# Can you Actually Have all Your Data Outside One Standard Deviation? (Chebyshev)

I was thinking about Chebyshev's Inequality when k=1, which gives an upper bound of up to 100% of the data lying outside one standard deviation.

I was trying to think of an example of a distribution where this was actually true though, and I can't.

Can someone give an example of a simple dataset or distribution in which none of the data lies within one standard deviation?

Or can I prove it's not possible? (Apologies if this is obvious, I am trying to self-teach statistics)

I feel like I am missing something obvious...

You can get as close as you like to 100% outside one standard deviation by placing $\epsilon$ probability at the mean and $(1-\epsilon)/2$ at two points equally spaced either side of it.
If you want to apply it to data, you need to treat the sample distribution (the ecdf) like a population distribution, so you'd have to use the $n-$denominator version of standard deviation, otherwise you can't actually attain the Chebyshev bounds.