### Critical Analysis

I've been wondering how to trade off various items with enhancements to spell damage. There are three basic types of these items. First, there are items which add direct damage to all spell damage (There is an uncommon form of this that adds to only particular types of spell damage, such as poison or mental). This type of enhancement is sometimes known as BoE, for Bolt of Energy, which was the name this effect was given when it first appeared.

The second kind of enhancement is an increase in critical chance, e.g., "+2 damage spell critical chance".

The third kind of enhancement is an item which procs a damage spell on a hostile spell.

I'd like to be able to do some math to get a rough idea of the contribution to total DPS each of these enhancements makes, as well as the contribution of increased INT. That is, is +1 crit chance better than +5 INT? This post will not resolve all these issues.

I gathered some data from my Ultraviolet Beam, a fast-casting, fast-recovery direct damage spell. I want to be able to relate the nominal damage on the spell to the actual damage, and try to figure out how much more damage, on average, a critical is worth.

I gathered data on 90 castings of Ultraviolet Beam. Examining the spell, it said that it would do "1120-1473" damage. I split the normal and critical hits apart, and sorted each subset, then plotted them on a graph, which looks like this:

There are a couple of clear outliers. This data was gathered in actual combat versus Ya-leih in Jarsath Wastes. I suspect that the outliers were due to unusual happenstances in combat, mobs with higher resistances, and so on.

Still, the data appears uniformly distributed, whether critical or normal hits. Here's a histogram plot of the same data:

Critical hits seem slightly non-uniform. This may simply be due to the effects of my debuffing mental resist, which I did not control for in this run. The whole dataset is online if you'd like to look at it.

Regardless, the normal hits had a mean value of 1426 with standard deviation 158, while the critical hits had a mean value of 1906, with standard deviation 207. I have seen reported elsewhere that a critical hit was worth 4/3 of normal spell damage, this data is consistent with that hypothesis.

How does the data relate to the "examine" data range, though? Recall that the "examined" range for this spell was 1140-1473. If we assume that damage is uniformly distributed within this range, that gives a mean damage of 1297, which is well below that of our data for normal hits, by roughly 130. This may be due to the effect of debuffs of mental resistance.

In any case, this gives us a rough formula for the value, in terms of DPS of an increase of one percent of critical chance. +1% critical chance is worth 4/3*.01*nominal DPS.

Incidentally, I have seen critical damage described as (max damage)*4/3. It turns out that the maximum "examine" damage is very close to the mean normal damage in our dataset, so this may be the source of that description.

Unfortunately this is too high. Criticals will only affect the first tick of a dot. So, if a large proportion of your dps comes from dots, criticals are going to be considerably less effective.

The next point to mention is that BoE damage seems to be incorporated into the "examine" value, since I had roughly a total of +500 spell damage equipped at the time. We don't really see a shift of 500 in the means from examine damage to normal damage. I think a further experiment on this is in order.

Labels: eq2, Everquest 2

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