FRM Part I · FRM Exam Part I · Sample Moments
A sample of five observations is 1, 1, 2, 4, 7. Using the sample estimator that divides central moments by n (not n-1), what is the skewness, defined as the third central moment divided by the cube of the standard deviation?
Skewness is about 0.81. The mean is 3, the variance with n in the denominator is 5.2, and the average cubed deviation is 9.6. Dividing 9.6 by the standard deviation cubed, 11.858, gives 0.81, indicating a positive skew.
- A0.81Correct
- B0.58
- C1.85
- D4.05
Explanation
Mean = 3. Squared deviations sum to 26, so variance = 5.2 and sd = 2.280. Cubed deviations sum to 48, so the third moment = 9.6. Skewness = 9.6 / 5.2^1.5 = 9.6 / 11.858 = 0.81. Using n-1 for variance gives 0.58; dividing by variance rather than sd gives 1.85; omitting division by n gives 4.05.
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