Basic Statistics · Unit 5
Moments, Skewness, and Kurtosis
Exam-focused notes for Moments, Skewness, and Kurtosis (Basic Statistics, STA154): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Central moments and raw moments
- Computation of moments from data
- Coefficient of skewness
- Mesokurtic, platykurtic, and leptokurtic distributions
- Coefficient of kurtosis
- Interpretation of skewness and kurtosis
Central moments and raw moments
The first four moments about point 5 are 3, 10, 40, and 500. Compute the four central moments. [5]
Moments about the point $a = 5$: - $\mu'1 = 3$ - $\mu'2 = 10$ - $\mu'3 = 40$ - $\mu'4 = 500$ Required: The four central moments $\mu1, \mu2, \mu3, \mu4$. The central moments are obtained from the moments about an arbitrary point using: $$\mu1 = 0$$ $$\mu2 =...
Full solved answer →Compute four central moments and hence find the mean, standard deviation, the measure of skewness, and the measure of kurtosis from central moments. Also, comment on the nature of the data. 120, 135, 150, 125, 140, 160, 170, 155, 165, 150 [10]
Data values (n = 10): $$120,\ 135,\ 150,\ 125,\ 140,\ 160,\ 170,\ 155,\ 165,\ 150$$ $$\bar{x} = \frac{\sum xi}{n} = \frac{120+135+150+125+140+160+170+155+165+150}{10} = \frac{1470}{10} = 147$$ $xi$ $di$ $di^2$ $di^3$ $di^4$ ---------------------------------...
Full solved answer →Coefficient of skewness
If the first four moments about mean are 0, 2.8, -2 and 24.5 respectively. Compute coefficient of skewness and kurtosis and comment upon result. [5]
- First moment about mean: $\mu1 = 0$ - Second moment about mean: $\mu2 = 2.8$ - Third moment about mean: $\mu3 = -2$ - Fourth moment about mean: $\mu4 = 24.5$ Using the beta coefficient: $$\beta1 = \frac{\mu3^2}{\mu2^3} = \frac{(-2)^2}{(2.8)^3} = \frac{4}{...
Full solved answer →Coefficient of kurtosis
The standard deviation of a symmetric distribution is 9. Compute the possible value of fourth central moment for the distribution to be (i) mesokurtic (ii) platykurtic, and (iii) leptokurtic. [5]
- Standard deviation: $\sigma = 9$ - Distribution is symmetric - Required: fourth central moment $\mu4$ for the distribution to be (i) mesokurtic, (ii) platykurtic, (iii) leptokurtic Key relation (coefficient of kurtosis): $$\beta2 = \frac{\mu4}{\mu2^2} = \...
Full solved answer →The standard deviation of a symmetric distribution is 7. Compute the possible value of fourth central moment for the distribution to be (i) mesokurtic (ii) platykurtic, and (iii) leptokurtic. [5]
- Standard deviation: $\sigma = 7$ - Distribution is symmetric - Required: value of the fourth central moment $\mu4$ for the distribution to be (i) mesokurtic, (ii) platykurtic, (iii) leptokurtic The coefficient of kurtosis is: $$\beta2 = \frac{\mu4}{\mu2^2...
Full solved answer →The standard deviation of a symmetric distribution is 7. Compute the possible value of fourth central moment for the distribution to be (i) mesokurtic (ii) platykurtic, and (iii) leptokurtic. [5]
- Standard deviation: $\sigma = 7$ - Distribution is symmetric - Required: fourth central moment $\mu4$ for (i) mesokurtic, (ii) platykurtic, (iii) leptokurtic The coefficient of kurtosis is: $$\beta2 = \frac{\mu4}{\mu2^2} = \frac{\mu4}{\sigma^4}$$ Kurtosis...
Full solved answer →Make Unit 5 stick
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