Adding standard deviations together
- Thread starter matt21
- Start date
callipygian
Well-Known Member
It's not, really. EV is expected value - expected value. Expected value correlates, but is difficult to determine from, actual winnings. Unless you have an absolute ton of data, your actual winnings will reflect variance more than your EV.matt21 said:
Your winnings can be used to estimate your EV, but that's hard to pin down an exact number.
Yes. First of all, there's a question of how you're calculating SD to begin with. If you're playing on a computer which is tracking your results by hand, that's probably okay. SD(total) = sqrt(SD(1)^2 + SD(2)^2 + SD(3)^2).matt21 said:
But more often, SD is calculated from session wins, which means you calculate SD(total) = sqrt((ActualWin(1)-ExpectedWin(1))^2 + (ActualWin(2)-ExpectedWin(2))^2 + ... (ActualWin(n)-ExpectedWin(n))^2).
callypigan, thanks for your reply.
The EV figures are based on simmed results for various playing conditions that i encounter. Likewise the SD is calculate from the sim results. Apologies for not mentioning this. On my results log I calculate EV for every session (using CVD) and then add together all the session EV's to calculate my EV for the day/week/month.
So can I check whether i got your point re adding the SD's?
Assume the following EV's and SD's, determined using CVD ahead of your weekend's playing schedule:
session A: EV 100/hr SD 2000/hr - for 3 hours
session B: EV 120/hr SD 2300/hr - for 1 hour
session C: EV 85/hr SD 1700/hr - for 10 hours
thus:
Session A:
EV= 3 x100 = 300
SD = sqrt(3) x 2000 = 3,464
Session B:
EV = 1 x 120 = 120
SD = sqrt(1) x 2300 = 2,300
Session C:
EV = 85 x 10 = 850
SD = sqrt (10) x 1700 = 5,376
Total EV (A,B,C) = 300+120+850 = 1,270
SD for the 14 hours of play...
sqrt [SD(a)^2+SD(b)^2+SD(c)^2]
= sqrt [3464^2+2,300^2+5,376^2]
= 6,796
so I have EV of $1,270 with SD of $6,796 for the proposed 14-hr playing schedule.
is that the right answer?
The EV figures are based on simmed results for various playing conditions that i encounter. Likewise the SD is calculate from the sim results. Apologies for not mentioning this. On my results log I calculate EV for every session (using CVD) and then add together all the session EV's to calculate my EV for the day/week/month.
So can I check whether i got your point re adding the SD's?
Assume the following EV's and SD's, determined using CVD ahead of your weekend's playing schedule:
session A: EV 100/hr SD 2000/hr - for 3 hours
session B: EV 120/hr SD 2300/hr - for 1 hour
session C: EV 85/hr SD 1700/hr - for 10 hours
thus:
Session A:
EV= 3 x100 = 300
SD = sqrt(3) x 2000 = 3,464
Session B:
EV = 1 x 120 = 120
SD = sqrt(1) x 2300 = 2,300
Session C:
EV = 85 x 10 = 850
SD = sqrt (10) x 1700 = 5,376
Total EV (A,B,C) = 300+120+850 = 1,270
SD for the 14 hours of play...
sqrt [SD(a)^2+SD(b)^2+SD(c)^2]
= sqrt [3464^2+2,300^2+5,376^2]
= 6,796
so I have EV of $1,270 with SD of $6,796 for the proposed 14-hr playing schedule.
is that the right answer?
callipygian
Well-Known Member
As far as I know, yes. I could be wrong, though, so get confirmation.matt21 said:
I think so.matt21 said:
What I would do, since both variance and EV are additive, is just add the variance and EV per physical round and take the square root of the total variance at the end to get SD and forget about the hourly stuff.
I think it would be the same but maybe stuff like not playing 100 or 80 or 60/hds/hr every hour, back-counting, spreading etc could maybe complicate it.
Or not lol.
So, I'd do like, in first example, 2000*2000=4000000 variance per hr * 3 hrs=12000000. Sqrt of that = 3464 like you say. Etc.
You liike stuff "per/hr". I like stuff "per/round" because I am more limited than you are. But I like you forcing me to think more in "hourly" terms.
All rounds are created equal but sometimes I'm less sure that all "hours" are, in "real-life" as equal lol.
i posted some of the discussion below on another thread, but am hoping that it will get more response in the Maths/Theory section!
Basically this is a real life scenario of adding standard deviations together, i.e. i am trying to calculate the std dev over a sample of actual completed sesssions.
