Continued at http://tamrielfoundry.com/topic/introduction-to-pve-damage-calculation-homestead/

**Acknowledgement**

A special thank you to addon developers for allowing me to understand the mechanics in the game in particular @Atropos for FTC, @SpellBuilder for LUI, @Kith for Srendarr, @coolmodi for MitigationPercent and TemplarExecute and all the other developers for the bunch of addons I use daily in ESO. I would also like to extend my gratitude to the community for the kind words and valuable discussion about ESO game mechanics

# Introduction

This post is divided into two main sections: Fundamental Equations and Application of Equations. Fundamental Equations covers the majority of calculations in the game while Application of Equations uses the derived equations to draw conclusions regarding what trait or mundus to use. While my initial focus was on Magicka Sorcerers, the focus has spread a bit and is probably of interest to most Magicka based classes. Note that a large number of equations can be applied to Stamina builds as well by simply substituting for the relevant stamina analogue. I am lacking healing related equations and I may delve into that in the future. Also include, at the end of this post, are spreadsheets that implements the Fundamental Equations as well as Application of Equations. The equations and values use are valid for the Orsinium PTS so some of it will be incorrect in the current live version 2.1.x. The spreadsheets can be used to determine the relative strengths of different sets. The spreadsheets are view only to prevent tampering but please feel free to make a copy for your own calculations. If you do use the spreadsheets I would appreciate feedback in terms of accuracy or ease of use.

# Fundamental equations

## Stat pool

Spoiler:The Base value at C160 is 8744 for Health and 7958 for Magicka and Stamina. Attribute points is the number of points spent in Health, Magicka or Stamina multiplied by 122 for Health and 111 for Magicka and Stamina. Gear covers most things like set bonuses and enchants. Gear1 appears to be a peculiarity unique to the 5 piece bonus of Destruction Mastery and Necropotence.

CPI is a cumulative percentage increase due to points spent in the corresponding constellation and can be calculated as follow

The Lord Mundus gives 1452 Health. The Mage and Tower Mundus gives 1320 Magicka and Stamina, respectively. Divines is the sum of divines. For example 4 pieces of green equipment with divines (4.5% each) means Divines = 1.18. Limited testing on the PTS suggest that Divines is rounded to 2 decimal places.

Example

I have a C160 Breton Sorcerer on the PTS. My gear gives me 7924 Magicka. This includes enchantments and gear bonuses. I have 64 points in Magicka, thus Attribute Points is 7104. I have 100 Champion Points in the Mage constellation which means that CPI is 1.134. I am using the Crown Fortifying Meal which gives 3617 Magicka. Skills is 1.31 (8% Bound Aegis, 5% Inner light, 2% Magicka Controller, 10% Gift of Magnus, 6% Undaunted Mettle). Putting this all into the formula, My in-game Magicka pool in the PTS is 38899.

## Spell damage

Spoiler:The Apprentice Mundus provides 166 Spell damage at C160.

## Ability tooltip value

Spoiler:The tooltip value for the majority of skill conforms to where

aandbare coefficients.bis typically around 10.5 andavaries for each skill. The range ofais typically between 0.02 to 0.2. It is quite challenging to get extremely accurate values foraeven with plane fitting over a large data range. We can fixbwithout much loss in the fitting forathus we can introduce to concept of effective poolThis allows a fast evaluation method for different builds with varying Magicka and Spell Damage.

A technically more accurate estimate can be obtained by using

However the first formula presented in this section usually provides sufficient accuracy and will be used for the remainder of this post.

