I doubt many people lose sleep over statistics. But I did. I keep wondering why the Law of Large Numbers is. The LLN says that the average weight of a series of samples bags of pebbles taken from a beach is a more accurate measure of the the weight of all the individual pebbles on the beach than one large bag of pebbles.
I
know the formula and I know the proof, but something has never gone
from my rational sphere to my gut, even after 50 years as a
statistician.
Everyone who teaches math says you can't prove a geometric theorem, like all angles of a triangle total 180 degrees, by cutting out and measuring triangles. People would laugh if you did. But that is geometry and that theorem is really a syllogism based on the definition of triangle and the definition of degrees.
The law of large numbers is the real world. So I did the equivalent of cutting out triangles. I made a list of 100 numbers from a random number table, the average of my list is 46.9. I did an average of every ten numbers on the list. The average of those ten averages is 46.9. The theoretical average for a column of random two digit numbers is 50. The LLN doesn't show anything in my test. Now my gut still has to depend on my rational sphere.
(A reader, Gene, below found an error in my calculations, right away. It is now corrected. Thanks Gene.)
Check out the fascinating and highly relevant comments below.