The attachment in which the data is contained is on this thread
http://www.blackjackinfo.com/bb/showthread.php?t=13026&page=5
My approach was:
The shoe-SD was calculated with my own probability models however the figures agree with simulations run in CVCX/CVD.
Next I calculated the session-SD by multiplying the shoe-SD by the square root of the number of shoes. If there was a mix of conditions then i squared the SD of each condition set to get the variance, and then took the square root of the sum of the variances to get the session-SD.
Finally I squared the session-SD to get session-variances, then summed all the variances and then took the square root of that sum – producing a 120-hr-SD based on actual playing conditions.
Example/Illustration
Session 332 – Condition1: EV/Shoe – 0.43 units, SD/shoe 18.26 units, SD (3 shoes) 31.63 units, EV (3 shoes) 1.3 units
Session 332 – Condition2: EV/Shoe – 0.59 units, SD/shoe 22.08 units, SD (5 shoes) 49.38 units, EV (5 shoes) 2.97 units
Session-SD for Session 332: Sqrt (31.63^2 + 49.38^2) = 58.64 units
Session-EV = 1.3+2.97= 4.27 units
Eventually I bundle all the session together. So for #332 I take 58.64^2 and add it to all the squared SD figures for all the other sessions. For the 120 hours, my EV was 413 units. I calculated the SD as 660 units. My actual result was a win of 712.1 units.
My ultimate goal is to determine whether my SD are in fact tighter than I have calculated - if they are then I would be more motivated to play in the future.
Can someone please tell me whether I calculated the SD correctly?
Basically this is a real life scenario of adding standard deviations together, i.e. i am trying to calculate the std dev over a sample of actual completed sesssions.
The attachment in which the data is contained is on this thread
http://www.blackjackinfo.com/bb/showthread.php?t=13026&page=5
My approach was:
The shoe-SD was calculated with my own probability models however the figures agree with simulations run in CVCX/CVD.
Next I calculated the session-SD by multiplying the shoe-SD by the square root of the number of shoes. If there was a mix of conditions then i squared the SD of each condition set to get the variance, and then took the square root of the sum of the variances to get the session-SD.
Finally I squared the session-SD to get session-variances, then summed all the variances and then took the square root of that sum – producing a 120-hr-SD based on actual playing conditions.
Example/Illustration
Session 332 – Condition1: EV/Shoe – 0.43 units, SD/shoe 18.26 units, SD (3 shoes) 31.63 units, EV (3 shoes) 1.3 units
Session 332 – Condition2: EV/Shoe – 0.59 units, SD/shoe 22.08 units, SD (5 shoes) 49.38 units, EV (5 shoes) 2.97 units
Session-SD for Session 332: Sqrt (31.63^2 + 49.38^2) = 58.64 units
Session-EV = 1.3+2.97= 4.27 units
Eventually I bundle all the session together. So for #332 I take 58.64^2 and add it to all the squared SD figures for all the other sessions. For the 120 hours, my EV was 413 units. I calculated the SD as 660 units. My actual result was a win of 712.1 units.
My ultimate goal is to determine whether my SD are in fact tighter than I have calculated - if they are then I would be more motivated to play in the future.
Can someone please tell me whether I calculated the SD correctly?
Hi Mattmatt21 said:i posted some of the discussion below on another thread, but am hoping that it will get more response in the Maths/Theory section!
Basically this is a real life scenario of adding standard deviations together, i.e. i am trying to calculate the std dev over a sample of actual completed sesssions.
The attachment in which the data is contained is on this thread
http://www.blackjackinfo.com/bb/showthread.php?t=13026&page=5
My approach was:
The shoe-SD was calculated with my own probability models however the figures agree with simulations run in CVCX/CVD.
Next I calculated the session-SD by multiplying the shoe-SD by the square root of the number of shoes. If there was a mix of conditions then i squared the SD of each condition set to get the variance, and then took the square root of the sum of the variances to get the session-SD.
Finally I squared the session-SD to get session-variances, then summed all the variances and then took the square root of that sum – producing a 120-hr-SD based on actual playing conditions.
Example/Illustration
Session 332 – Condition1: EV/Shoe – 0.43 units, SD/shoe 18.26 units, SD (3 shoes) 31.63 units, EV (3 shoes) 1.3 units
Session 332 – Condition2: EV/Shoe – 0.59 units, SD/shoe 22.08 units, SD (5 shoes) 49.38 units, EV (5 shoes) 2.97 units
Session-SD for Session 332: Sqrt (31.63^2 + 49.38^2) = 58.64 units
Session-EV = 1.3+2.97= 4.27 units
Eventually I bundle all the session together. So for #332 I take 58.64^2 and add it to all the squared SD figures for all the other sessions. For the 120 hours, my EV was 413 units. I calculated the SD as 660 units. My actual result was a win of 712.1 units.