Some skills notably Hardened Ward and Annulment scale of only Magicka. In some cases the coefficient

ais modified by Champion Points, at the time of writing, Thaumathurge appears to increase the tooltip for all abilities even non-DoTs. Elemental Expert is correctly being applied to tooltips. In the remainder of this post, it will be assumed that tooltip refers to the base tooltip unmodified by Champion Points.A very well made skills browser by @uesp

http://esoitem.uesp.net/viewSkills.php

A list of skill coefficients for the PTS 2.3.2 can be found at

https://docs.google.com/spreadsheets/d/1YN8YWDpi1-d4CfoagRy1F9ath2w2nb-TniL4MjdJdz4/edit?usp=sharing

A list of all skill coefficients and the weird ones by @uesp

## Recovery

Spoiler:The recovery formula was altered after PTS 2.5. Thanks to @Reorx_Holybeard for the following equation

Base Magicka and Stamina recovery at C160 is 514. Base Health recovery at C160 is 309. The Atronach mundus provides 198 Magicka recovery at C160. Examples of Skills are Magicka Controller [Mages’ Guild Passive], Major Intellect, Recovery [Light Armour Passive], and Spellcharge [Altmer Passive].

## Spell Cost

Spoiler:The Base Cost of a spell is the tooltip cost value, without any points in Magician and without any equipment or skills that provide either a percentage or flat cost reduction. Flat Cost Reduction is typically in the form of enchantments on jewellery and % Cost Reduction comes from skills and passives. Note that the 2 piece Molag Kena is a 33% cost increase when activated.

Example

I am calculating the cost to cast Force Pulse. The Base Cost is 2700 at C160. I have 92 points in Magician (15.2%) and 1 legendary reduce spell cost enchantment (203 each). %Cost Reduction is 0.23, 15% from Evocation [Light Armour Passive], 5% from Unholy Knowledge (Sorcerer Passive) and 3% from Magicka Mastery [Breton Passive]. The cost for casting Force Pulse is

This matches the in-game cost for Force Pulse in the PTS.

The base spell cost varies with level and @uesp has explained how it works

http://tamrielfoundry.com/topic/pts-2-1-2-sorcerer-arithmagic/page/25/#post-645278

http://tamrielfoundry.com/topic/pts-2-1-2-sorcerer-arithmagic/page/27/#post-646518

## Spell Critical

Spoiler:Simply add up all sources that increase Spell Critical. 219 Spell Critical rating is equivalent to 1% Spell Critical

## Critical modifier

Spoiler:This formula has been updated due to changes in [2.2.4]

where Fl is the floor function, Rd is the round function and Elfborn_Real is similar to the tooltip value of Elfborn but when it is used no unexpected rounding errors are found. Rd(x, 2) rounds a number to 2 decimal places and Fl(x, 0.01) truncates a number at the 2nd decimal places. Here are examples of both functions in action, Rd(23.458,2) = 23.46, Fl(23.458,0.01) = 23.45. Skills tested were the Piercing Spear passive and Trap Beast (Minor Force). Thanks to @Beltan3 and @hofawd with some help in getting this formula down. It appears that Critical multiplier is floored after Major Force is calculated but further test are needed to confirm this.

By the way, if you don't mind some error in your calculation, a simpler formula is

Due to rounding errors, Elfborn still suffers from jump. Any points in between jumps do not increase your critical modifier. @Erraln has kindly listed all the jump points in this thread but for convenience, I'll put them here as well. The Elfborn jump points are at

1,2,4,7,9,12,15,18,22,26,29,33,38,42,46,51,56,61,66,71,76,81,87,92, and 98.

## Resistance

Spoiler:Your spell resistance can be calculated with the following formula

where Gear is the sum of tooltip armour values, Resolve is a Heavy Armour Passive, Defending is a weapon trait. Other includes Breton Spell Resistance Passive, Balanced Warrior Passive (Templars), Major Ward, Spell Warding and Spell Shield CP.

Similarly, physical resistance is calculated as follows

The Lady mundus provides 1980 Physical Resistance at C160 and is put into the variable Other. The Reinforced trait increases the tooltip armour value and will be included when calculating Gear. The Shield Expert passive under the Steed increases the tooltip value of the shield thus is also included in Gear.

Example

I have 5 pieces of heavy armour, 1 medium and 1 light. The sum of all my tooltip armour values, Gear, is 16666. Resolves grants 1811 resistance and Spell Warding grants 363 Spell Resistance. I have a legendary defending weapon equipped (6%). My set bonus for physical resistance is 5805. I have 100 points in Heavy Armour Focus (5281). I also have the Spell Resistance (Breton passive) and Balanced Warrior (Templar passive) passives. With Major Resolve and Ward active, I estimate my physical and spell resistance to be

My actual physical and spell resistance are 35951 and 31828, respectively.

