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What are the most common misconceptions about the law of large numbers?
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I teach introductory statistics and I'm always surprised by how many students misunderstand the law of large numbers. They seem to think it means that if you flip a coin and get heads 10 times in a row, tails is "due" to come up soon.
But that's actually the gambler's fallacy, not the law of large numbers. The real law says that as you increase sample size, the sample mean approaches the population mean.
What other law of large numbers misconceptions have you encountered? And how do you explain the difference between this and the gambler's fallacy statistics concept?
The biggest misconception I see is people thinking the law of large numbers guarantees short-term correction." Like if a coin comes up heads 5 times in a row, they think tails is somehow "overdue." That's actually the gambler's fallacy statistics trap.
The law just says that over a very large number of trials, the proportion will approach 50/50. But it doesn't say anything about what happens in the next few flips. Each flip is still independent.
I have to explain this to clients all the time in finance. Just because a stock has been down for several days doesn't mean it's "due" for a rebound. That's not how probability works.
Another common law of large numbers misconception is thinking it applies to small samples. People will say I flipped a coin 10 times and got 7 heads, so the law of large numbers isn't working!"
But 10 flips isn't large! The law is about what happens as n approaches infinity, not about small samples. Even 100 or 1000 flips might not be enough to see the proportion stabilize, especially if there's any bias in the coin or flipping method.
I think part of the problem is that introductory stats courses sometimes oversimplify this. They show the classic graph of proportion stabilizing, but don't emphasize enough how many trials it actually takes.
I see this confusion between the law of large numbers and regression to the mean all the time. They're related but different concepts.
Regression to the mean is about extreme measurements being followed by less extreme ones. The law of large numbers is about sample means converging to population means.
But people mix them up. Like if a student gets a really high score on one test and a lower score on the next, they might say oh that's the law of large numbers at work." No, that's regression to the mean!
Both are important counterintuitive statistics concepts, but they work differently.
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