Chebyshev's Inequality

Small question concerning chebyshev’s inequality, what if you take 1 standard deviation? Formula is 1-(1/k square), so if k is one doesn’t the formula result in 0? That would mean no observations lie between 1 standarddeviation which cannot be correct. Can someone help me see what I am overlooking?

It is possible that no observations lie within 1 std dev of the mean so I don’t think you overlooked anything.

The general probability statement for Chebyshev’s inequality includes “for all k>1”. This is because for all values of X when -11, the formula returns a number greater than 1 which is not a probability and therefore is nonsensical.