What’s the law of large numbers?

Question

Pretty much the title.

In: 388

Answer 1

Statistician here. Others have given good answers about *what* the law of large numbers is, I want to give perspective on *why it matters*.

If you’re trying to find some underlying truth about numbers, you need to gather a lot of data points to eliminate “randomness” and chance.

– consider a baseball team. The best baseball team might win 100 games and lose 62 games during the long regular season. We can confidently say they are a *good team* because we’ve seen them play enough that we are confident we’ve seen enough wins and losses to know they are truly good, *they didn’t just get lucky*. When the baseball team gets to the playoffs, they might lose 2 out of 3 games to not advance to the next round. *We cannot say this makes them a bad team*, because they may have just been unlucky or had a couple fluke games.

– or consider rolling a six sided die. It’s not unreasonable you roll a 2 then another 2. Does this mean the average roll of a six sided die is 2? No of course not! You need to roll the die a lot more to get the actual average; the more you roll, the closer you’ll get to the actual underlying value.

– or in political polling – if you ask 3 random people their political preference and they all say they’re going to vote for the same political party, you can’t say “oh this means that party is guaranteed to win the next election!” because you have randomness in that small sample. You’d need to ask lots more people before you start to get an accurate guess about who will actually win.

– or say you’re playing poker with a friend and he deals himself 4 aces (a very good hand). Should you accuse him of cheating? No, he probably just got lucky. But if he deals himself 4 aces every time he deals 20 times in a row should you accuse him of cheating? Probably, because you’ve seen enough deals to know this probably isn’t random, he’s stacking the deck somehow. Don’t play games for money with this friend, he’s a cheater.

This is why the law of large numbers matters. With large enough data, the actual underlying truth is revealed.