A sale-price distribution has fourth central moment m4 = 196,830,000 in (thousands of dollars)^4 and second central moment m2 = 8100 in (thousands of dollars)^2. Compute Pearson moment kurtosis m4 / m2². Use moments with the same population (N) divisor, not an unbiased sample variance; report two decimals.
Correct Answer
C) 3.00
196830000 / (8100)² = 3. 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 m4 / m2² = 196,830,000 / (8,100)² = 196,830,000 / 65,610,000 = 3.00. Both moments use the same population (N) divisor, as the question specifies. A value of 3 matches the kurtosis of a normal distribution.
Why the Other Options Are Wrong
Option A: 0.00
Option A, 0.00, is what you get by subtracting 3 from the correct value (3.00 − 3 = 0.00), which is excess kurtosis. The question asks for Pearson moment kurtosis m4 / m2², which is 3.00.
Option B: 0.30
Option B, 0.30, comes from a place-value slip in squaring m2: 8,100² is taken as 656,100,000 instead of 65,610,000, and 196,830,000 / 656,100,000 = 0.30. With 8,100² = 65,610,000, the kurtosis is 196,830,000 / 65,610,000 = 3.00.
Option D: 30.00
Option D, 30.00, comes from dropping a zero in squaring m2: 8,100² is taken as 6,561,000 instead of 65,610,000, and 196,830,000 / 6,561,000 = 30.00. With 8,100² = 65,610,000, the kurtosis is 196,830,000 / 65,610,000 = 3.00.
Fourth over second squared
Kurtosis uses the fourth moment on top and the second moment squared on the bottom, so the units cancel to a pure number.
How to use: Use this to set up the kurtosis fraction correctly before dividing.
Exam Tip
Match the power in the denominator to the moment on top: fourth moment over the standard deviation to the fourth power, which equals m2 squared. Pearson kurtosis takes no adjustment of 3; only excess kurtosis subtracts 3.
Common Mistakes to Avoid
- -Subtracting 3 when the question asks for Pearson kurtosis, not excess kurtosis
- -Dropping or adding a zero when squaring m2
- -Using an N−1 sample variance in place of the second central moment
Concept Deep Dive
Analysis
This question tests calculating Pearson moment kurtosis, the standardized fourth central moment. The formula is m4 divided by the square of m2, which is the same as dividing by the standard deviation to the fourth power. Here 8,100 squared is 65,610,000, and 196,830,000 / 65,610,000 = 3.00.
Background Knowledge
Kurtosis is the standardized fourth central moment and describes how heavy the tails of a distribution are relative to its spread. A normal distribution has Pearson kurtosis of 3. Some sources subtract 3 to report excess kurtosis, which is 0 for a normal distribution.
Real-World Application
An analyst reviewing a large set of sale prices computes kurtosis to see whether extreme sales are more common than a normal distribution would suggest. A value near 3 suggests tail behavior similar to a normal distribution, though the analyst still examines a histogram.
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