An appraiser arranges twenty sale prices in ascending order: $290,000, $310,000, $330,000, $350,000, $370,000, $390,000, $410,000, $430,000, $450,000, $470,000, $490,000, $510,000, $530,000, $550,000, $570,000, $590,000, $610,000, $630,000, $650,000, $670,000. What is the 10th percentile by the nearest-rank method, which takes the lowest price at or below which at least 10% of the sales fall?
Correct Answer
D) $310,000
r=ceil(0.1×20)=2; observation 2 is $310,000. 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(0.10 x 20) = ceil(2) = 2. The second observation in the ordered list is $310,000. That value is the 10th percentile under the nearest-rank rule.
Why the Other Options Are Wrong
Option A: $290,000
Option A, $290,000, is the first observation, as if the rank were 1 instead of ceil(0.10 x 20) = 2. Under the stated rule the 10th percentile is the second value, $310,000.
Option B: $330,000
Option B, $330,000, is the third observation, as if 1 were added to the rank or 2 were rounded up again. Since 0.10 x 20 is already the whole number 2, the rank stays 2, giving $310,000.
Option C: $480,000
Option C, $480,000, is the median, the average of the 10th and 11th values ($470,000 + $490,000) / 2, which is the 50th percentile. The 10th percentile is at rank ceil(0.10 x 20) = 2, which is $310,000.
Multiply, ceiling, count over
For nearest rank, multiply p by n, round up only if needed, then count over to that position.
How to use: Use these three steps whenever a question specifies the nearest-rank percentile method.
Exam Tip
When p x n is already a whole number, the ceiling leaves it unchanged; do not round up again.
Common Mistakes to Avoid
- -Rounding up a rank that is already a whole number
- -Taking the minimum value as the 10th percentile
- -Confusing a low percentile with the median
Concept Deep Dive
Analysis
This question tests applying a stated percentile rule exactly. With the nearest-rank method, you compute r = ceil(p x n) and take the observation at that one-based position. Because percentile conventions vary, the stated rule controls the answer.
Background Knowledge
A percentile marks the value below which a given share of observations falls. There is no single universally accepted way to compute sample percentiles, so a question must name a method such as nearest rank.
Real-World Application
An appraiser describing the lower end of a market sorts 20 sales and uses the nearest-rank rule to identify the 10th percentile at $310,000. Stating the method in the report lets a reviewer reproduce the figure.
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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Sale prices are $380,000, $400,000, $420,000, $440,000, $460,000. Find the arithmetic mean of their absolute deviations from the sample mean. Report to the nearest dollar.
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