I was wondering how reliable is a large standard deviation whose negative values are below 0? If we were calculating 2 standard deviations away from the mean to catch at least 75% of the observations over a non-normal distribution and end up with (say) -600 standard deviation over a mean of 300 knowing that the measure we are applying the standard deviation to does not accept negative values(i.e. min value is 0), we would end up with something like 300 - (600*2) on the left of our distribution and 300 + (600*2) on the right. Does it even make sense to have those negative values below 0? Or in this case, is it imperative to normalize the data? Thanks
Standard deviation is always positive, so a std of -600 doesn't make sense.
Chebyshev's inequality is just that: an inequality. It doesn't say that to get 75% of the data, you have to go out 2 std. It says you have to go out at most 2 std. In your examples, at least 75% of the data has a value greater than -900. Now, you may know, from sources other than Chebyshev's inequality, that all of the data has a value greater than 0, and hence greater than -900. So in that case, Chebyshev's inequality doesn't give you any more information about the lower bound than what you already had. In that case, Chebyshev's inequality isn't particularly useful, but it is still valid.
Given you said in your comment
none of the low values below the mean would be considered outliers
I wouldn't worry about the lower end (mean - std*2).
I use Chebyshev's inequality in a similar situation-- data that is not normally distributed, cannot be negative, and has a long tail on the high end. While there can be outliers on the low end (where mean is high and std relatively small) it's generally on the high side. We use 3 std for excluding outliers prior to doing some forecasting, so that no more than 11% of observations will be excluded per the inequality. In practice when we first implemented we found that only about 1% of observations were excluded. Chebyshev's was a nice theoretical bound to give us some justification for setting the threshold at 3 std's instead of needing to go higher.