My ultimate goal is to determine whether my SD are in fact tighter than I have calculated - if they are then I would be more motivated to play in the future.
Can someone please tell me whether I calculated the SD correctly?
Yes I get the same 660 unit SD you did for that sheet you posted.
FWIW, I get an N0 of 59,351 hands for all those sessions as a whole. (I think you asked that but I don't see it here?) Where EV and 1 SD will both be 1052 units.
To KewlJ's point of playing so many hds/hr, I can only assume that you are spreading to multiple hands at certain times. Surely you must be, to some extent, in sessions 335, 336, 339, 342 etc where your variance per hand triples or quadruples?
So my general question is, if in fact you may be spreading, what do you mean by "hands"? When, and if, you spread, do you count that as 2 "hands" or 1 "round"? Is unit-SD and unit-EV expressed in "hands" or "rounds"?
Don't ask me for answers lol as I don't even know, for sure, how QFIT's software expresses stuff - sometimes I get it in a posted spreading sim and sometimes I don't lol.
I know you seem to think in terms of "shoes" and derive "hands played" from that, which I'm sure is fine, but I tend to think in terms of "physical rounds played" (number of dealer upcards played against) even if spreading. Or backcounting. Do you ever not play a full shoe or back-count? Like whether spreading to 2,3 or more hands in a round is still only 1 round and I like to express SD per round not per hand. Maybe just me - and I'm not saying there isn't a way to express it per "hand" either.
Guess I'm just trying to say as long as you understand fully what's up, I'm down with it lol.
Also, just another question to you or anyone (Sonny? help lol) about you using "units" in your sheet, in those different "conditions" that occur from time to time does your $unit never change? If it were to change from time to time would one have to also, say in the sheet you posted assuming such a thing occurred, also have to derive some avg $unit on the off-chance one wanted to know how many dollars 660 units of SD represented?
Hi Kasi, here's answers to your questions
Like this discussion & look forward to your next reply (see question in next post!)
In those sessions I exploited a promotion that was run by a local joint. The EV effectively doubled, however the SD increase (and thus variance) was less than double. I.e. the N0 for the promotion 'abuse' was significantly better than for my normal counting. I had actually consolidated counting and promotion abuse within the same sitting.Kasi said:
I generally play 1 or 2 hands per round. If I play 1 round with 2 hands, then that counts as 2 hands. Thus far, i have never expressed EV or SD on a hand-basis or round-basis. I have always calculated it on a shoe-basis with my own models and then cross-referenced to CVD to ensure that the values were pretty similar.Kasi said:
For me 'shoes' works best because i tend to play-all. No backcounting.Kasi said:
In my case I had actually done all calculations in $ but then divided the $ by a nominal amount to arrive at a unit-expression. This also worked here because my bet ramp has been the same throughout the 120 hours of play.Kasi said:
Like this discussion & look forward to your next reply (see question in next post!)
kasi many thanks for your reply!
i copied part of my post on the 'my final conclusion'-thread over here in the Maths section - thats why some of my earlier questions have disappeared.
ok, glad that you got the same result as me ie SD=660 units.
how did you calculate N0? i understand that it's the number of hands where EV=1SD, but how did you calculate it given there is no fixed number of hands associated with the sample of 120 hours of play?
now my next questionfor you kasi is this:
over the 56 sessions, in only 3 of these sessions was the actual result more than one standard deviation away from the session EV. I had thought that about 32% of sessions (say about 16) should be more than one standard deviation from the session EV. Does this throw doubt onto the accuracy of the calculated SD figures. (the 32% is calculated by the area under the normal curve that lies outside +/- 1SD from the mean = 100%-68%)
does it surprise that so few sessions exhibit significant deviation from the EV?
i copied part of my post on the 'my final conclusion'-thread over here in the Maths section - thats why some of my earlier questions have disappeared.
ok, glad that you got the same result as me ie SD=660 units.
how did you calculate N0? i understand that it's the number of hands where EV=1SD, but how did you calculate it given there is no fixed number of hands associated with the sample of 120 hours of play?
now my next questionfor you kasi is this:
over the 56 sessions, in only 3 of these sessions was the actual result more than one standard deviation away from the session EV. I had thought that about 32% of sessions (say about 16) should be more than one standard deviation from the session EV. Does this throw doubt onto the accuracy of the calculated SD figures. (the 32% is calculated by the area under the normal curve that lies outside +/- 1SD from the mean = 100%-68%)
does it surprise that so few sessions exhibit significant deviation from the EV?
