Perhaps, but due to the sampling of the distribution you would likely never know. If 95% of your samples fit in the top 3 bins, you can’t say much at all with certainty. Poisson, Gaussian, binomial, Boltzmann, gamma…

That is not IMHO what he is trying to say, you don't shift the distribution, you measure if somebody passed a test. I the test is "passable" then one side of "distribution" is at least cut off. E.g. it's normal (and sometimes expected) that the whole class will pass without issues.

if your scale doesn't have the atomic values at the top end to differentiate the data it's not a normal, it's Pareto or Zipf or some other power law.