A sale-price distribution has fourth central moment m4 = 229,635,000 in (thousands of dollars)^4 and second central moment m2 = 8100 in (thousands of dollars)^2. Compute excess kurtosis m4 / m2² − 3. Use moments with the same population (N) divisor, not an unbiased sample variance; report two decimals.
Correct Answer
C) 0.50
229635000 / (8100)² − 3 = 0.5. The denominator is the square of the second central moment; using an N−1 sample variance would be a different statistic.
Why This Is the Correct Answer
Option C is correct because Pearson kurtosis is 229,635,000 / 65,610,000 = 3.50, and excess kurtosis subtracts 3: 3.50 − 3 = 0.50. Subtracting 3 makes a normal distribution's value zero, so a positive result indicates heavier tails than normal.
Why the Other Options Are Wrong
Option A: 3.50
Option A, 3.50, is Pearson kurtosis, m4 / m2² = 229,635,000 / 65,610,000, before subtracting 3. The question asks for excess kurtosis, which is 3.50 − 3 = 0.50.
Option B: 3.00
Option B, 3.00, is the reference value subtracted in the formula, which is the Pearson kurtosis of a normal distribution, not the result for these moments. Excess kurtosis here is 3.50 − 3 = 0.50.
Option D: 6.50
Option D, 6.50, adds 3 to the Pearson kurtosis instead of subtracting it: 3.50 + 3 = 6.50. Excess kurtosis measures how far kurtosis exceeds the normal value of 3, so it is 3.50 − 3 = 0.50.
Excess means minus three
Excess kurtosis is the amount by which kurtosis exceeds the normal value of 3, so subtract 3 from m4 / m2².
How to use: Use this to know when, and in which direction, to apply the 3 on a kurtosis question.
Exam Tip
Read the label carefully: "excess kurtosis" requires the minus 3, while "kurtosis" or "Pearson kurtosis" does not.
Common Mistakes to Avoid
- -Forgetting to subtract 3 when excess kurtosis is requested
- -Adding 3 instead of subtracting it
- -Dividing by m2 instead of m2 squared
Concept Deep Dive
Analysis
This question tests excess kurtosis, which is Pearson kurtosis minus 3. First compute m4 / m2² = 229,635,000 / (8,100)² = 229,635,000 / 65,610,000 = 3.50. Then subtract 3 to get 0.50.
Background Knowledge
Kurtosis is the standardized fourth central moment. Because a normal distribution has kurtosis of 3, some sources subtract 3 so that a normal distribution has excess kurtosis of 0. Positive excess kurtosis suggests heavier tails than normal, and negative suggests lighter tails.
Real-World Application
An analyst finds excess kurtosis of 0.50 for a set of sale prices. This suggests somewhat more extreme sales than a normal distribution would produce, so the analyst reviews the outlying transactions before relying on mean-based summaries.
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