Ordered sale prices are $120,000, $160,000, $200,000, $280,000, $360,000, $440,000, $600,000, $720,000, $900,000. The first, second and third quartiles are taken as the 3rd, 5th and 7th observations, respectively. 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
B) 0.20
Q1=200000, Q2=360000, Q3=600000; (600000+200000−2×360000)/(600000−200000)=0.2. The stated quartile convention removes ambiguity between sample-quantile conventions. Common errors: dividing by Q3 + Q1 instead of Q3 − Q1 gives 0.10; dividing the upper half-spread by the lower half-spread gives 1.50; inverting the formula gives 5.00.
Why This Is the Correct Answer
Option B is correct because Q1 = $200,000 (observation 3), Q2 = $360,000 (observation 5) and Q3 = $600,000 (observation 7). The numerator is $600,000 + $200,000 − 2 × $360,000 = $80,000, and the denominator is $600,000 − $200,000 = $400,000. Dividing gives 0.20, a mild positive skew.
Why the Other Options Are Wrong
Option A: 0.10
Option A, 0.10, comes from dividing by Q3 + Q1 instead of Q3 − Q1: $80,000 ÷ ($600,000 + $200,000) = $80,000 ÷ $800,000 = 0.10. The correct denominator is $600,000 − $200,000 = $400,000, giving 0.20.
Option C: 1.50
Option C, 1.50, comes from dividing the upper half-spread by the lower half-spread: ($600,000 − $360,000) ÷ ($360,000 − $200,000) = $240,000 ÷ $160,000 = 1.50. Bowley skewness is ($80,000) ÷ ($400,000) = 0.20.
Option D: 5.00
Option D, 5.00, comes from inverting the formula: ($600,000 − $200,000) ÷ ($600,000 + $200,000 − 2 × $360,000) = $400,000 ÷ $80,000 = 5.00. The correct order puts $80,000 in the numerator and $400,000 in the denominator, giving 0.20.
Gap difference over IQR
Bowley skewness equals the upper gap (Q3 − Q2) minus the lower gap (Q2 − Q1), divided by the IQR (Q3 − Q1).
How to use: Use this restatement to rebuild the formula and confirm the denominator is the IQR.
Exam Tip
Bowley skewness must lie between −1 and +1, so results such as 1.50 or 5.00 signal a formula error.
Common Mistakes to Avoid
- -Dividing by Q3 + Q1 instead of the interquartile range
- -Inverting the numerator and denominator
- -Using the ratio of the two half-spreads as the skewness
Concept Deep Dive
Analysis
This question tests Bowley (Galton) skewness, a quartile-based measure of asymmetry. Using the stated positions, Q1, Q2 and Q3 are identified from the ordered list and placed into (Q3 + Q1 − 2Q2) ÷ (Q3 − Q1). A positive result indicates the upper quartile is farther from the median than the lower quartile.
Background Knowledge
Bowley skewness compares the distance from the median to the upper quartile with the distance to the lower quartile, scaled by the interquartile range. It ranges between −1 and +1, with 0 indicating a symmetric middle half. Because sample-quantile conventions differ, stating which observations are used as quartiles removes ambiguity.
Real-World Application
An analyst studying a resort-town price distribution finds the upper quartile sits much farther above the median than the lower quartile sits below it. She reports a Bowley skewness of 0.20 to show the moderate stretch toward higher prices.
More Statistics Questions
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