## Critical resistance

Spoiler:Critical resistance is not needed in PvE since monsters do not do critical damage. An enemies’ critical modifier can be reduced by equipping gear with the Impenetrable trait or by spending points in the Resilient champion point sign. Every percent in Resilient decreases the enemies’ critical modifier by the same amount and every 250 points of critical resistance reduces an enemies’ critical modifier by 3.5%

Example

If you are PvP’ing against an enemy with a critical modifier of 0.5 and you have 500 critical resistance (2 legendary equipment) and 48 points (15%) in Resilient, then

## Miscellaneous equations

Spoiler:

Block CostThe block formula is

Example

If you have 100 points in Shadow Ward, 8 pieces of Legendary Sturdy gear, 3 Legendary Shield play enchants (203 block cost reduction each), 2 points in Fortress (36% block cost reduction) and Defensive Posture slotted your block cost will be

The in-game block cost is 88.

Base Cost vs Level@uesp has found how most cost scale with level. The link is provided below.

http://tamrielfoundry.com/topic/pts-2-1-2-sorcerer-arithmagic/page/32/#post-651311

Sneak Damage@MrTarkanian48 has reported the PvE sneak damage formula to be

where Sneak Bonus is 3.75 for Melee attacks and 1.46 for Ranged attacks. I have verified this using Wrecking Blow, 2H Heavy Attack, Bow Heavy Attack, Focused Aim and Hidden Blade.

In PvP the Sneak Bonus is 0.27.

# Damage calculation

## Base damage

Spoiler:The base damage formula is where

Attacker Bonus includes relevant champion point signs, Minor Berserk [Combat Prayer] and Elemental Talent [Altmer Passive]. These stack multiplicatively. Defender bonus refers to relevant champion point signs. Battle Spirit can be included by simply multiplying by 0.5. In 2.2.3, the impact of Thaumathurge is included in the tooltip but the impact of Elemental Expert is not included in the tooltip, so be wary you’re not double counting Thaumathurge. Resistance is the relevant physical or spell resistance. Penetration is the sum of percentage based penetration. For magicka builds, this is either 18% for a legendary nirnhoned weapon or 28% when using a legendary nirnhoned weapon and casting a Destruction Staff spell due to the Penetrating Magic passive. Examples of Flat Penetration are Concentration for Light Armour users, Spell Erosion and Piercing. The base Flat Penetration is 100. Veteran rank 16 corresponds to level 66. For PvE, most mobs have a level of 50. Examples of Armour Debuff are Major and Minor Fracture and Breach, 5 piece bonus of Night Mother's Gaze and Glyphs of Crushing. The resistance of some bosses in 2.1.x can be found at [2.1] Project Resistance.

PvE Example

A C160 Breton cast Crystal Fragments on Slimecraw (Wayrest Normal Dungeon). Slimecraw's spell resistance is 18200.The tooltip value without any points in Elemental Expert is 7832 (Refer to the section

Ability tooltip valueif you're confused). I have 75 points in Elemental Expert (20.4%) and 25 points in Spell Erosion (1492 Flat Spell Penetration). My Flat Spell Penetration is 100[base]+4884[Concentration]+1492[Spell Erosion] = 6476. I'm using an epic sharpened staff (12% penetration). I have combat prayer active. Given these valuesand

The actual damage recorded by CLS is 8243.

## Average damage

Spoiler:The average damage when taking into account critical damage is

# Healing calculation

## Base Healing

Spoiler:The base healing formula for a healer using a variety of sources of Healing Done and Healing Taken and Healing Received isThe Tooltip value is increased by Restoration Master, Soul Siphoner, Major Mending and the Ritual Mundus. These add additively. Healing Done was tested with Blessed. Healing Taken was tested with Tormentor and Leeching sets. Healing Received was tested to be additive with Quick Recovery [Champion Point], Rapid Mending [Heavy Armour Passive], Minor Vitality [Swallow Soul & Coagulating Blood], Burning Heart [Draconic Power Passive] and Quick to Mend [Argonian Passive].

