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He wants to find out what fraction of the playerbase tends to end up on each rank, by season's end.
@OP
Nothing official as far as I'm aware.
I'd expect that over 95% of active players to be at least Gold in any mode they play in by Season's end, as matches below that are pretty much always bot matches.
Aside from that, anything else is hard for me to speculate. I'd imagine the bigger concentration of players is in Diamond for solos and random duos, and Platinum for premade teams (as most people will have fewer premade games, so not always enough games to climb).