Comprehensive Stat Analysis: Crit vs. DMG [Update pending]

  • Written by: imakebirdiescry


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    Hi, welcome to my guide to CRIT vs DMG. Hopefully this will be part of a series that will include HP/DEF, and possibly more (if anyone can think of a comparison they would like, please let me know).


    Since I want this guide to be accessible and useful for everyone, I will start off with the conclusions of my math.

    IF YOU DON'T WANT TO READ A WALL OF MATH, JUST READ THE FIRST PART.


    How should I build?


    Long story short: high crit damage with a mix of DMG and CRIT will give you the most damage. However, if you don't have the luxury of building alot of crit damage (i.e. for a bard or guardian), it's better to ignore CRIT altogether.


    If you're focusing on maximizing your damage output, build as much crit damage as you can without sacrificing major stats. Toto's Water blade, Comet set, and Envoys are the easiest sources, with secret stones coming in second (get the 6% crit dmg ones). If you really want more, you can use Fia's Fairy Dust, but that sacrifices quite alot of stats and is better replaced with a higher-level trophy (Saint Hale's Horn, crafted Toto's Tusk, crafted Argus Heart, etc). Crafting a Deadly weapon (+15% crit damage) is another option, though quite expensive.


    Your stat points should generally be split between CRIT and DMG as follows:


    Offensive set=Aria gear and offense-focused envoys

    Defensive set=Requiem gear and defense-focused envoys


    If you are below 195% crit damage on an offensive set or 164% crit damage on a defensive set, go pure DMG. If you are above 230% crit damage on a defensive set, go pure crit.


    For anything else, depending on your particular crit damage, you will want to split your stats between crit and DMG according to the following charts (right click-->"view image" to get a closer look):


    a4pPHAe.png


    Crit damage is on the bottom axis, and for 100 stat points, the number of points to put into DMG is on the left axis (the other points should go into crit). The lines represent, from left to right:

    Harp->Dual Gun->Grimoire->Dual Sword->Sword/Shield->Staff->Cannon->Axe.


    Quote

    To read the chart: first pick the chart that represents your gear setup: offensive set = Aria gear, defensive set = Requiem gear. Next, find the curve that represents your weapon: from left to right, Harp, Dual Guns, Grimoire, Dual Swords, Sword/Shield, Staff, Cannon, Axe. Then, look on the bottom to find your crit damage (1.9 = 190% crit damage, 2.0 = 200%, etc) and follow the dotted line upwards until it hits the curve for your weapon. Now follow the horizontal dotted line to the left and read the number there. That is the percentage of points that you want to put into DMG (assuming the rest are in CRIT).


    For example, I use a cannon at 236% crit damage, and I have full Aria gear. I look at the left-hand chart, and the second curve from the right. I follow the dotted line halfway between 2.3 and 2.4 up to the curve, then follow it to the left side and see "60" there. That means I want a 60:40 ratio between DMG and CRIT. More simply, I want 3 points in DMG for every 2 points in CRIT.


    Calculations Intro


    Since crit runs on RNG, we will use the law of large number averages to calculate an average damage. The simple formula for average damage is therefore as follows:


    total damage = DMG * [1 + critrate * (critdmg - 1) ]


    ex: You have 15000 DMG, 50% crit chance, and 170% crit damage. Your average damage is therefore 15000 * (1 + 0.5 * (1.7 - 1) ] = 20250.


    Now, most of your damage comes from skills. Skills have a base damage and a damage ratio (these can be found here). Say you're a level 45 wizard and your Meteor skill is level 48. You have a DMG stat of 15000. Then its damage is 1.4 * (9103 + 15000).


    Now one thing to note here is that increasing your DMG will increase the bonus damage, but not the base damage, of the skill. A crit, however, will increase the damage of both the base and the bonus components. We can thus give the damage formula for a skill as:


    skill damage = skillbonus * (base + DMG) * [1 + critrate * (critdmg - 1) ]


    Now that we have a formula for skill damage, let's take a look at how adding points to CRIT or DMG will affect your overall damage.


    Stats


    Eidolon bonuses are always applied at the end. They are not affected by any % increases.


    DMG


    Each stat point in damage increases your damage by 0.35%. This applies to both your base damage, weapon damage, and any bonus damage from gear. Other percentage increases in damage (from Envoys and from %DMG stats on gear) stack additively, not multiplicatively, with this. For example, say you have 10 points in damage (3.5% increase) and a full set of damage gear (8% increase). This is an 11.5% increase (3.5 + 8 = 11.5).