Example

A C160 Argonian cast Healing Springs. This character has 12% Healing Taken from the Tormentor and Leeching Sets and has 100 points in Blessed (25% Healing Done) and 100 points in Quick Recovery (16% Healing Received). In addition, this character is wearing 7 Heavy armour pieces (7% Healing Received), has the Minor Vitality buff, has a Draconic Power ability active and has 3 points in the Quick to Mend passive. This tooltip value includes the bonuses from Restoration Master, Major Mending and the Ritual Mundus.

The recorded in-game healing is 3264.

# Application of equations

## Champion Point System

### Warrior

Spoiler:Most players will be looking to spend points in either Hardy, Elemental Defender or Thick-Skinned. Sorcerers will also be looking to spend points into Bastion. Not much can be said for Armour Focus or Spell Shield as a 100 points provides only 5281 resistance which corresponds to 8% mitigation at C160 which corresponds to 20 points in Hardy or Elemental Defender. In addition, Armour Focus and Spell Shield are affected by percentage penetration and are not taken into account when a damage shield is used.

### Thief

Spoiler:For Magicka builds, the decision is on how to spread points between Magician and Arcanist. Essentially, we're trying to optimise the function

This function represents the average magicka gain per second. Since this spreadsheet is rather complicated, I recommend selecting your

RaceandClassand then filling in all the grey filled cells. Most of them should be reasonably easy to understand and I've made comments for most of them. Ensure that theFinal Magicka RecoveryandFinal Spell Costmatches your ingame values to make sure all inputs are correct. The two inputs that require further explanation areAverage Base Spell CostandCast per second. Both can be determined from a parse. I'll provide an example in the future. But for now assume that I casted 10 Funnel Healths and 1 Crippling Grasp over 20 seconds. Look up theBase Spell Costof both spell which are 1233 and 2923 respectively. TheAverage Base Spell Costcan be determine fromInput - Spell Costor from theSkill Dataspreadsheet. So myAverage Base Spell Costis (1233*10+2923)/11=1387.Cast per secondis 11/20 = 0.55. Look over to the right and find Recovery 0, Cost Reduction 0 to determine the ideal CP distribution.If you wanted to mess around with the number of jewellery to enchant remove the impact of jewellery from

Magicka Recovery - GearandSpell Cost - Gear (Flat CR)then just look at the corresponding output line. The spreadsheet can be found athttps://docs.google.com/spreadsheets/d/1Yp90QzJnsL5PNqSgKqDGv9IpYzJajG_tnlslFNPOf9g/edit?usp=sharing

### Mage

Spoiler:For a Magicka damage dealer, there are five signs of interest, Elemental Expert, Thaumathurge, Spell Erosion, Elfborn and Staff Expert.

The mechanics of all these except Staff Expert have been discussed in the sections on

Critical ModifierandBase Damage. Staff expert increases the damage of light and medium attacks that is your weaving damage. Light and medium staff attacks are also increased by Elemental Expert.In order to determine the optimal Champion Point distribution, we first need to consider the ratio of Elemental, DoT and Staff attacks. Then we can optimise the following function

One way to approach this is to enumerate all possible CP distributions and then calculate the function. However, this is quite a challenging task since with 167 Mage CP and 5 CP stars to consider the total possible of combinations is 30507895 (166C4). Although, some of these can be eliminated through some insight on the relative strengths of the CP stars. While this can be done, I have decided to make the equation a bit more accessible with the loss of a tiny amount of accuracy. I've included a spreadsheet that will do this

https://docs.google.com/spreadsheets/d/1Zp9v1Vp4Z7X6zfDfcxTwyAnejv-tEC5LujbXYBiVMDk/edit?usp=sharing

