Xayma's answer does cover the major point here. Chains will grow until a termination event occurs. The chances of such an event are low when there is a lot of unreacted monomer around, and increase as the monomer gets used up. Since this is governed by chance (probability), there will not be a single weight observed, but rather a range. Some chains will terminate quickly (unlikely, but possible). Others will grow much longer - Hence, a range.
The fact that there are several different types of termination possible will also cause polymer chains of different molecular weight to be obtained.
For example, (taking Init as being the radical initiator), you can get termination by a growing chain meeting a radical initiator, and undergoing a radical-radical combination reaction (unlikely, but possible)
Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n</sub>CH<sub>2</sub>CH<sub>2</sub><sup>.</sup> + <sup>.</sup>Init ---> Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n+1</sub>-Init
or by undergoing a radical-radical combination with another growing chain
Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n</sub>CH<sub>2</sub>CH<sub>2</sub><sup>.</sup> + Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>m</sub>CH<sub>2</sub>CH<sub>2</sub><sup>.</sup> ---> Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n+m+2</sub>-Init
or by undergoing a disproportionation with another growing chain
Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n</sub>CH<sub>2</sub>CH<sub>2</sub><sup>.</sup> + Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>m</sub>CH<sub>2</sub>CH<sub>2</sub><sup>.</sup> ---> Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>n</sub>-CH<sub>2</sub>=CH<sub>2</sub> + Init-(CH<sub>2</sub>CH<sub>2</sub>)<sub>m</sub>-CH<sub>2</sub>-CH<sub>3</sub>
or by abstraction of a hydrogen from a terminated chain (thus producing branching) or ...