The following is something I sent a WKNH (any gueses?) over PM, after he asked a similar question...
OK... Standard Deviation is the measure of spread in a normal population.
So, that asks the first question: whats a normal population?? A normal population is a set of data, that basically fits into a bell curve. I pulled this picture off of the web:
This is a very run of the mill normal curve.
Im not sure exactly what this represents, but lets assume it measures earthworm length (I am a bio major afterall).
The X-axis (the numbers on the bottom (2-18)) represent length. The Y-axis (they dont have numbers on it) represents the probability of picking an earthworm of that length (from a large pool or earthworms)
Do you see how it has a maximum at 10? That is because 10 is the mean of this population. That is, there are more earthworms that are 10 inches long than any other length.
OK... now, how does standard deviation relate to this?
Here are two normal curves I made up...
The first has a mean of 2 BB/100, with a SD = 16BB/100. The second has the same mean, but a SD = 8BB/100.
(the colours will be explained later)
What these graphs measure is the probability that you will win X BBs in your next 100 hands.
If you notice, at the peak of the first you will find that there is about a 2.5% chance that you will win 2BB in your next 100 hands. But, in the second, it is close to 5%. Also, in the first you will notice that it is possible to lose 30 BBs in 100 hands, but in the second, it becomes essentially impossible.
With a bigger standard deviation, simply put, you have bigger swings.
Now, what about the colours?
The dark red represents one standard deviation about the mean.
So, for the first... since the mean = 2BB/100, and SD = 16BB/100, this range =
2 +/- 16 =
-14BB -> 18BB
You can expect data to fall within this range about 68% of the time.
Put another way: if you have a winrate of 2BB/100 and a SD of 16BB/100, there is about a 68% chance that, in your next 100 hands, you will win between -14BB and 18BB.
The second colour represents two standard deviations about the mean. For the winrate = 2, SD = 16, this represents the range:
-30BB --> 34 BB.
There is about a 95% chance that the data falls in here. Put another way: there is a 95% chance that, in your next 100 hands, you will win between -30BB and 34BB.
Now... on calculating a range of your winrate...
In general, 95% confidence is about what most people use, so we will use that.
Here is the forumla:
winrate +/- 2*SD / ((#hands/100)^1/2)
Which looks complicated, but its really not.
Lets say you have played 51,000 hands.
Your winrate = 2BB/100.
Your SD = 16BB/100
here's how the forumla works...
Take your # of hands, and divide by 100. This = 510.
Take the square root of this number (510). This = 22.58
Now, calculate 2*SD/(this number)...
that is, 2*16/22.58 = 1.417
This is the number you add and subtract...
therefore, this player can be 95% sure his winrate is in the range:
2 +- 1.417
0.583 -> 3.417
We really cant be all that sure of anything at this point, ya know?
We can be pretty sure he is a winning player, but thats about it.
Just goes to show what those people who claim massive winrates after 15k hands really know...
PM me back if you have any questions / anything I didnt cover.
Also, I hope the images work, I've had problems in the past
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