On the first sheet you'll see a range of inputs including

- Number of Champion Points
- Critical Chance
- Critical Modifier

- Defined as Critical Damage/Non-Critical Damage - 1
- Target Resistance
- Target Level
- Percentage Penetration
- Flat Penetration
- Elemental Ratio
- DoT Ratio
- Staff Ratio
It will output the ideal CP distribution in Elemental Expert, Thaumathurge, Staff Expert, Elfborn and Spell Erosion. It does not take into account the Elfborn jump points in the calculation so if it suggest a non-Elfborn jump point it will also display the nearest Elfborn jump points. Also all CP passives are ignored. This means that there may be situations where it will not recommend putting 30 points into the Apprentice first.

[b]How it works?[/b]

It starts by assuming you have 0 in all 5 CP stars. It then calculates the following function

for an increase of 1 point in each star. It selects the optimum distribution then keeps going until it reaches your stated number of Champion Points. I had to use a continuous equation to model how the CP stars vary with points spent. The equation used can be seen in Sheet 3. Because a continuous equation was used some deviation from my previous discrete optimisation will be present. In addition, I could not include the impact of Elfborn jump points thus jump points are suggested at the end of the optimisation. While the previous discrete optimisation is probably better to model the jump points, I hope that this method of presenting the ideal mage CP distribution for magicka builds will be easier to use thus increasing it's accuracy as you can put in your own relevant values instead of looking for the closest table match.

## Nirnhoned, Precise and Sharpened

Spoiler:When evaluating between these three item traits it is imperative to recall the average damage formula

where

The skill coefficient,

a, depends on the skill in question and an excellent list of skill coefficients can be found at http://esoitem.uesp.net/viewSkills.php.bis around 10.5 for most abilities. Attacker Bonus refers to certain Champion Points like Elemental Expert, Mighty or Thaumathurge and includes certain skills and passives such as Major or Minor Berserk. For simplicity, we can just set this to one and ignore it for the large part.Armour debuffs include Major/Minor Breach, 5 piece bonus of Roar of Alkosh and the Crushing Enchantment. An example of percentage penetration is Penetrating Magic (Destruction Staff). Flat Penetration includes the base penetration of 100, Concentration and Spell Erosion/Piercing.

In order to compare the three traits, we can convert both the additional flat penetration offered by Sharpened and the extra critical chance into Spell/Weapon Damage. Finally we can convert the Spell/Weapon damage equivalence to an increase in an arbitrary ability tooltip.

I’ll use a generic magicka build as an example, I’ll assume a Magicka pool of 40k and Spell Damage of 3k which is reasonably typical for endgame builds. Only legendary trait values are considered.

NirnhonedIt increases the tooltip value of a weapon by 11% which in turn increases Spell/Weapon damage by 11% before any buffs. For a staff, this corresponds to a base increase in spell damage of 146 Spell Damage and for dual wielding swords it corresponds to 175 Spell Damage. Typically this base spell damage is buffed by 25% by Major and Minor Sorcery. So you would expect 183 Spell Damage for a staff and 219 Spell Damage for dual wielding swords. Thus if we’re dual wielding swords, Nirnhoned leads to an increase ability tooltip of roughly 3.2%.

PreciseThe 7% increased weapon and spell critical can be converted to an equivalent spell damage with the following equation

A reasonably common spell critical for endgame builds is around 60% and a critical modifier between 0.6-0.7 is typical for most magicka builds. This assumes around 30 points in Elfborn which is a fairly common recommendation from my Champion Point optimisation spreadsheet (https://docs.google.com/spreadsheets/d/1Zp9v1Vp4Z7X6zfDfcxTwyAnejv-tEC5LujbXYBiVMDk/edit?usp=sharing) thus the spell damage equivalence for Precise is about

This translates to an increase in average ability tooltip by 3.5%. Note that we can reach this answer quicker by noting that

