Standard deviation
Suppose you roll the dice 6 million times. The expectation is 1 million of those rolls will be a 7. In reality, it will usually be slightly more or less than 1 million. That's the deviation.
For a more detailed, complete explaination, go to http://en.wikipedia.org/wiki/Standard_deviation
When you get done with that, check out http://en.wikipedia.org/wiki/Variance
What I don't really get is how to use the information. If the standard deviatiion is X, what does that tell me? Oh, squared it's the variance? [g] To me it's the case of the naked number. The number is 3, let's say, but 3 what Mister?
LOL
For example: if a fair coin is flipped 100 times, it should be fairly obvious that the average number of heads that will appear is 50. Let's say I offer you an even-money bet that the number of heads will be between 40 and 60 for a particular trial. If it's within this range, I win; outside, you win. Should you accept this bet? In order to make an informed decision, you need to know something about the uncertainty of the result, and standard deviation is the tool you would use. In this case, the standard deviation is 5. So it is likely that the number of heads will be between 45 and 55 (within one SD of the mean), extremely likely that the result will be between 40 and 60 heads (within two SD's of the mean), and almost certain that the result will be between 35 and 65 heads (within 3 SD's of the mean). For this particular game, there is about a 95% probability that the number of heads will be between 40 and 60, so it would not be smart for you to accept an even money bet. You should be looking for something like 20:1 odds for a fair game.
The particular value of standard deviation is that the units are the same as the quantity that you are measuring. What's the historical temperature distribution in Chicago on April 1? The average is 53 degrees F, with a standard deviation of 8 degrees F. How much does an adult man weigh? The average is 210 lbs, with a SD of 23 lbs. How much money does a beekeeper with 10 years of experience make? The average is $42K, with a SD of $3500. BTW, I'm making all these number up. Just providing some examples of how SD is used and what the units are.
Quote: RaleighCrapsWell done PapaChubby. Based on my other readings I thought that was how it worked, but you said it in plain English for us dummies.
Praise likewise from me. I subscribed to this thread so I can easily find this again. Thanks!
Quote: PapaChubby
For example: if a fair coin is flipped 100 times, it should be fairly obvious that the average number of heads that will appear is 50. In this case, the standard deviation is 5.
For this particular game, there is about a 95% probability that the number of heads will be between 40 and 60, so it would not be smart for you to accept an even money bet. You should be looking for something like 20:1 odds for a fair game.
BTW, I'm making all these number up. Just providing some examples of how SD is used and what the units are.
Actually, you did correctly calculate the Standard deviation for 100 coin tosses. It is exactly 5. The general formula is SQUARE ROOT ( 0.25*N) where N is the number of coin tosses.
The probability of being within 2 standard deviations (i.e. 40 to 60 ) is also correctly calculated as 95%.
In probability of being within 1 standard deviations (i.e. 45 to 55 ) is about 68% so there would be hefty house advantage if you take an even money bet where your position is that it will fall outside of that range.
Try this variant. Flip a coin 400 times (or use the =INT(2*RAND()) in 400 cells in a long column in Excel which is much faster. What are the odds of getting a streak of 7 or more in row? Which side of an even money bet should you take (yes or no). If you are a little bit clever you can program auxiliary columns to tell you what the longest streak is for a calculation (the F9 button on PC's recalculates the spreadsheet and gives you new random numbers). If you program it, it saves you the trouble of looking down the column and counting the length of the streaks.
When you've done that consider the similar question of a streak of 8 or more in a row.
Quote: pacomartinActually, you did correctly calculate the Standard deviation for 100 coin tosses. It is exactly 5. The general formula is SQUARE ROOT ( 0.25*N) where N is the number of coin tosses.
The probability of being within 2 standard deviations (i.e. 40 to 60 ) is also correctly calculated as 95%.
In probability of being within 1 standard deviations (i.e. 45 to 55 ) is about 68% so there would be hefty house advantage if you take an even money bet where your position is that it will fall outside of that range.
Try this variant. Flip a coin 400 times (or use the =INT(2*RAND()) in 400 cells in a long column in Excel which is much faster. What are the odds of getting a streak of 7 or more in row? Which side of an even money bet should you take (yes or no). If you are a little bit clever you can program auxiliary columns to tell you what the longest streak is for a calculation (the F9 button on PC's recalculates the spreadsheet and gives you new random numbers). If you program it, it saves you the trouble of looking down the column and counting the length of the streaks.
When you've done that consider the similar question of a streak of 8 or more in a row.
OK, ya got me. I was only making up the numbers in the last paragraph about temperatures, weights and salaries.
As for your puzzle, it doesn't have anything to do with standard deviations, but I think I'd calculate this way:
The chances that a streak of 7 heads in a row starting on any particular flip is (1/2)^7 = 1/128.
The chances that this streak does not start on any particular flip is 1 - (1/128) = 127/128.
The chances that this streak does not start on any of the 393 flips where it could start is (127/128)^393 = 4.585%
So the chances of having at least one streak is 100% - 4.585% = 95.415%
You should definitely take an even money bet that at least one streak will occur.
For a streak of 8, the probability is 78.4%. Still a good bet.
For a streak of 9, the probability is 53.4%. Now a reasonably fair bet.
Quote: RaleighCrapsWell done PapaChubby. Based on my other readings I thought that was how it worked, but you said it in plain English for us dummies.
Quote: odiousgambitPraise likewise from me. I subscribed to this thread so I can easily find this again. Thanks!
Thanks for the kind words. I'm new here (but have been an avid follower of the Wizard for many years). I'm happy I could help!
silly
