Information for "The Case for Kurtosis in the Classroom"

Mathematica Notebook
Mathematica notebook which generates relative frequency approximations of sampling distributions for various statistics. Note: You must have Mathematica on your computer for this to work. If you are using Internet Explorer, simply click on the link to directly open the notebook. If you are using another browser, you may have to download the notebook and then open it.


Data
Reported heights of 559 college students in both text and Minitab formats.


Figures
Figure 1. Graphs of gamma, normal, beta, and uniform distributions all with means of 1/2 and standard deviations of 1/4. $\beta_2$=4.5, 3, 2, and 1.8, respectively.

Figure 2. Dell stock prices over a two-year period (a) and a relative frequency histogram of the logarithm of the ratio of consecutive stock prices (b). Note that the distribution is more peaked than the corresponding normal distribution ($b_2=5.2$).

Figure 3. Enron stock prices over a two-year period (a) and a relative frequency histogram of the logarithm of the ratio of consecutive stock prices (b). Note that the distribution is extremely peaked ($b_2=88$).

Figure 4. Frequency histogram of weights (in pounds) of 72 12-ounce cans of (a) Pepsi and 72 12-ounce cans of (b) Coca-Cola. The sample kurtoses (1.3 and 1.4, respectively) are very small, indicating little variation around $\bar{x} \pm s$.

Figure 5. Graphs of density functions for three different normal mixtures (solid): (a) M=0, S=.2, p=.5 (b) M=4, S=1, p=.5 and (c) M=2, S=1, p=.6 along with the corresponding normal densities (dashed). The kurtoses are approximately 5.6, 1.7, and 2.6, respectively.

Figure 6. Graph of kurtosis as a function of p and M with S=1. The curves visible on the surface are curves of constant $\beta_2$. The thick curve separates those parameter values which give unimodal and bimodal mixtures.

Figure 7. Frequency histogram of reported heights (in inches) of 559 college students.

Figure 8. Relative frequency approximation (1,000 repetitions) of the sampling distribution of $b_2$ under the null hypothesis that the student height data are sampled from a normal distribution with mean 68 and standard deviation 4.2. The area of the lightly shaded region is the corresponding P-value (.01).

Figure 9. Relative frequency approximations (1,000 repetitions) of the sampling distributions of (a) $(n-1)S^2/\sigma^2$ with $n-1=19$ and (b) $\nu S^2/\sigma^2$ with $\nu=4.9$ for an exponential random variable with unit mean. Note how much better the $\chi^2$ distribution with the modified degrees-of-freedom approximates the appropriate sampling distribution.