A review appraiser reads two market summaries built from the same eight closed sales: $300,000, $320,000, $350,000, $380,000, $400,000, $430,000, $470,000 and $520,000. One analyst reports a 25th percentile of $320,000 using the nearest-rank method. The other reports $342,500 using a spreadsheet's inclusive percentile function, which interpolates between ordered prices. What can the reviewer conclude?
Correct Answer
D) Both figures are correct
Nearest rank gives rank 0.25 × 8 = 2, which is $320,000. Interpolation places the percentile at position 1 + 7 × 0.25 = 2.75, which is $320,000 + 0.75 × $30,000 = $342,500. Both figures follow correctly from their methods, and no single percentile convention is universally accepted, so a report should state which one it used.
Why This Is the Correct Answer
Option D is correct because each figure follows from its own method: rank 2 gives $320,000 and position 2.75 gives $342,500. Percentile calculations have more than one accepted convention, so the same data can yield different results. Stating the method lets a reader understand and reproduce the figure.
Why the Other Options Are Wrong
Option A: The first analyst's figure is incorrect
Option A calls the first analyst's figure incorrect. Nearest rank selects an observed price by design, and rank 0.25 × 8 = 2 correctly gives $320,000.
Option B: The second analyst's figure is incorrect
Option B calls the second analyst's figure incorrect. Interpolating conventions can land between observed prices, and position 2.75 correctly gives $320,000 + 0.75 × $30,000 = $342,500.
Option C: Both figures are incorrect
Option C calls both figures incorrect. Recomputing each from the eight prices reproduces it exactly, $320,000 by nearest rank and $342,500 by interpolation, so neither shows an error.
Same data, different rule
Before calling a percentile wrong, ask which rule produced it. A different rule can give a different answer from the same array.
How to use: Use this when two reported percentiles for one data set disagree, to check the conventions before assuming an error.
Exam Tip
Recompute each figure with its own method. If both check out, the difference lies in the method and not in the arithmetic.
Common Mistakes to Avoid
- -Assuming there is one standard percentile formula
- -Treating an interpolated percentile as an arithmetic error
- -Omitting the percentile method from a report
Concept Deep Dive
Analysis
This tests the concept behind the calculation: a percentile depends on the convention used to find it. Checking each analyst's arithmetic shows that both follow their stated rules. The right response is therefore to explain the difference and disclose the method rather than to declare an error.
Background Knowledge
Percentiles are ordered-position measures, but statistical references describe several ways to convert a percentage into a position and to handle fractional positions. Differences among conventions tend to matter most in small data sets, where neighboring observations can be far apart.
Real-World Application
A lender's reviewer notices that a market study and an appraisal report different lower-quartile prices for the same closed sales. The appraiser shows that one figure uses nearest rank and the other uses interpolation, and states the convention in the report. The difference is then understood as one of method, not of data.
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.