I'm not sure if this is useful but a few years ago I tried simulating tournament results with my own finish distribution for 180 seat tournaments (old - 18 paid).
Finish Dist in %. (0.712, 0.821, 0.876, 0.986, 0.821, 0.67, 1.041, 0.438, 0.602, (10 to 18) 0.575 each)
I did 500,000 simulations at each amount of games using the above finish position probabilities and plotted the the results. The right hand plot is a
P-P plot of normal vs the simulated, the closer to a 45% straight line the closer the data to normal.
Even with the very spiked results of tournaments this simulation shows the results seem to come from a normal shaped distribution by the time I played about 100 games, and just about exact by 1000 games.