In a set of sale prices, the third quartile is $700,000, the median is $480,000 and the first quartile is $300,000. Skewness is measured as the distance from the median up to the third quartile minus the distance from the first quartile up to the median, divided by the interquartile range. What is this skewness measure, to two decimals?
Correct Answer
C) 0.10
The third quartile is $220,000 above the median and the first quartile is $180,000 below it. ($220,000 − $180,000) ÷ ($700,000 − $300,000) = $40,000 ÷ $400,000 = 0.10. Common errors: dividing by the sum of the quartiles instead of the interquartile range gives 0.04; dividing the interquartile range by the median gives 0.83; assuming the two distances are equal gives 0.00.
Why This Is the Correct Answer
Option C is correct because the third quartile is $220,000 above the median and the first quartile is $180,000 below it. The difference is $40,000, and the interquartile range is $700,000 - $300,000 = $400,000. So the measure is $40,000 / $400,000 = 0.10.
Why the Other Options Are Wrong
Option A: 0.00
Option A, 0.00, would require the median to sit exactly halfway between the quartiles at $500,000, which does not follow from the facts. With a median of $480,000, the measure is $40,000 / $400,000 = 0.10.
Option B: 0.04
Option B, 0.04, divides by the sum of the quartiles instead of their difference: $40,000 / ($700,000 + $300,000) = $40,000 / $1,000,000 = 0.04. The interquartile range is $700,000 - $300,000 = $400,000, so the measure is 0.10.
Option D: 0.83
Option D, 0.83, divides the interquartile range by the median ($400,000 / $480,000 = 0.83) instead of dividing the difference in distances by the interquartile range. The correct result is $40,000 / $400,000 = 0.10.
Upper plus lower minus twice middle
The numerator reads upper plus lower minus two times the middle, and the denominator is upper minus lower.
How to use: Use this phrasing to put the values in the right places when a question measures skewness from the quartiles and the median.
Exam Tip
Find the distance above the median and the distance below it, subtract in the order the question states and watch the sign.
Common Mistakes to Avoid
- -Dividing by the sum of the quartiles instead of their difference
- -Dividing by the median instead of the quantile spread
- -Assuming symmetry and reporting zero
Concept Deep Dive
Analysis
This question tests a quartile-based skewness measure, known as Bowley skewness. You compare the distance above the median with the distance below it and divide the difference by the interquartile range. The distractors come from using a wrong denominator or assuming symmetry.
Background Knowledge
Quantile-based skewness measures, such as the Galton or Bowley skewness, compare the distance of the upper and lower quantiles from the median. A positive value means the upper quantile lies farther from the median than the lower one, suggesting right skew.
Real-World Application
An appraiser describing a market's price distribution finds the upper quartile farther above the median than the lower quartile is below it. Computing a positive quantile skewness of 0.10 supports describing the distribution as mildly right-skewed.
More Statistics Questions
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Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
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The assessment-to-sale ratios are 0.88, 0.92, 0.96, 0.98, 1.02, 1.04, 1.08, 1.12. 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.