Hi Mattmatt21 said:
As far as N0 goes, not sure what you mean by saying there were no fixed number of hands associated with the 120 hrs of play. It seems you assumed 23,359 hands were associated with that overall play? In any case I assumed that number of hands lol. So "=(SQRT(435371/23359)/(413.9/23359))^2"= 59364 hds for N0. Basically, avg SD/hd/ divided by avg EV/hd squared, same as always. I guess you could also express it in shoes, hrs whatever. Basically an N0 treating the 120 hrs as if they were one game, even though the 120 hrs represent many different games, each of which has its own N0, all blended into one.
So, while you say you never do things per "hd", in fact, you do by inference.
As far as your number of sessions that fall outside 1 SD, I didn't really address that because I am clueless lol. But, yes, it seems a little unusual to me. You might be over-stating "hds played" for example. I still have a bad feeling on this spreading stuff although I don't know why lol. QFIT's stuff does not do things by shoe so I don't know why bother converting everything to shoes in the first place lol. Also, when spreading, when QFIT's stuff says "SD/hr" does he mean using "SD/rd" or "SD/hd"? When he says "EV" is it per "hd" or "rd"? (Not that I know lol).
Over time, being inaccurate about physical rds played or mis-interpreting what a sim may be telling you can certainly effect where one's actual results apparently fall on the curve.
Even in the simplest case, no spreading, alot of sims express stuff "hrly", assuming 100 "hds/hr", but the player may only keep track of hours for his records yet play from full tables to heads-up and make no adjustment. 300 "hrs" later God alone knows how many "hds" he has played.
What if you were to re-do these 118.5 hrs on a "round" basis assuming Qfit's stuff, when spreading, may be expressed on a "round" basis rather than a hand basis?
Also, I'd hate combining promo stuff into the play of my counting system. Just as I'd hate counting some $1000 bet I happened to have won into my overall results when such a bet was never called for by my "system". Maybe set such results on the side but don't confuse them with "system" results, maybe lol.
So, maybe, yes, lol, when results are maybe way outside of expected, as your winning sessions seem to be, perhaps, somewhere, somehow, there is something rotten in Denmark.
I'd probably at the very least re-examine everything from the ground up with an open mind to re-assure yourself all is kosher.
All this "sometimes 1 hand, sometimes 2", in the same shoe apparently, is so confusing and makes it so potentially difficult to measure results later.
Is it so impossible to ask of a counter to at least play a whole shoe consistently to make one's "measuring" later so much easier simply because such shoe and betting scheme relates to a sim? Or, at least, if not, have the confidence to know what he is doing before he does it?
What with this being theory and all, maybe, since this spreading stuff is what's bugging me, maybe post a sim, or sims, for some theoretical shoe you spread in or for a shoe you both spread and don't in.
Nothing on you Matt - you're doing it and I ain't lol.
Hats off to you.
Hey kasi! sorry a day has lapsed and i have not replied!
And I do actually play each shoe consistently. This has allowed me to be aware of my EV. However it is important that I keep track of the game variables, such as pen #shoes and #other players.
Ok great, thanks for explaining!Kasi said:
I am tracking #of shoes played and calculating 'hands played' by pen, #players and average cards per hand - if i am tracking the variables correctly then my 'hands played' figure is correct. But maybe i am tracking one or more variables incorrectly?Kasi said:
I was able to get QFIT's software to express results per shoe (by adjusting the number of rounds per hour and knowing how many rounds there would be in a shoe in any given playing condition), and then assumed i was playing just 1 shoe per hour - this way i got the simulation to give me exactly what i was looking for.Kasi said:
Agree 100%Kasi said:
That would be an interesting exercise i think.Kasi said:
Actually this wasnt a problem at all. It was very easy to consolidate the two into one from both a playing and a modelling perspective. And it was a BIG FAT WINNER :joker::joker: unfortunately probably a rare situation.... but I'm on the look-out!Kasi said:
Yes, you are right. I mean overall my winnings are well within expectation, but the individual session results just vary much less than expected. If I say, consistently overstated pen, then I would be overstated session-SD and this would explain why my session-results dont appear to vary much, however that would also mean that i overstated EV, and that my actual result for the 120 hours is then much better than expected, rather than within one SD. Hmmm, something to think about.Kasi said:
Sometimes 1, sometimes 2 - sorry i was meaning 2 hands at TC2 or above, 1 hand otherwise. With the SIM i am aware of the TC frequency distribution and therefore will know how often i will be betting 1 or 2 hands. Of course this impacts on the total number of hands for the shoe - as the player is now taking a larger proportion of the cards against the dealer. Provided for all that in my model and CVD obviously does too.Kasi said:
LOL Now you have made me laugh :grin:Kasi said:
And I do actually play each shoe consistently. This has allowed me to be aware of my EV. However it is important that I keep track of the game variables, such as pen #shoes and #other players.