    So the overall formula for damage stat is:


    DMG = [(base + weapon + gear) * (stat% + gear% + envoy%)] + title + eidolon


    This is the damage stat listed in your character window. The only difference between your primary and secondary weapons is the main damage stat listed on the respective weapons. All other % bonuses are applied the same way.


    CRIT


    When dealing with CRIT, I will refer to crit stat and crit chance. These are different. Gear/stats dealing with crit are generally added to your crit stat (some exceptions exist). Crit chance is your percentage chance to land a crit derived from your crit stat relative to the target's level. Crit stat is linearly related to crit chance - that is, increasing your crit stat by 10 gives you the same increase in crit chance whether your crit stat is going from 100->110 or 3500->3510.


    Each stat point you invest in CRIT will give you enough crit stat to raise your crit chance by 0.25%. That means the stat points you invest in CRIT give you a constant crit chance as you level, but their crit stat value increases as you level. It also means that points invested in CRIT are affected by enemy level as well.


    Most things that give CRIT+X% just add directly to your crit chance without affecting your crit stat. These incluce costumes, envoy path, and secret stones (?unsure, I don't have any crit chance stones to test). However, a few gear bonuses (crafting cores, the 3% bonus that most staves have) will actually add a % of your crit stat. These % crit stat bonuses will ignore any crit stat you get from adding stat points, but include all crit stat gained from gear, your base crit stat, and titles.


    The overall formula for CRIT% is:


    CRIT% = %CRITBonus + (0.25 * statpoints) + {K * [ %CRITStat * (base + gear) + title + eidolon] }.


    statpoints is the number of attack points you have invested into crit. %CRITBonus is the total sum of all your %CRIT bonuses that apply directly to your crit chance (envoy, costume, secret stone, gear). %CRITStat is the total sum of the %CRIT bonuses that apply to your crit stat instead of directly to your crit chance (ultimate skill, some gear). K is a constant of proportionality between your crit stat and your crit chance. This changes based on level. At level 53 my value of K is around 0.007, which means every point of crit stat gives me 0.007% crit chance.


    A nice explanation of K by Mythyc:


    A more simplified version of the crit formula if you just want to check values quickly:


    CRIT% = %CRITBonus + (K * critstat).



    Simple Maths on Total Damage


    Using level 48 Meteor as an example, we have:

    skill damage = 1.4 * (9103 + DMG) * [1 + critrate * (critdmg - 1) ]

    Sample starting stats are 15000 DMG and 20% crit chance. I plotted a change of 50 stat points' worth in crit (red line) and in damage (blue line) at different levels of crit damage (130%, 190%, 200%, 210%, 300%). The following plot is what results:


    Kqu4x3p.png


    Here we see that at low crit damage, adding points to DMG will increase your damage more. However, once you go over roughly 200% crit damage, adding points to CRIT will add more to your damage. Also, we can see that you will get more out of building CRIT+crit damage than you could ever get out of going pure damage.


    This gives us a rough idea of what to do: go for crit damage, and then add CRIT or DMG accordingly. However, I only looked at adding pure DMG vs pure CRIT. It turns out that in most cases, it is actually better to split between the two. The next section will analyze the optimal stat split for different build setups.



    Optimizing Your Stat BuildSerious math involved


    First off, it will be helpful to define a few new quantities to simplify our analysis. We will need to separate our DMG stat into scalable damage and non-scalable damage. Scalable damage is the portion of the DMG stat that is affected by your stat points (i.e. your base damage, weapon damage, and flat gear damage). Non-scalable damage, conversely, is not affected by raising stat points (i.e. any damage you get from %DMG increases, as well as the base damage of your skills). Thus the portion of the skill damage formula that used to read


    skillbonus * (skillbase + DMG)


    becomes


    skillbonus * (non-scalable + scalable)


    Now, we can easily apply the 0.35% DMG increase per stat, even if we have other %DMG increases already in effect. However, we now have turned what used to be one variable (DMG) into two variables (non-scalable and scalable damage). To fix this, we define a new quantity, scalability ratio S, as the ratio of scalable damage to non-scalable damage, or


    S = scalable / non-scalable = (base + weapon + gear) / [ skillbase + %DMGbonus * (base + weapon + gear) ]

    where %DMGbonus is the sum of all your %DMG bonuses (a 10% DMG bonus = 0.10)


