Consider any survey.  If you want to be specific, think about polling people if they prefer Coke or Pepsi.  However, the content of the survey is not relevant to this question.   You do want some accuracy and so you decide to be 95% confident that the error is at most 3% in measuring the proportion (of people who prefer Coke over Pepsi or some other proportion).  This is the most common error estimate reported on tv.    So how many people do you need to survey to be this accurate? The formula in the book for no preliminary estimate spits out the number 1068 (after rounding up). This is also the number of times you should toss your coin to be 95% sure that your error is at most 3% (in the last discussion).   If you want to restrict your survey to only residents of Charleston, a population of 20,000, you only need to poll 1068 residents of Charleston.  Sounds reasonable.   If you only want to conduct your survey in Illinois, 13 million residents, you still only need to survey 1068 residents of Illinois!   The population of the U.S. is 320 million.  You would still only need to survey 1068 people!   Now, if you wanted to extend your survey to the world, population of 7.5 billion, you would still only need to survey 1068 people in the world!   What is going on here?  How can just the opinion of 1068 people always determine the opinion of the entire population, no matter how large it is? Is Statistics broken? Is 1068 just a magical number?

Consider any survey.  If you want to be specific, think about polling people if they prefer Coke or Pepsi.  However, the content of the survey is not relevant to this question.

You do want some accuracy and so you decide to be 95% confident that the error is at most 3% in measuring the proportion (of people who prefer Coke over Pepsi or some other proportion).  This is the most common error estimate reported on tv. 

So how many people do you need to survey to be this accurate? The formula in the book for no preliminary estimate spits out the number 1068 (after rounding up). This is also the number of times you should toss your coin to be 95% sure that your error is at most 3% (in the last discussion).

If you want to restrict your survey to only residents of Charleston, a population of 20,000, you only need to poll 1068 residents of Charleston.  Sounds reasonable.

If you only want to conduct your survey in Illinois, 13 million residents, you still only need to survey 1068 residents of Illinois!

The population of the U.S. is 320 million.  You would still only need to survey 1068 people!

Now, if you wanted to extend your survey to the world, population of 7.5 billion, you would still only need to survey 1068 people in the world!

What is going on here?  How can just the opinion of 1068 people always determine the opinion of the entire population, no matter how large it is? Is Statistics broken? Is 1068 just a magical number?