Most of the larger stairway balusters installed in the USA are 1 3/4 inches wide at the squares (the smaller standard balusters are usually 1 1/4 inches wide at the squares). I’m not sure why this has become an unspoken standard, but I suspect it came about because of the availability of the size of raw materials, what customers were willing to pay for and pleasing look of the prevailing baluster sizes. It may have to do with the reality of how many balusters will fit on a stair tread.
I do, however, come across customers that prefer a larger size baluster. The immediate problem with installing a larger baluster is how to fit them on a stair tread. The solution: a closed stringer – a closed string means that the balusters will attach not to the stair tread but to the closed stringer that rises above the treads. The second problem with installing the larger balusters is how to attach them to the smaller handrail. Stair codes have become more restrictive in recent years. As such, handrails must be sized to be grasped by children and people with small hands. The handrail bottom is no longer wide enough to receive the larger baluster. The solution here is to make a handrail with wider bottom and smaller top so that the hand can grasp the handrail easily. The larger bottom will adequately receive the larger baluster. The two images I have posted here should explain both solutions.