Let be the normalized average of n independent U(–1, 1) random variables. Based upon your intuition:

a. How large do you think n must be for the PDF of to have the same general shape as the PDF of a standard normal distribution?

b. How large do you think n must be for the kurtosis of to match the 3.0 kurtosis of a normal distribution to one decimal place?

a. The following exhibit illustrates PDFs of for n equal to 1 through 5.

For n = 3, the distribution already has the “bell” shape of a normal distribution, although its tails are modest. For n = 4 or n = 5, the PDFs could easily be mistaken for the standard normal PDF. For comparison, the PDF of a standard normal distribution is illustrated below.

b. A normal distribution has a kurtosis of exactly 3. The following exhibit lists the kurtoses of for n = 1 through n = 40. All values are rounded to four decimal places. If we round to a single decimal place, the first kurtosis that equals 3.0 is for n = 24.