Displaying results of “random” events on a graph displaying a bell curved distribution is mathematical definition and often used as a test for randomness.
No, it is not a test for “randomness”. It is a measure of how likely a single event is to end with a specific result.
For example, if two people were each given a pair of dice and are told to come up with a result. One person is told on each result he is to roll the dice, the other person is told that he can set the dice to any result he wants with the restriction that he cannot set the dice to a combination that was used in any of the previous 35 tests.
The first persons results will be random. It will be impossible to predict with 100% accuracy any result. The second persons results will be non random, you would be able to tell, with 100% assuracnce what the result of the of the 36th test (and all subsequent tests) will be.
Yet taking all of the results from both tests (and given enough tests) both will give a bell curve graph.