Sale-price mean is $480,000, median $420,000 and standard deviation $90,000. Calculate Pearson's second skewness coefficient 3(mean − median) / standard deviation, to two decimals.
Correct Answer
C) 2.00
3 × (480000 − 420000) / 90000 = 2. The factor of three belongs to this coefficient; the first coefficient uses the mode instead.
Why This Is the Correct Answer
Option C is correct because 3 × ($480,000 − $420,000) / $90,000 = 3 × $60,000 / $90,000 = 2.00. The mean exceeds the median, so the coefficient is positive. The factor of three belongs to this second coefficient.
Why the Other Options Are Wrong
Option A: 0.67
Option A, 0.67, comes from dropping the factor of 3: ($480,000 − $420,000) / $90,000 = 0.67. The formula requires 3 × $60,000 / $90,000 = 2.00.
Option B: 0.43
Option B, 0.43, divides by the median instead of the standard deviation: 3 × $60,000 / $420,000 = 0.43. The coefficient is scaled by the standard deviation: 3 × $60,000 / $90,000 = 2.00.
Option D: 0.00
Option D, 0.00, would require the mean to equal the median, which is not the case here; it does not follow from the facts. The correct result is 3 × $60,000 / $90,000 = 2.00.
Three times mean minus median
Second coefficient: 3 times (Mean minus Median) over SD. Mean first, so a mean above the median gives a positive number.
How to use: Use this to recall both the factor of 3 and the order of subtraction.
Exam Tip
Keep the factor of 3 and divide by the standard deviation; dividing by the median or dropping the factor gives a different number.
Common Mistakes to Avoid
- -Leaving out the factor of 3
- -Dividing by the median instead of the standard deviation
- -Using the mode, which belongs to the first coefficient
Concept Deep Dive
Analysis
This question tests applying Pearson's second skewness coefficient, 3(mean − median) / standard deviation. The candidate must keep the factor of 3, subtract in the correct order and divide by the standard deviation. A positive result shows the mean is pulled above the median, indicating right skew.
Background Knowledge
Skewness describes asymmetry in a distribution. Pearson's second coefficient compares the mean and median, scaled by the standard deviation, while the first coefficient uses the mode instead of the median. In sale-price data, a few high-priced sales typically pull the mean above the median and produce positive skew.
Real-World Application
An appraiser reviewing a market with a handful of very expensive sales finds the mean price well above the median. Computing a positive skewness coefficient confirms the right skew, so he relies more on the median when describing typical prices.
More Statistics Questions
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