Standard Deviation Calculation

[DSchles]

Huh. In my copy of BJA3 (ISBN 0-910575-20-7), the footnote on p.20 reads only, "* Note: h = number of simultaneous hands played." The header for the fifth (of six) columns is labeled "Var+(h-1)*Cov."

As I wrote, the book was reprinted last year, and that footnote was changed to correspond precisely to Griffin's page 142 formula, on which you haven't commented.

Interesting. Then it's worth noting for Norm that CVCX is also computing overall variance in the same incorrect way as described in the book (although the individual variances conditioned on true count, which seem to be the focus of OP's confusion, are correct).

I can remember Michael Canjar's once writing something to the effect that third and fourth moments (skew and kurtosis) were virtually nonexistent for blackjack, so I'm wondering if the effect you're commenting on is just of theoretical interest or if it has any practical importance.

Don

[ericfarmer]

As I wrote, the book was reprinted last year, and that footnote was changed to correspond precisely to Griffin's page 142 formula, on which you haven't commented.

Sorry, I did gloss over the reference to Griffin's formula, which is indeed correct-- at least in form. That is, the 0.50 covariance seems to be dragged along as a "constant" into current calculations, when it isn't clear to me that this is justified (and from your follow-on comment about Wong's data, it probably isn't).

I can remember Michael Canjar's once writing something to the effect that third and fourth moments (skew and kurtosis) were virtually nonexistent for blackjack, so I'm wondering if the effect you're commenting on is just of theoretical interest or if it has any practical importance.

Don

This is a good point; the difference depends on how the variance is being used. As we've discussed in past email exchange, if it's being used, for example, to compute SCOREs, win rates, etc., then it's definitely of practical importance, causing a difference of roughly 6-7% errors in win rate for the 6D study from a couple of years ago.

What is interesting is that, unfortunately, the error in the table (and apparently thus in CVCX) is not easy to correct without more detailed data. That is, we really need the *distributions* of outcomes in each true count bin, not just the EVs and variances, to correctly compute the overall variance.

E

[DSchles]

Sorry, I did gloss over the reference to Griffin's formula, which is indeed correct-- at least in form. That is, the 0.50 covariance seems to be dragged along as a "constant" into current calculations, when it isn't clear to me that this is justified (and from your follow-on comment about Wong's data, it probably isn't).

Right. It wasn't unknown to me or Peter that both the per-hand variances AND covariances change a bit from line to line. To be more precise, one should use the values that are appropriate for each true count in question. Clearly, that is not what was done on page 20, nor for Griffin's one-size-fits-all formula. But, to do so would have greatly complicated both demonstrations, and, apparently, neither of us deemed it necessary to do so.

This is a good point; the difference depends on how the variance is being used. As we've discussed in past email exchange, if it's being used, for example, to compute SCOREs, win rates, etc., then it's definitely of practical importance, causing a difference of roughly 6-7% errors in win rate for the 6D study from a couple of years ago.

I'm not getting this part. The overall variance shouldn't cause an error in win rates or SCOREs, because, as you mentioned earlier, the line-by-line individual variances are correct. And again, there are minor differences in the covariances, as true count varies, but I don't think that's what your point is. So, the line-by-line optimal bets are correct (if the edges and individual variances are). Therefore, the win rates should be right, as they are simply the aggregate of the line-by-line results. SCORE is the hourly win rate for an optimal bettor with a $10,000 bankroll, so how is SCORE affected by your change in overall variance?

Don

[ericfarmer]

I'm not getting this part. The overall variance shouldn't cause an error in win rates or SCOREs, because, as you mentioned earlier, the line-by-line individual variances are correct. And again, there are minor differences in the covariances, as true count varies, but I don't think that's what your point is. So, the line-by-line optimal bets are correct (if the edges and individual variances are). Therefore, the win rates should be right, as they are simply the aggregate of the line-by-line results. SCORE is the hourly win rate for an optimal bettor with a $10,000 bankroll, so how is SCORE affected by your change in overall variance?

Don

You are absolutely right, I misspoke. The root problem is really in computing RoR, and thus only indirectly on *optimal* (risk-constrained) win rate. That is, suppose that we have fixed our playing strategy, and want to select a betting ramp to maximize win rate, subject to a pre-specified risk of ruin. If we are computing (as we have done in recent studies) the ordered pair (RoR, win rate) for each of a collection of possible betting ramps, and are getting that RoR calculation wrong (due to incorrectly computing per-round variance), then as a result we select the wrong "best" win rate as well. How much the incorrect variance matters depends on the "slope" of feasible win rate vs. RoR... in the recent study this made about a 6-7% difference.

[DSchles]

You are absolutely right, I misspoke. The root problem is really in computing RoR, and thus only indirectly on *optimal* (risk-constrained) win rate. That is, suppose that we have fixed our playing strategy, and want to select a betting ramp to maximize win rate, subject to a pre-specified risk of ruin. If we are computing (as we have done in recent studies) the ordered pair (RoR, win rate) for each of a collection of possible betting ramps, and are getting that RoR calculation wrong (due to incorrectly computing per-round variance), then as a result we select the wrong "best" win rate as well. How much the incorrect variance matters depends on the "slope" of feasible win rate vs. RoR... in the recent study this made about a 6-7% difference.

OK, understood. But, as you state, this is a somewhat different consideration.

My main interest now would be knowing, for any calculation that follows the page 20 methodology, what the lack of using the cross terms contributes to the ultimate calculation of overall variance. My sense is that the discrepancy ought to be minor or negligible. But, I'm getting that you think otherwise, and so I'd like to determine the magnitude of any error.

Obviously, I understand that this will vary with different scenarios. Note, as well, that this doesn't affect the Chapter 10 studies, as they don't involve playing two hands.

Don

[ericfarmer]

OK, understood. But, as you state, this is a somewhat different consideration.

My main interest now would be knowing, for any calculation that follows the page 20 methodology, what the lack of using the cross terms contributes to the ultimate calculation of overall variance. My sense is that the discrepancy ought to be minor or negligible. But, I'm getting that you think otherwise, and so I'd like to determine the magnitude of any error.

Obviously, I understand that this will vary with different scenarios. Note, as well, that this doesn't affect the Chapter 10 studies, as they don't involve playing two hands.

Don

My memory hasn't been very good-- as I work through these calculations again, I find that I need to correct myself yet again: this *does* affect SCORE (since we're talking about computing variance). Note that this implies that it *would* affect Chapter 10 results as well, at least assuming that variance is computed in the weighted-sum manner indicated in Table 2.1-- this is about partitioning the space of outcomes by true count, not how many hands are played within the round.

Having said that, the magnitude of the error is small, typically only affecting the decimal places of SCORE. I have a single anecdote from a past study where this error came up, where the SCORE of 31.1135 was corrected to 31.1018.

E

[DSchles]

My memory hasn't been very good-- as I work through these calculations again, I find that I need to correct myself yet again: this *does* affect SCORE (since we're talking about computing variance). Note that this implies that it *would* affect Chapter 10 results as well, at least assuming that variance is computed in the weighted-sum manner indicated in Table 2.1-- this is about partitioning the space of outcomes by true count, not how many hands are played within the round.

Having said that, the magnitude of the error is small, typically only affecting the decimal places of SCORE. I have a single anecdote from a past study where this error came up, where the SCORE of 31.1135 was corrected to 31.1018.

E

As I said, Eric, I appreciate the drive for accuracy, but I'm not going to lose any sleep over this. :-)

Don