SharpenedIn a similar fashion to the previous section, the increased physical and spell penetration of Sharpened can be converted to an equivalent spell damage or more directly into an increase in ability tooltip

where Mitigation has been separated into a Sharpened component and everything else. The key concern here is that the Resistance of a target cannot go below zero. Let us consider a typical 4 person dungeon, bosses have around 18k resistance and trash have anywhere from 10 to 18k resistance (http://tamrielfoundry.com/topic/2-1-boss-resistance/). The main source of armour debuff is Major Breach (5280) and Flat Penetration for Magicka Characters is at least 4984 (100 base + 4884 Concentration). Flat Penetration is typically higher by around 1-2k depending on the number of points assigned into Spell Erosion. This means the average boss will have about 6.5k resistance and 100% of the Sharpened bonus is used meaning that it increase ability tooltip by

Unfortunately, Sharpened is useless on any mob with less than about 12k resistance as its resistance is already reduced to zero from other sources. So at first glance, the optimal trait seems to be Sharpened for bosses and Precise for trash, similar to the meta on Live.

But there may be another way around this, given that we are reaching unmitigated damage it is reasonable to remove all points from Spell Erosion and apply it all into Elfborn instead. Then a more interesting question would be how much resistance does Sharpened need to remove for it to be equivalent to Precise. Precise provides 3.5% increase in ability tooltip, this value can be achieved if Sharpened removes at least 1750 resistance. My optimal champion point spreadsheet tends to suggest around 30 points in Elfborn which corresponds to this value. Thus it is possible to simply use Sharpened for most trash and all bosses and put 66 points in Elfborn (67 points in Elfborn works the same as 66 points so spent the last point in something else like Staff Expert. In this way Sharpened is acting at worst equivalent to Precise but you get additional points in Elfborn thus increasing your average damage or it significantly outperforms Precise on bosses.

Although I've only discussed magicka builds primarily, this conclusion of using Sharpened in most situations holds true for stamina builds as well. Stamina builds do not have a Flat Penetration skill but this is compensated by several armour debuff methods.

## Divines and Infused

Spoiler:The current meta advice suggest using Infused on large pieces (Head, Chest, Legs and Shield) and Divines on small pieces (Shoulders, Waist, Hands and Feet). There are no other viable traits for optimising damage. However, with the buffs to the Thief and Shadow mundus stones this advice is called into question. In this section, I will derive the conditions where Infused or Divines should be used. It is quite laborious mathematics and this section has been implemented in my spreadsheet. The equations here are to explain how the calculation is done.

The amount of magicka gained from using Infused on a legendary large piece instead of Divines, Inf, is173 is the difference between in enchantment of a C160 legendary Magicka enchant on an Infused large piece compared to a non-Infused large piece. CPI and Skills were defined in

Stat Pool.The average damage when using an Infused piece is

The coefficient,

k, in this section isDmg_Inf can be separated into two parts, Dmg_Base and Dmg’_Inf. The former is the damage component from not using any trait and the latter is the bonus damage coming from using infusedThe average damage increase when using Divines depends on the mundus stone. For completeness, I’ll analyse the Apprentice, Mage, Thief and Shadow mundus stones.

Apprentice

The average damage increase when using Divines with the Apprentice mundus isSkill_SD refers to abilities and buffs that increase spell damage, notably Minor and Major Sorcery and the Expert Mage passive.

Again this can be separated into two components, Dmg_Base and Dmg_Divine^App’

Mage

The average damage increase when using Divines with the Mage mundus isSkills refers to abilities and passives that increase maximum Magicka and was defined in more detail in <b>Stat Pool</b>.

Again this can be separated into two components, Dmg_Base and Dmg_Divine^Mage’

Thief

The average damage increase when using Divines with the Thief mundus is

Again this can be separated into two components, Dmg_Base and Dmg_Divine^Thief’

Shadow

The average damage increase when using Divines with the Shadow mundus is

I’m using a simplified form of the Critical Modifier equation here but in my implementation of the equation in my spreadsheet I use the more exact form of the Critical Modifier. Skill_S refers to skills that increase the critical modifier such as Piercing Spear [Templar passive], Hemorrhage [Nightblade passive], Trap Beast [Minor Force] and Aggressive Warhorn [Major Force].

