This must be a standard problem, but I can't find an answer.
I'm assuming that particle weights follow a Normal frequency distribution, that is the number of particles with weight < x is given by N(x), which is a CDF, the integral of the pdf n(x).
I want the weight of particles with weight < x. I think this CDF will be the integral of x*n(x), but I've no idea how to calculate that.
Edited to add: As pointed out by @scherm the total weight particles with weight < x depends on the sample size. I should have said that the total weight of all particles is known, so I want the proportion of the total weight which is made up of particles with weight