Throwing dice

We take N dice, where each one is a standard 6-sided, unloaded die. We define a trial as throwing all N of them once. Alternately it could also be throwing a single die N times. We record the sum of these N throws. The trial is repeated many many times. The probability of obtaining a certain sum can be computed relatively easily (at least when N is small). How many trials do we need so that the observed distribution closely resembles the theoretically expected distribution? Below are a few animations that illustrate this:

N=1: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.

N=2: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.

N=5: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.

N=10: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.

N=20: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.

N=50: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 and 10,000,000 trials.