Ordered sale prices are $390,000, $400,000, $410,000, $430,000, $450,000. Using the inclusive percentile method, in which the lowest observation is the 0th percentile and the highest is the 100th, with linear interpolation between observations, what is the 80th percentile, to the nearest dollar?
Correct Answer
A) $434,000
n=5; h=1+(5−1)×0.80=4.2. Interpolate $430,000+0.2×($450,000−$430,000)=$434,000. The inclusive method locates the position as h=1+(n−1)p. Common errors: taking the lower surrounding observation without interpolating gives $430,000; taking the upper surrounding observation without interpolating gives $450,000; using h = (n + 1)p instead of 1 + (n − 1)p gives $446,000.
Why This Is the Correct Answer
Option A is correct because h = 1 + (5 - 1) x 0.80 = 4.2, so the percentile lies 0.2 of the way from the 4th value, $430,000, to the 5th value, $450,000. That gives $430,000 + 0.2 x ($450,000 - $430,000) = $434,000. This is the position formula for the inclusive method.
Why the Other Options Are Wrong
Option B: $430,000
Option B, $430,000, takes the lower surrounding observation without interpolating. The fractional part of h = 4.2 requires moving 0.2 of the $20,000 gap, giving $434,000.
Option C: $450,000
Option C, $450,000, takes the upper surrounding observation without interpolating. The position 4.2 is only slightly past the 4th value, so the result is $434,000.
Option D: $446,000
Option D, $446,000, uses h = (n + 1)p = 6 x 0.80 = 4.8 instead of 1 + (n - 1)p, which gives $430,000 + 0.8 x $20,000 = $446,000. The inclusive method uses h = 1 + (n - 1)p = 4.2, giving $434,000.
Position, then interpolate
Compute the position h with the formula for the named method, then move the fractional part of h between the surrounding ordered values. The fraction tells how far along the gap to go.
How to use: Use it for any percentile or quartile question that calls for interpolation.
Exam Tip
Use the position formula for the method named; switching formulas is the planted trap. Confirm the data are sorted before locating positions.
Common Mistakes to Avoid
- -Rounding the position to a whole observation instead of interpolating
- -Using a different position formula than the inclusive method requires
- -Locating positions in unsorted data
Concept Deep Dive
Analysis
This calculation tests applying the inclusive percentile method with linear interpolation. With n = 5 and p = 0.80, the position is h = 1 + (5 - 1) x 0.80 = 4.2, which falls between the 4th and 5th ordered values. Because there is no universally accepted percentile method, the question names the method, and the distractors either skip interpolation or use a different position formula.
Background Knowledge
A percentile is the value below which a given share of ordered observations falls. Several formulas exist for locating a percentile and interpolating between observations, and there is no single universally accepted method, so results differ by convention. When a question specifies the method, apply that method exactly.
Real-World Application
An appraiser summarizing a small set of comparable sales wants to describe the upper part of the price range. Following a stated percentile convention, the appraiser computes the 80th percentile with interpolation and notes the method in the workfile. Stating the method matters because other conventions would give a different figure.
More Statistics Questions
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Previous Question
The assessment-to-sale ratios are 0.85, 0.9, 0.95, 1, 1, 1.05, 1.1, 1.15. 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.
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