I love the theoretical aspects of the game. When I started playing with these I realised that these things are one set of perfomance predictors (bet ramp and ROR is what we didnt discuss above) and i realised how much difference there was between the different playing conditions. So I just started looking for playing conditions that gave me the statistical variables that I wanted - my EV and actual results have increased dramatically since then.Kasi said:
No big deal Matt.matt21 said:
Guess I still wonder a little about "rounds" vs "hands" since, like you say, when you talk about QFIT's software you say "rounds" but your sheet only says "hands".
So how do you convert QFIT's "rounds" SD to your sheet's "hands" SD what with co-variance and all?
If Qfit's stuff is in "rounds" why not stick with "rounds" in your analysis of your play?
Good chance maybe I make too much of these semantics lol.
Maybe you post a QFIT sim of your choice and how you would translate that into your sheets just for, say, 1 shoe assuming you actually played that way for a whole shoe lol.
I'm never quite sure what QFIT's stuff is expressing when spreading is involved in the first place lol. So, no doubt, I'd learn something.
Like it's all about you lmao.
hi kasiKasi said:
i am using QFIT's software to get the SD/EV per shoe, not per round. Although i list the hands in the attachment, they are not actually used in calculating the SD. Both the number of hands and the SD in the attachment are derived from the number of shoes. So I dont have to do any converting.
The software is actually great to work with. I have input by bet ramp to spread to 2 hands at TC2+, and the software automatically takes this into account in calculating the shoe or hour SD. So I wouldnt get hung up on understanding the SD per round or per hand.Kasi said:
You know it would be great if you and i could sit down next to each other one day and talk about this in all its details :grin:.
okay here's another question:
let's say you play a game with given conditions
50% of the time you use approach A
Approach A - EV 0.35u SD 16.25u
50% of the time you use approach B
Approach A - EV 1.53u SD 24.5u
Does this mean that your EV/SD for this game will be:
EV = 0.5x0.35 + 0.5x1.53 = 0.94u
SD = sqrt (Sum of Variances x probability of occurring)
= sqrt (0.5*16.25^2 + 0.5*24.5^2)
= sqrt (432.15)
= 20.8u
Thanks in advance to anyone that can confirm this!
Happy counting!
Matt
let's say you play a game with given conditions
50% of the time you use approach A
Approach A - EV 0.35u SD 16.25u
50% of the time you use approach B
Approach A - EV 1.53u SD 24.5u
Does this mean that your EV/SD for this game will be:
EV = 0.5x0.35 + 0.5x1.53 = 0.94u
SD = sqrt (Sum of Variances x probability of occurring)
= sqrt (0.5*16.25^2 + 0.5*24.5^2)
= sqrt (432.15)
= 20.8u
Thanks in advance to anyone that can confirm this!
Happy counting!
Matt
My cat just walked across my keyboard and sent me in to a different world lol so I have no idea what I may have posted and where.matt21 said:
Anyway, if you're happy, I'm happy.
All I think I know is CVCX does not change his descriptions of "win/hd" or "sd/hd" whether one spreads or not.
If you understand exactly what you're looking at when you look at a CVCX sim, great - just seemed an area to explore as to why your number of winning "sessions" within 1 SD of EV is so high. It (your results) does seem a little out of the ordinary to me, as it also seems it does to you, so, you tell me why lol. Or, we just chalk it up to "variance" and tune back in with another 120 (and 240) hours of results lol.
I still don't get, why, if you have, somehow, have CVCX expressing stuff in terms of "shoes" why you even bother converting to "hands" in your sheet. Why not just express everything in terms of "shoes"?
Whatever, I have never been able to play jack without, somewhere in the back of my mind, knowing how'd I'd go about comparing what actually happened to what was suppose to happen.
Very easy for me to do on the internet - how you real-life guys do it remains a subject of great interest to me lol.
Don't worry - it's only me who has alot of questions on being able to understand better "spreading sims" by CVCX.
OK - I lied - glad you're happy - I'm miserable lmao.