This can be separated into two components, Dmg_Base and Dmg_Divine^Shadow’

The way to decide between Infused and Divines is then to evaluate Dmg_Inf’ - Dmg_Divine’ with the corresponding mundus stone. Since several variables need to be taken into account, I’ve simply implemented my calculation into my spreadsheets. On a personal note, in the majority of calculations that I have performed Divines with Thief or Shadow outperforms Infused. However, if the Apprentice or Mageis used then Infused on large pieces is preferred.

## Mundus stone: Apprentice, Mage, Thief and Shadow

Spoiler:There are four mundus stones of interest to optimising Magicka based damage dealers. They are the Apprentice, Mage, Thief and Shadow. I’ll begin with the average damage equation for each mundus. Astute readers might notice a striking similarity with the section <b>Divines and Infused</b>.

ApprenticeIn this section, the coefficient,

k, is defined to beSkill_SD refers to abilities and buffs that increase spell damage, notably Minor and Major Sorcery and the Expert Mage passive.

Dmg_App can be separated into two components, Dmg_Base and Dmg_App’. The former is the damage component without any mundus active and the latter is the damage from using the Apprentice mundus.Mage

Skills refers to abilities and passives that increase maximum Magicka and was defined in more detail in

Stat Pool. Separating into a base and Mage component yieldsThiefOnce again, this can be separated into a base and Thief component.Shadow

I’m using a simplified form of the Critical Modifier equation here but in my implementation of the equation in my spreadsheet I use the more exact form of the Critical Modifier. Skill_S refers to skills that increase the critical modifier such as Piercing Spear [Templar passive], Hemorrhage [Nightblade passive], Trap Beast [Minor Force] and Aggressive Warhorn [Major Force].

This can be separated into two components, Dmg_Base and Dmg_Shadow’From this, we can easily conclude that the Apprentice mundus is preferred to the Mage mundus in nearly all cases for increasing damage sincewhereAt C160 the Apprentice provides 166 Spell Damage, <i>b</i> is approximately 10.5 and Skill_SD is typically 1.2 due to the Major Sorcery buff. The Mage gives 1320 Magicka at C160 and Skills is typically around 1.31 for a Sorcerer and is lower for Templars. Putting this into the equation, we obtainThus the Apprentice is preferred. The Mage mundus is sometimes preferred due to increasing Magicka pool for stronger shields and higher pet damage as these scale solely off Magicka.

For the comparison between the Apprentice and Thief there are no easy simplifications and one is left to evaluateThere are many variables and no obvious simplifications thus I have simply implemented the laborious calculation in my spreadsheet. Similarly the comparison between Apprentice and Shadow is very involved and is implemented in the spreadsheet.

While the equations presented for the Thief are a crude approximation due to the complicated rounding in the more accurate formula for Critical Modifier, it is possible to make a rough comparison between the Thief and Shadow mundus stones.Thus the Thief is better ifThis is equivalent toIn the Orsinium PTS, the Shadow mundus increases Critical Modifier by 12% and the Thief increases Critical Chance by 11%. Putting this values in, we get the following inequalitywhich means that Thief is better than Shadow if your Critical Modifier is at least ~10% greater than your Critical Chance.

## Percentage Penetration and Spell Damage Equivalence

Spoiler:Since Maelstrom weapons cannot come in Nirnhoned, it is natural to ask how does a non-set Nirnhoned destruction staff compare to a Sharpened Maelstrom destruction staff.

To evaluate this, let us consider T1 and T2, where T1 is the base damage with S1 extra spell damage and T2 be the base damage with no extra spell damage but 4% extra penetration. Then

where M is Max Magicka, S0 is the base spell damage, S1 is the extra spell damage for T1 and

Note that any penetration that is common to T1 and T2 can be seen as just a reduction in the Resistance.

