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Question 589:



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The idea here is that with small samples your percentages are not as accurate a reflection of the expected value as a large sample—you'll then see more fluctuations with small samples.  So, lets say you flip a fair coin 20 times. We'd expect 10 heads and 10 tails and the expected percent of heads is 50%. Now, lets say after flipping we actually see 15 heads and 5 tails. We observed 75% heads when we expect 50% and we observed 15 heads when we expected 10. So the difference between the expected percent is 25% and the difference in number is 5.


Now, lets say this time we flip the coin 100 times. We expect to get 50 heads and 50 tails. Lets say we instead get 60 heads and 40 tails giving us an observed 60% heads and 40% tails. The difference between our expected percent has now dropped from 25% to 10% (75% - 50% to 60% - 50%) but the actual number of tosses different than expected has INCREASED from 5 to 10 (15 - 10 to 60 - 50). It's sort of a brain twister because it's obvious and yet counterintuitive. But that's our friend, the Law of Large numbers.

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