Sale-price mean is $510,000, mode $420,000 and standard deviation $90,000. Calculate Pearson's first skewness coefficient (mean − mode) / standard deviation, to two decimals.
Correct Answer
C) 1.00
(510000 − 420000) / 90000 = 1. Each monetary unit cancels; the coefficient is dimensionless.
Why This Is the Correct Answer
Option C is correct because ($510,000 − $420,000) ÷ $90,000 = $90,000 ÷ $90,000 = 1.00. The monetary units cancel, so the coefficient is dimensionless, and the positive sign indicates the mean lies above the mode.
Why the Other Options Are Wrong
Option A: 0.18
Option A, 0.18, divides the difference by the mean instead of the standard deviation: $90,000 ÷ $510,000 = 0.18. Pearson's first coefficient scales by the standard deviation, so ($510,000 − $420,000) ÷ $90,000 = 1.00.
Option B: 3.00
Option B, 3.00, matches multiplying the result by 3, as if applying the factor of 3 from Pearson's second (median-based) coefficient: 3 × ($90,000 ÷ $90,000) = 3.00. The first coefficient has no factor of 3, so the answer is ($510,000 − $420,000) ÷ $90,000 = 1.00.
Option D: 0.21
Option D, 0.21, divides the difference by the mode instead of the standard deviation: $90,000 ÷ $420,000 = 0.21. Pearson's first coefficient scales by the standard deviation, so ($510,000 − $420,000) ÷ $90,000 = 1.00.
Mean minus mode, over SD
Pearson's first coefficient is mean minus mode, divided by the standard deviation. The order mean-then-mode sets the sign.
How to use: Use this to get both the order and the sign right on skewness questions.
Exam Tip
Keep the order mean minus mode and divide by the standard deviation, not by the mean or the mode. Do not import the factor of 3 that belongs to the median-based version.
Common Mistakes to Avoid
- -Dividing by the mode instead of the standard deviation
- -Applying the factor of 3 from the median-based coefficient
- -Dividing by the mean instead of the standard deviation
Concept Deep Dive
Analysis
This question tests Pearson's first skewness coefficient, (mean − mode) ÷ standard deviation. Substitute the given values in the correct order; the sign shows the direction of skew and the result has no unit.
Background Knowledge
Pearson's first skewness coefficient measures asymmetry by comparing the mean with the mode, scaled by the standard deviation. A positive value suggests right skew, where higher-priced sales pull the mean above the mode. Because the units cancel, the coefficient can be compared across data sets.
Real-World Application
An appraiser notices that the mean sale price in a neighborhood is well above the most common price. A positive skewness coefficient confirms that a few higher-priced sales are pulling the average upward.
More Statistics Questions
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