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?
Correct Answer
A) $290,000
r=ceil(0.7×8)=6; observation 6 is $290,000. A different percentile convention can yield a different answer, so the stated nearest-rank method controls.
Why This Is the Correct Answer
Option A is correct because r = ceil(0.70 × 8) = ceil(5.6) = 6, and the sixth ordered price is $290,000. The nearest-rank method rounds the rank upward to the next whole position and takes that observation without interpolation. The list is already ordered, so counting to position 6 gives the answer.
Why the Other Options Are Wrong
Option B: $275,000
Option B, $275,000, is the fifth observation, which results from rounding 5.6 down to 5 instead of up. The stated rule uses the ceiling, ceil(5.6) = 6, which gives $290,000.
Option C: $310,000
Option C, $310,000, is the seventh observation, which would result from using n + 1 instead of n, since ceil(0.70 × 9) = ceil(6.3) = 7. The stated rule uses n = 8, so ceil(0.70 × 8) = 6 and the answer is $290,000.
Option D: $267,500
Option D, $267,500, is the average of the fourth and fifth values, ($260,000 + $275,000) ÷ 2, which is the median (50th percentile), not the 70th percentile. The nearest-rank rule gives position 6, $290,000.
Multiply, round up, count
For nearest rank, multiply p by n, round up to the next whole number, then count that many positions into the ordered list.
How to use: Use this whenever a question asks for a percentile by the nearest-rank method.
Exam Tip
Confirm the list is ordered and that you are using n, not n + 1 or n − 1, before computing the rank.
Common Mistakes to Avoid
- -Rounding the rank down instead of up
- -Using n + 1 in place of n when the rule specifies n
- -Confusing the 70th percentile with the median
Concept Deep Dive
Analysis
This question tests applying a specified percentile convention, the nearest-rank method r = ceil(p × n). With n = 8 and p = 0.70, the rank is 5.6, rounded up to 6. Because different percentile conventions give different answers, the stated rule controls, and the correct answer is the sixth ordered value.
Background Knowledge
Percentiles locate the value below which a given share of ordered observations falls. Several conventions exist, including nearest-rank and various interpolation methods, and they can produce different results on the same data. When a problem specifies a method, apply that method exactly.
Real-World Application
An appraiser summarizing a neighborhood's sale prices reports the 70th percentile to show where the upper-middle of the market sits. Using the nearest-rank rule on eight sales, the appraiser identifies the sixth-ranked price and states the method so readers can reproduce it.
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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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?