The following sale prices are in ascending order: $300,000, $320,000, $340,000, $360,000, $380,000, $400,000, $420,000, $440,000, $460,000, $480,000, $500,000, $520,000. What is the 25th percentile by the nearest-rank method, which takes the lowest price at or below which at least 25% of the sales fall?
Correct Answer
C) $340,000
r=ceil(0.25×12)=3; observation 3 is $340,000. A different percentile convention can yield a different answer, so the stated nearest-rank method controls.
Why This Is the Correct Answer
Option C is correct because r = ceil(0.25 × 12) = ceil(3) = 3, and the third ordered price is $340,000. Because 3 is already a whole number, the ceiling leaves it unchanged. Using one-based positions, counting to the third value gives $340,000.
Why the Other Options Are Wrong
Option A: $355,000
Option A, $355,000, applies an interpolating convention, rank = 1 + 0.25 × (12 − 1) = 3.75, then $340,000 + 0.75 × $20,000 = $355,000. The stated nearest-rank rule uses r = ceil(0.25 × 12) = 3 with no interpolation, giving $340,000.
Option B: $360,000
Option B, $360,000, is the fourth observation, which results from wrongly bumping an exact rank of 3 up to 4, or from using n + 1 (0.25 × 13 = 3.25, rounded up to 4). The ceiling of exactly 3 is 3, giving $340,000.
Option D: $410,000
Option D, $410,000, is the average of the sixth and seventh values, ($400,000 + $420,000) ÷ 2, which is the median, not the 25th percentile. The nearest-rank rule gives position 3, $340,000.
Whole rank stays put
If p × n is already a whole number, the ceiling does nothing, so take that exact position.
How to use: Use this when a nearest-rank calculation produces an integer rank.
Exam Tip
Do not round up a rank that is already a whole number; the ceiling of an integer is the integer itself.
Common Mistakes to Avoid
- -Rounding an integer rank up to the next position
- -Interpolating between values when the rule calls for nearest rank
- -Confusing the 25th percentile with the median
Concept Deep Dive
Analysis
This question tests the nearest-rank percentile rule when p × n is already a whole number. With n = 12 and p = 0.25, the rank is exactly 3, so no rounding is needed. The answer is the third ordered value, and the main trap is shifting the position by one.
Background Knowledge
The first quartile corresponds roughly to the 25th percentile, marking the lower quarter of ordered data. Different percentile conventions may interpolate or shift positions, so the specified rule determines the answer. Under nearest rank, an exact whole-number rank is used as is.
Real-World Application
An appraiser reports the lower-quartile sale price for a subdivision of twelve recent sales to show the entry-level price point. Applying nearest rank, the appraiser takes the third-ranked sale and identifies the method used.
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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Previous Question
Eight sales in a market area, in ascending order of price, sold for $210,000, $230,000, $245,000, $260,000, $275,000, $290,000, $310,000, $340,000. By the nearest-rank method, which takes the lowest price at or below which at least 70% of the sales fall, what is the 70th percentile?
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