The problem:

I have some data and I have to find the probabilty for the data to stay below a certain threshold X or the other way around. I have to find the probabilty for the data not to exceed the threshold. What I have done so far.

1) I have been analysing data and trying to find a distribution (like normal, extreme value,...), which fits the data. The test I used to find out whether the data fits a distribution is chi-square test.

2) I have found out, with a confidence level of 99% (95% was not any better), that the data does fit none of them.

Let us say, I had found out, the data is normally distributed. Then I would have been able to say the probabilty for the data to be within e.g. +-1.96 standard deviation would have 95% (0.95).

My question:
Since I haven't found any distributionb to to fit the data, how can I know give a statement on the probabilty it does not reach a certain threshold?

Thanks in advance.