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...


Note that Chebyshev applies to distributions and population quantities (probabilities, population means, population standard deviations).

Also note that it's not "outside", since the statement in Chebyshev's inequality is that the proportion at least one standard deviation away is less than or equal to 1. That is, it can be exactly at one standard deviation away.

There is a distribution that does this: 50% of the probability at two distinct points, both of which are then one standard deviation away from the mean.

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.