In math, randomness mean a measured variance from either a mean, median, mode, or predicted outcome. This can be calculated by three fundamental distributions: Gauss (bell curve) , Cauchy ('heartbeat'), and Levi ('flight of the albatross').
In science, the word designates an observation that's unaccounted for in terms of a viable explanation. For example, one of ten photons will escape water, the other nine being absorbed. As to which of the ten is unpredictable--therefore 'random'.