Well, if we take the 3.5 sigma at face value and bound it using Chebychev's inequality [1], it seems that the discovery of the Higgs Boson is bounded by 1/3.5^2, which is 8%. In fact, the standard inequality is two-tailed. If we're measuring just from one tail, we can use the single-sided inequality to yield 7.5%. Remember, we can't always assume a distribution is normal, a mistake that even respected researchers often make.
Hmm, I don't actually think chebyshev applies (in a meaningful way) in this context. 3.5 sigmas refers to the probability of the observation being significant, not the average mass of the higgs. Or did I misunderstand what you meant?
3.5 sigmas refers to the probability of some statistic being as severe as they're observing under a model where the Higgs does not exist. Regardless of the distribution, Chebyshev applies solely by assumption that sigma is a meaningful unit. It may be overly conservative though.
Thanks for the correction. That link worked when I checked...
Well, Chebyshev inequality always applies. It just might not be the tightest bound possible. Unless we know the distribution better, this is the best estimate (tightest bound that's provably correct given the single assumption about standard deviation)
[1] http://en.wikipedia.org/wiki/Chebyshevs_inequality