    Which turns this portion of the formula into


    skillbonus * non-scalable * (1 + S)


    Note that if all we're doing is comparing stat distributions for a given gear setup, we can divide skillbonus and non-scalable out of the equation without affecting our results. Yes, this changes the raw numbers, but it doesn't change the optimal ratio of points in DMG to points in CRIT. Thus the formula we will work with is:


    relative damage = (1 + S) * [1 + critrate * (critdmg -1) ]


    Let's say we have 100 offense points to spend in either crit or DMG. The number of points put into DMG will be Dp, and the number of points put into crit will be Cp = 100 - Dp. Putting these into the relative damage formula we had before gives us


    relative damage = [1 + S + (S * 0.0035 * Dp) ] * [1 + (critrate + 0.0025 * Cp) * (critdmg - 1) ]


    This gives us relative damage as a function of points put into DMG. To find the maximum relative damage, we invoke some basic calculus and find where the derivative with respect to Dp is equal to 0. After expanding, differentiating, and solving for Dp, we get the following equation (CR is starting crit rate, CD is crit damage, S is scalability ratio):


    lkoi5Rr.png


    Any other mathies out there can just plug their numbers into that formula and find out their personal optimal point spread for 100 stat points. Unfortunately, the optimal ratio between DMG/crit does not stay constant as you increase the number of available stat points, so you will have to re-derive the formula for any different number of stat points.


    I will be using level 50 weapon data found here: . I will assume the weapons are all 130%, no fortification.


    I will use two different equipment sets for this: an offensive set with full Aria gear and Comet/Strategist accessories + crit and damage envoys, and a defensive set with no bonuses to DMG or crit. Since we don't have skill data for level 50/53, I will be using the skill base damage that I calculated for my level (57, skill level 60), which is 14157. At this level, base character DMG is 849. Also, I will ignore title bonuses for DMG as they not significant on the scale we're working with (+700ish to non-scalable damage, when we're dealing with well over 20000 DMG in most cases).


    The offensive set gives around 7000 extra DMG, with a 14% damage boost (8% from gear, and 6% from envoys) and has a starting crit chance of 35% (which includes gear stats and envoy bonuses). The defensive set, again, has no DMG or crit bonuses, and the starting crit rate is assumed to be 12% (the base is somewhere around 10%, but titles are significant at low crit rates like this).


    Now, we plot optimal number of points in DMG versus crit damage using the above equipment setups. The graphs were too cramped to label every single line, so the order from left to right is Harp, Dual Gun, Grimoire, Dual Sword, Sword/Shield, Staff, Cannon, Axe.


    a4pPHAe.png


    This data is actually quite surprising. For the high damage set, even at the crit damage breakpoint, you still want to build a significant amount of DMG over crit because your starting crit rate is so high. The defensive set wants to build around half-half DMG/crit at the breakpoint.


    Anything below around 195% crit damage on the offensive set or 164% on the defensive set is still best with pure DMG. Anything above 230% crit damage on the defensive set is still best with pure crit. However, the offensive set will never want to build pure crit, even at 300% crit damage (this is probably the most surprising part of this data).


    One thing to keep in mind after all of this: if you're gearing offensively, you still want to get crit damage over damage increases on your gear. Remember the first graph I showed you? If you look back to it, the jump from 130% crit damage to 190% crit damage (a 60% change) was worth a good 75 points of DMG. That's 26.25% DMG. In all the slots that crit damage and %DMG compete for, you generally get 3% crit damage per 1%DMG. Not only that, but crit damage scales with both DMG and crit. %DMG scales only with crit (remember, %DMG does not scale with stat points in DMG). This section was only to show you the optimized stat distribution given a certain crit damage.


    General Trends


    Some general trends to keep in mind:

    1. Offensive gear favors DMG more than defensive gear

    2. Higher weapon damage (including fortification) favors DMG (this increases the scalabilitiy ratio)

    3. Lower skill level favors DMG; higher skill level favors CRIT (I'm talking about the actual levels of your skills, not how good you are)

    4. Anything with +%DMG actually favors building CRIT over building DMG (counterintuitive, I know, but this is because %DMG increases non-scalable damage, which decreases the scalability ratio S)

    5. When fighting higher level enemies, building CRIT becomes weaker; when fighting lower-level enemies building CRIT becomes stronger