Law of large Numbers and Standard Deviation

[Ty H]

I have a couple of questions relatingto the Law of Large Numbers, and Standard Deviation.


FYI – I will stipulate in advancethat when I took my college entrance exams, ( lo these many years ago), my verbal-related scores were maxed out and my math-related scoreswere in the low 50's. Historically, for me that means that if youcommunicate your responses very clearly and very simply there is achance that my mathematically-retarded brain might understand it. Iwill probably need to ask a few more questions to understand anyresponses. Thanks in advance for any help you fine folks might offerin overcoming my probabilistic obtuseness.


The Law of Large Numbers states that theaverage of the results obtained from a large number of trials shouldbe close to the expected value, and will tend to become closer asmore trials are performed.


Well and good. That makes sense to me.If I flip a fair coin 10 times it would be no surprise that eventhough the expected odds are 50% for each possible outcome came outto be 6-4 or even 7-3 biased to one outcome or the other. Thatdoesn't mean that the coin is biased. It is just expected deviation.If I understand correctly though, if I flip that fair coin 100 times,the odds of deviation from the expected 50-50 outcome drop. And if Iflip the coin 1000 times the odds of deviation drop even more,correct?


From googling around on the internet, Ihave determined that out of a sample size of 100 events, ( 100 faircoin flips ), the standard deviation from the expected result wouldbe 5 and that there is a 68.2% chance of the observed results fallingwithin that deviation range. I have also learned that two standarddeviations would be 10 and that there is a 95.45% chance of theobserved results falling within that deviation range.


So here is my first question- whatwould be the standard deviation on 1000 flips of that fair coin, andwhat would be the chances of the observed results falling within thatdeviation range ( within one standard deviation of the expectedresult ). What would be the chances of the observed results fallingwithin TWO standard deviations of the expected results? If Iunderstand the Law of Large Numbers correctly, then I assume that thestandard deviation should be a smaller percent of the total number ofevents but I do not know how to calculate that standard deviation.


Next question- given that the observedresults are supposed to get closer to the expected average as thesample size of events grow, how large of a sample size would you needto have before you were able to look at the results and conclude thatthe coin itself was biased or that the events were non-random? Sayfor instance, if I had a coin and I flipped it 1000 times and theresults came out to be 550 Heads and 450 tails, what would be theodds against that happening if the coin were truly non-biased and theevents were truly non-random? That would be a deviation of 5%, whichfor 100 flips would be standard deviation and should be expected tooccur 31.8% of the time. How often should one expect that 5%deviation with 1000 flips? Odds of 5% deviation on 2000 flips?


I have some observed results out of alarge sample size of real-life events that would seem to intuitivelyindicate that they are non-random. ( a 5% bias towards one result outof 1000 events that should theoretically have an equally likelybinary solution set / outcome one way or the other). I am trying todecide whether there is a likely bias or if I am just seeing standarddeviation and imputing bias where there may be none. I know thatintuition is often disproved by mathematical reality. Perhaps beingpatient and increasing the sample size of events to 2000 would clearthis all up.


Thank You in advance!

 

[tkhunny]

First, this is very confusing. Do a little extra reading and learn to differentiate between "the probability of" and "the chance that" and "the odds are". More precise language will help you to ask better questions. For example, if something has a 20% chance of happening, then the odds for it happening are 1:4. You see, they aren't even written in the same sort of notation! The odds against that thing happening are 4:1.

Second, forget the entrance exams. Just pursue learning without assuming a handicap. That's just silly.

Third, the reason it is called a "Standard" deviation is that it is standard. If your Distribution is reasonably Normal (and the Law of Large Numbers tends to get you there). the same rules apply, no matter how large the sample or dataset.

Fourth, there isn't a magic number of outcomes that proves, beyond any doubt, that a coin or die is biased. You must decide how much risk you are willing to take. Read up on "Type I Error" and "Type II Error". This will help you on your way.

There are two very important results that also should help you on your way:

1) The Empirical Rule - It doesn't care how big the sample. The more Normal the Distribution, the more meaningful it is.
2) Chebyshev's Theorem (or Inequality) - This does a lot of what the Empirical Rule does, but it doesn't care nearly as much about having a Normal Distribution.

Good luck with your studies.

 

[Ty H]

OK... I appreciate your response, but referring me to Chebyshev's Theorum and the Empirical Rule and saying "good luck with my studies" doesn't really help. As you request, I will try and be more precise with my language for future respondants who might have a more precise answer. How about this-

What would be the standard deviation on1000 flips of a fair coin? What would be the probability of theobserved results falling within onestandard deviation of the expected result? What would be the chances of the observed results falling within TWO standarddeviations of the expected results?

That's really what I need to figure out.

 

[tkhunny]

There is a mistake. Rather than suggesting that a response is not helpful, why not check it out? You might not know all there is to know to help you along your way. Why did you ask a question? Just to control the conversation? That's no good.

Flipping a coin exactly constitutes a Binomial Distribution. Just a few flips does not produce anything very close to a Normal Distribution, particularly if the individual probabilities are not very close to 1/2. 30 is close with probability = 1/2. 100 is nice. 1,000 certainly is enough to be Normal enough, with all but the most extreme individual probabilities. You can then apply the Empirical Rule.

Using 1000 flips of a fair coin and the "Normal Approximation", we get

1) Mean 1000 * 0.5 = 500 or simply a 50% chance that we obtain 500 or more Heads.
2) Standard Deviation = \(\displaystyle \sqrt{1000 \cdot 0.5 \cdot 0.5} = 15.811\)
3) 68% probability that we will obtain between 500-15.811 and 500+15.811 Heads.
4) 95% probability that we will obtain between 500 - 2*15.811 and 500 + 2*15.811 Heads.
5) ... and similarly for 99.7% and 3 standard deviations.

This is the Empirical Rule. Had you not resisted, you could have shown me how it is done and what it means.

The question is, what # of Heads is enough to establish bias? What probability OUTSIDE one of these ranges is enough to suggest that something is biased? The company GE, under Jack Welch, very strongly implemented programs to establish failure probabilities as far out as SIX standard deviations. (Read up on \(\displaystyle 6\sigma\)). This leads us again to your personal tolerance. How small a probability will suggest to you that your coin is biased?