Question

The mean of a data set is $400 and the standard deviation is $40. According to Chebyshev's Theorem, what percent of the values will lie within plus or minus 1.6 standard deviations of the mean?

Answer 1

Chebyshev's theorem says

`P\left(|X-\mu|\le k\sigma\right)\ge1-\frac1{k^2}`

That is, the probability that a data point
`X` falls within
`k` standard deviations of the mean
`\mu` is at least
`1-\frac1{k^2}`.

In this case, we have

`P\left(|X-400|\le1.6\cdot40\right)\ge1-\frac1{1.6^2}\approx0.6094`

so that at least 60.94% of the values can be expected to lie within 1.6 standard deviations of $400.