Chapter 15, Problem 75E

Chebyshev’s theorem. The Russian mathematician P.L. Chebyshev (1821-1894) showed that for any data set and any constant k greater than 1, at least $1-\left(1/k^2\right)$ of the data must lie within k standard deviations on either side of the mean A. For example, when $k=2$ this says $1-14=34$ (i.e., 75%) of the data must lie within two standard deviations of A (i.e., somewhere between $A-2\sigma$ and $A+2\sigma$)

a. Using Chebyshev’s theorem, what percentage of a data set must lie within three standard deviations of the mean?

b. How many standard deviations on each side of the mean must we take to be assured of including 99% of the data?

c. Suppose that the average of a data set is $A$. Explain why there is no number $k$ of standard deviations for which we can be certain that 100% of the data lies within k standard deviations on either side of the mean $A$.