the nearest-rank percentile method.
Exam Tip
Check whether positions are counted from one or zero and whether the rank needs rounding. The question's stated convention decides the answer.
Common Mistakes to Avoid
- -Counting positions from zero instead of one
- -Confusing a percentile with the median
- -Rounding the rank incorrectly
Concept Deep Dive
Analysis
This question tests the nearest-rank percentile method. Compute r = ceil(p × n) using one-based positions, then read that observation from the ordered list. Percentile conventions differ, so the method named controls.
Background Knowledge
A percentile indicates the value below which a given share of observations falls. Different methods, including nearest-rank and interpolation methods, can give different results for the same data. When a question specifies the method, apply it exactly.
Real-World Application
An appraiser describing a submarket's price range reports the 20th and 80th percentiles of recent sales using a stated nearest-rank method. Documenting the method lets a reviewer reproduce the figures.
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.
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