(*Update 3/8: I originally completely forgot to talk about what happens with different winrates. I added this analysis to the bottom of the post.*)

In two of my previous posts (1 2), I crunched some numbers to show that the variance in large-field MTTs is pretty damn crazy. Now I’m going to turn my attention to NLHE 6-max cash games. (Sorry for the delay. I could claim to have been busy, but mostly I’ve just been lazy.)

This is actually pretty easy thanks to the statistician’s best friend, the central limit theorem. For sample sizes of at least a few thousand hands, you can just take your standard deviation and winrate and use the normal approximation. (For sample sizes of less than a few thousand hands…. Who cares?) This makes the math really easy, and in a way, it’s responsible for the relatively cushy lifestyle of cash players–The normal distribution is a lot cleaner than the distribution that I found for MTT players. (Proof that the distribution is normal)

(Another consequence of this mathematical convenience is the fact that some other people have done this analysis already. While my posts on MTTs were, as far as I know, the first honest attempt at tackling that problem correctly, this post will mostly just explain what’s already known to the nerdy contingent of the poker world and anyone else with a basic understanding of statistics. I’m just bothering to share this information with our less nerdy brethren in a way that I feel is reasonably clear. To that end, I’ll mostly just leave out any explanations, but suffice it to say that all of this comes from very basic facts about the normal distribution. However, I think everybody who plays poker should be able to do some basic statistics, so I might make a tutorial explaining where these numbers come from at some point.)

To see how that works out in practice, let’s look at a basic example. Take a solid 6-max NLHE grinder with a 5 bb/100 win rate (I’m going to use bb/100, not PTBB/100 in this post. An unfortunate tradition ported from LHE leads many people–including PTR, PT3, and sometimes me–to call a BB or a PTBB twice a bb.) with a fairly typical standard deviation of 90 bb/100. What happens if 10,000 clones of this guy play 50,000 hands each? Well, this does: