Question

chebysshev's theorem

Answer 1

Chebyshev’s Theorem:

at least $1-\frac{1}{k^2}$ of the data lie within   standard deviations of the mean, that is, in the interval with endpoints $\bar{x}-ks \ \ and \ \ \bar{x}+ks$   for samples and with endpoints $\mu-k\sigma \ \ and \ \ \mu+k\sigma$ for populations, where k is any positive whole number that is greater than 1.

For any numerical data set,

1.

1. at least $\frac{3}{4}$ of the data lie within two standard deviations of the mean, that is, in the interval with endpoints

$\bar{x}-2s$  and $\bar{x}+2s$ for samples and with endpoints $\mu-2\sigma$ and $\mu+2\sigma$ for population

2.

at least  $\frac{8}{9}$ of the data lie within three standard deviations of the mean, that is, in the interval with endpoints $\bar{x}-3s$   and $\bar{x}+3s$ for samples and with endpoints $\mu-3\sigma$ and $\mu+3\sigma$   for populations;