My class did an turning pen experiment in class,where we turn a pen 5 times eventually, each time there is a possibility of the pen pointing to 1,2,3 or 4 equally.We need to sum up the accumulated sum(e.g 1+3+1+2+4+1=12). My class has are 100 people.The results showed up and 21% got 12, 12% got 11, 9% got 10, 8% got 9, 11% got 13 etc.. It is like a bell shaped curve due to CLT after 5 turns of the pen.Initially with just one turn the shape is not a normal distribution.

I know the CLT is due to the increase in turning of pens(5 times), not the number of students in class(The number of students is just to show the distribution). I then compute the sample mean using (0.21X12+ 0.12x11+0.09x10......) from the percentage of people who got the respective results.It is roughly ard 12.5. I then compute the expected value by ((1+2+3+4)/4) X5=12.5 which is 12.5. Is it the correct way to do it?

Does this shows the Law of Large numbers? Im rlly confused if the number of students(100) or the number of turns of pens(5) cause the Law of Large numbers. Can someone clarify which is the deciding factor, the number of turns of pens or the number of students?

Thanks