Of course there is a rigorous way to prove that with larger sample sizes the sample statistic converges to the population statistic, if you pick the mean then the average value of a sample set will converge to its expected value over larger and larger sample sizes. Take a dice. Take a dice, denote 6 as best and 1 as worst. Then roll it 10 times, then calculate the average face number you rolled across the 10 rolls. Let's say you got a 5.xx. So, does this mean the dice is not a fair dice because you scored so high? Maybe, maybe not. You are not sure, luck may be in place here. It actually most certainly is. Now roll the same dice 1000 times. If you dont get an average close to 3.5 you can be almost certain that the dice is not fair. You can now adjust your confidence interval the larger you make the sample size. Pretty easy and basic concept imho. Hope it makes sense. Here the proof:

Take a look at the concept of the law of large numbers: http://en.wikipedia.org/wiki/Law_of_large_numbers

The proof is on the same page and directly relates to your problem.

By the way, I concur with Nassim Taleb on this one (though I disagree with his style of trading and some of his opinions on options trading). George Soros has proven many more times than Buffet that his investment and trading skills are highly skilled and that his returns were not the outcome by mere luck. Its that most people only know about his BOE play but hardly anything else. Buffet in turn has done a lot bigger deals but fewer in between, simply because his investment style is much more longer term oriented than Soros's.

Of course there is a rigorous way to prove that with larger sample sizes the sample statistic converges to the population statistic, if you pick the mean then the average value of a sample set will converge to its expected value over larger and larger sample sizes. Take a dice. Take a die, denote 6 as best and 1 as worst. Then roll it 10 times, then calculate the average face number you rolled across the 10 rolls. Let's say you got a 5.xx. So, does this mean the die is not a fair die because you scored so high? Maybe, maybe not. You are not sure, luck may be in place here. It actually most certainly is. Now roll the same die 1000 times. If you don't get an average close to 3.5 you can be almost certain that the die is not fair. You can now adjust your confidence interval the larger you make the sample size. Pretty easy and basic concept imho. Hope it makes sense. Here the proof:

Take a look at the concept of the law of large numbers: http://en.wikipedia.org/wiki/Law_of_large_numbers

The proof is on the same page and directly relates to your problem.

By the way, I concur with Nassim Taleb on this one (though I disagree with his style of trading and some of his opinions on options trading). George Soros has proven many more times than Buffet that his investment and trading skills are highly skilled and that his returns were not the outcome by mere luck. Its that most people only know about his BOE play but hardly anything else. Buffet in turn has done a lot bigger deals but fewer in between, simply because his investment style is much more longer term oriented than Soros's.

Of course there is a rigorous way to prove that with larger sample sizes the sample statistic converges to the population statistic, if you pick the mean then the average value of a sample set will converge to its expected value over larger and larger sample sizes. Take a look at the concept of the law of large numbers: http://en.wikipedia.org/wiki/Law_of_large_numbers

The proof is on the same page and directly relates to your problem.

By the way, I concur with Nassim Taleb on this one (though I disagree with his style of trading and some of his opinions on options trading). George Soros has proven many more times than Buffet that his investment and trading skills are highly skilled and that his returns were not the outcome by mere luck. Its that most people only know about his BOE play but hardly anything else. Buffet in turn has done a lot bigger deals but fewer in between, simply because his investment style is much more longer term oriented than Soros's.

Of course there is a rigorous way to prove that with larger sample sizes the sample statistic converges to the population statistic, if you pick the mean then the average value of a sample set will converge to its expected value over larger and larger sample sizes. Take a dice. Take a dice, denote 6 as best and 1 as worst. Then roll it 10 times, then calculate the average face number you rolled across the 10 rolls. Let's say you got a 5.xx. So, does this mean the dice is not a fair dice because you scored so high? Maybe, maybe not. You are not sure, luck may be in place here. It actually most certainly is. Now roll the same dice 1000 times. If you dont get an average close to 3.5 you can be almost certain that the dice is not fair. You can now adjust your confidence interval the larger you make the sample size. Pretty easy and basic concept imho. Hope it makes sense. Here the proof:

Take a look at the concept of the law of large numbers: http://en.wikipedia.org/wiki/Law_of_large_numbers

The proof is on the same page and directly relates to your problem.

By the way, I concur with Nassim Taleb on this one (though I disagree with his style of trading and some of his opinions on options trading). George Soros has proven many more times than Buffet that his investment and trading skills are highly skilled and that his returns were not the outcome by mere luck. Its that most people only know about his BOE play but hardly anything else. Buffet in turn has done a lot bigger deals but fewer in between, simply because his investment style is much more longer term oriented than Soros's.