I'm reading the book Gambling theory and other topics and there was a quote I didn't quite understand, and was wondering if someone could please explain it?

What about that quote don't you understand?

Does it mean if my standard deviation is say 2.5 that it can go up to three times higher like 7.5?

Values from a probability distribution must follow certain "guidelines" dictated by its underlying standard deviation. You can look up Chebyshev's inequality and a myriad of central limit theorems on the internet if you are interested.

The three-sigma rule you quoted says that a certain (high) percentage of "mass" (observations) must occur within three standard deviations from the mean of a distribution.

I don't know if that helps or not.

No.

I suggest your real question, even if you don't realize it, is "what does standard deviation mean, and where does it come from?"

Thank you for explaining but to be honest a lot of those words that you used are way over my understanding sorry, I'll watch some YouTube videos to see if I can understand it that way.

Is this video the Chebyshev's inequality and a myriad of central limit theorems you mention?

Yeah its a bit confusing tbh as I don't have much maths understanding.

But I been playing around with excel to try and understand it like entering 1, -1 stdev = 1.41, and 1 -2 = 2.12


So from that and google, I understand its how much the members of a group differ from the mean value for the group.

And I assume mean is 0? As in 1,1 = 0

If I understand the basics correctly.

Yes, standard deviation reflects how spread out are the data you observe (or the underlying probability distribution "governing" how data are generated).

So, as you say, if I play in a low-stakes poker game, my daily winnings over a 7-day period may be:

Me: +5, -3, +2, +4, -1, +1, -1.

You, on the other hand, may play in a high-stakes poker game with the following daily winnings over a 7-day period:

You: +500, -300, +200, +400, -100, +100, -100.

Clearly, your daily winnings are more spread out than mine. So the standard deviation of your daily winnings will be much larger than the standard deviation of my daily winnings.

The three-sigma rule says that, if we each play for a large number of days, around 99% or more of each of our respective daily winning amounts will occur within three standard deviations of our respective mean (average) daily winning amount.

This result comes from the definition of standard deviation and the nature of how distributions behave with a sufficiently large number of observations.

If you have any further questions, do not hesitate to ask.

OK this above I understand.

5 500
-3 -300
2 200
4 400
-1 -100
1 100
-1 -100
2.89 288.68





But this below I think I do understand but I'm not certain I do




So you're saying its only a definition, and not some extra added calculations on top of the standard deviation calculations I need to include /calculate further?

As I thought it was some more calculations I need to add to the already standard deviation that I use in excel, but as I'm understanding it now, its just a definition only.

To follow this example further, those sigma-rules are helpful for some people to better understand the ranges of their likely daily winnings.

For example, take "You" in the example above. Say your average daily winning is 100 with a standard deviation of 300.

The "one-sigma" rule suggests that around 68% of the time your daily winnings will be between -200 and +400.

The "two-sigma" rule suggests that around 95% of the time your daily winnings will be between -500 and +700.

The "three-sigma" rule suggests that around 99% of the time your daily winnings will be between -800 and +1000.

TYVM!

I was thinking about this example you mentioned..

5 500
-3 -300
2 200
4 400
-1 -100
1 100
-1 -100
2.89 288.68


Where the smaller the betting equals smaller std dev and bigger the betting equals bigger std dev.

And was wondering why are poker players std dev similar for same game type between smallest stakes and biggest stakes unlike the top example.

For example Heads Up Pot Limit Omaha, most players std dev is around the 200, for stakes played at plohu $0.25/$0.50 or plohu $200/$400?

sorry not betting but values/numbers

but isn't that just in game swings and pots going back and fourth like say in a $1/$2 game

6, -10, 6, 42, -28 etc?

meaning the player won a $6 pot followed by losing a $10 pot then won a $6 pot and so on