General Formula for Standard Deviation
Warning, I have spend a good while preparing this post. It is fairly long and fairly heavy in algebra, but my hope is that everyone can learn from my thinking and hopefully I can get a formula to find Standard Deviation for any betting style, counting system, or anyone.
Ok. Here it goes. I've tried to make a generic formula to calculate Standard Deviation for entire playing sessions based on TC frequencies, for any given bet ramp or playing style.
First, I'm going to define my variables:
b = bet (This will change with each TC.
Where b*b simply means b squared, I didn't care to try to write the superscript 2.)
v = (1.33 + (n-1)*0.5)*n (With n = the number simultaneous hands of one bet, b. I am assuming that the simultaneous hands will be the same bet i.e. 2 hands of 6 units, 3 hands 4 units. Both simultaneous hands will have the exact same bet. You must calculate v before plugging it into the formula. Also, note that if you don't spread to two hands than v = 1.33.)
h = hands played (If you want to use hours simply multiply hours by an average of 100 hands per hour. I think the general consensus is that 100 hands per hour is an accurate estimate.)
fc = the frequencies of c, where c = the actual true count (Here is where it gets tricky. This number will represent the percentage of hands play at a certain TC for your count. Since you are betting according to the TC and every TC will have a different bet out, you want the correct bet to be attached to the correct frequency. Also, since most negative counts will be played with 1 unit, you can simply add all of the negative count frequencies together and use the total negative count frequency as the coefficient. If you don't get this now, I'll explain it again after the entire formula is presented.)
Now that all of the variables have stated here is the actual formula:
Total Session Standard Deviation = (f <=-3 * sqrt( b * b * v *h)) + (f-2 * sqrt( b*b*v*h)) + (f-1 * sqrt(b*b*v*h)) + (f0 *sqrt(b*b*v*h)) + (f1 * sqrt(b*b*v*h)) + (f2 * sqrt(b*b*v*h)) + (f3 * sqrt(b*b*v*h)) + (f4 * sqrt(b*b*v*h))+ (f5 * sqrt(b*b*v*h)) + (f6 * sqrt(b*b*v*h)) + (f7 * sqrt(b*b*v*h)) + (f >=8 * sqrt(b*b*v*h))
So, if I wanted to calculate the Total Standard Deviation for 10 hands of flat betting 1 unit, simply plug in:
b = 1
v = 1.33 (since we won't ever be spreading to two hands its simply 1.33)
h = 10
f = 100% (since 100% of the hands will have the same bet, 1 unit.)
Total Session Standard Deviation = 1 * sqrt(1*1*1.33*10)
Total Session Standard Deviation = 3.6469 units.
If I wanted to calculate a 1-10 bet spread for Zen after playing 10 hours, betting 1 unit at a TC of <=1, 2 units at TC of 2, 3 units at TC of 3, 4 units at TC 4, 5 units at TC 5, and spreading to two hands of 5 units at TC >= 6 (I know this isn't an optimal/realistic spread I'm just using it as an example). I'm using the frequencies provided by bjcount above. The formula would look like this:
b = bet at each TC. So in this example, we will be betting 1 unit 74.17% of the time, 2 units 7.58% of the time, 3 units 4.65% of the time, 4 units 3.97% of the time, 5 units 2.42% of the time, and 2 hands of 5 units 7.19% of the time (the frequencies add up to 99.98% of hands due to rounding error, but I'm ok with that for the example). Now we want to make sure each bet corresponds with each frequency.
v = 1.33 (Except when we spread to two hands shown below, we also want to make sure that the right variance will be corresponding with each frequency.)
For the 7.19% of hands we play two hands of 5 units, we calculate v like this:
v = 1.33 + (n-1)*0.5
Since we only spread to two hands, n = 2:
v= (1.33 + (2-1)*0.5) * 2
v= (1.33 + (1)*0.5) * 2
v= (1.83) *2
v= 3.66
Remember v only equals 1.83 for when f = .0719
h = 10 hours * 100 hands per hour
h= 1000 hands
Now f = the frequency (in decimal form) of each bet.
So our formula looks like this:
Total Session Standard Deviation = (.7417*sqrt(1*1*1.33*1000)) + (.0758*sqrt(2*2*1.33*1000)) + (.0465*sqrt(3*3*1.33*1000)) + (.0397*sqrt(4*4*1.33*1000)) + (.0242*sqrt(5*5*1.33*1000)) + (.0719*sqrt(5*5*3.66*1000))
Which gives you:
Total Session Standard Deviation = 69.618 units
You simply add as many functions as needed i.e. (fc*sqrt(b*b*v*h))+
Please let me know if my thinking is flawed or if I have any arithmetic issues. If my thinking is flawed please tell me how so I can learn from my mistakes. Also note that I have spent several hours coming up with this logic, it has been well thought out and not just slapped together.