Note: this post is a short, pseudo-statistical look at how the recent nerf to Smoldering Heart affects our damage variance. It is not a discussion on the implications of this change.
For the uninformed, our new legendary gloves were just nerfed from 0.2% to 0.1% per point of Maelstrom spent. This is a big but deserved hit to an overly powerful legendary. However, in reading Earthshrine earlier I saw a discussion of how this change might affect our damage variance. One person argued that because of a lower chance to proc, we’ll have higher variance than before, by virtue of a bigger gap between our mean (average number of procs) and our max (proc on every Earth Shock). At first this might seem like a reasonable thing to say, but we can easily disprove the opinion with the power of qualitative pseudo-statistics!
Chance to proc
Imagine you’re in a vacuum where you can cast Earth Shock at 100 Maelstrom back to back for an indefinite period of time. Eventually, you’d expect to average out at the expected 20% proc chance pre-nerf (10% post-nerf). However, this is a very simplified view of how independent experiments work. In reality, you’d have a distribution of probabilities for the number of procs in a given number of casts (I use the words proc and cast to mean success and trial, respectively). To illustrate this, I powered up RStudio and plotted the binomial distribution that would result from a 20% chance proc to occur over 50 casts (Note: 50 has been used to illustrate this effect more easily. It could be 50,000 or 50,000,000):
Awesome! That looks right. With a 20% chance per cast, we’d expect 10 procs from 50 casts. Note that the distribution includes probabilities of fewer or more than 10 procs. There’s roughly an equivalent chance to get 0 procs or 20 procs. Look at the width and height of the distribution; much of the probability lies between 5 and 15 procs over 50 casts. Now, when we compare it to the distribution of the post-nerf proc chance (10% chance per 100 Maelstrom), we start to see the effect this might have on our damage variance.
Notice how the distribution has gotten much “skinnier” and “taller” (also known as a “fat-tailed distribution”)? In the most basic of explanations, this means that we’re more likely to result in our mean number of procs (5), as we’re plotting probability on the y-axis. What does a fat-tailed distribution mean for my DPS? The lower chance will of course mean a lower number of procs, but at 0.1% per Maelstrom spent, the number of procs you get is more consistent than at 0.2%.
“That’s fine slanderman, but this sounds like borderline feelycrafting to me. This is just theory!” Of course I’ll back this up with data! I simulated the T20M Gamble spec with and without the proc chance nerf to demonstrate this, and to ensure that my theory correctly translated to the expected effect in simcraft. Here’s the normal distribution of the two separate trials:
It’s a bit difficult to make a qualitative distinction between the shape of these two distributions like we did before, but it can be done. The blue (post-nerf) distribution shows a higher peak and is a fat-tailed distribution (at least in comparison to the other one). Using the thought process from before, a fat-tailed distribution is more consistent than a normal or light-tailed distribution. Just to prove I’m not making this up, I had a look at the Statistics & Data Analysis section in the simcraft report. The pre-nerf sim had a standard deviation of 87K, while the post-nerf had a standard deviation of 79K. This is a small difference, but I was right! The proc chance nerf has made our DPS variance with gloves smaller, without affecting the minimums (no procs) or maximums (proc on every Earth Shock cast). I hope this helped educate somebody out there!