We then proceed to solve T1-T2=0 for S1

We can rearrange this to get

Now we have to put some typical endgame values, I'll let M=43487, S_0=3764 and Mit=0.19 (17k boss resistance, 14% penetration, 4984 Focus, 0 Spell Erosion). Mit'=0.04*0.34

Thus for the stats assumed a Sharpened Maelstrom Destruction Staff is better than a non-set Nirnhoned Destruction Staff as the 4% additional penetration is equivalent to 133 Spell Damage which is less than the Maelstrom enchantment of 189 Spell Damage

## Julianos and Twice-Born Star

Spoiler:Due to changes to Elfborn and skills that increase critical damage, TBS is no longer optimal from a DPS point of view. In the spreadsheets below, I have introduced a new metric called the

CombinedMetric. This metric was introduced because staff attacks scale differently from abilities. For most abilities 10.5 Max Magicka ~ 1 Spell Damage but for staff attacks 40 Max Magicka ~ 1 Spell Damage. To obtain theCombinedMetric, I assume approximately 15% of total dps comes from Heavy (Medium Attacks) and 85% comes from abilities, then the weighted average of theAbilityandAttackmetricresults in theCombined Metric.We see then that for the Combined Metric on staffs, Law of Julianos is better by about 1.5%. Previously, my calculations showed that Law of Julianos was ~0.5% better than TBS but that was without taking into account the higher Attack Metric. On the dual wield bar, we should use the Ability Metric since no weaves are used and in this situation Julianos is better by around 0.5%.

If you were to replace one magicka enchantment on a large piece in favour of health so that the Health with Julianos and Twice-Born Star are comparable then on the staff bar Julianos is better by 0.1% but worse by 0.1% on the dual wield bar. To help put all these percentage differences into context, my rough calculations suggest that not having Divines on one piece (maybe you have been unlucky and have a bank full of Well-Fitted Molag Kena shoulders/helm) equates to a loss of ~0.5%

I've heard of people saying that Twice-Born Star is better with lower CP but I have yet to see extensive calculations that demonstrate this. Using the spreadsheets below, I varied the amount of CP by adjusting both the number of points invested into the Mage Tree and assumed that the first 100 points will be put into Elemental Expert followed by all points (up to 66) into Elfborn. While this CP distribution is not absolutely optimal, it is reasonably close. In this case, even with 100 Mage CP (300 total CP) Julianos is better than Twice-Born Star on the staff bar (0.3%). On the Dual-Wield bar, Julianos outperforms Twice-Born Star at 129 Mage CP (387 total CP)

Note: Ignore the Magicka Recovery and Spell Cost boxes. I was too lazy to move them away before I took the pictures.

Twice-Born Star Staff

Twice-Born Star Dual Wield

Law of Julianos Staff

Law of Julianos Dual Wield

## Damage distribution

Spoiler:Damage calculation in ESO is rather straightforward you either crit or don't crit. This leads to a binomial distribution. If you repeat the event (of using an ability) a sufficient number of times you can approximate the binomial distribution with a normal distribution with the following statistics

Since stats can already be estimated along with tooltip values and base damage numbers, it is possible to create total damage normal distributions (or DPS distributions) for different item sets.

So if I were to redo my section on

Julianos and Twice-Born Star, I could use the calculated stats to calculate the tooltip value of Force Pulse. Here is a summary of the data required to create the normal distributionsand the distributions themsevles

It's rather interesting because initially I thought that while the average of Julianos was greater than Twice-Born Star, I felt that the higher crit modifier of Twice-Born Star would mean that there would be a good chance of it doing more damage than Julianos if crits went your way. However, in the case of 300 Force Pulses which is 100 cast or about 100 seconds of just using Force Pulse, not only is the average damage of Julianos greater than Twice-Born Star but the maximum damage for Twice-Born Star is rather similar to Julianos. It is still true that the SD of Julianos is smaller than the SD of Twice-Born Star. Even if Force Pulse were casted only 10 times (30 counts), you would expect similar'ish distribution.

From here it's straightforward to calculate the difference of the distributions

We can setup damage thresholds and calculate the chance that a single test will be incorrect, inconclusive or correct. Someone a bit more motivated than me could then calculate the minimum sample size to get a reasonable estimate of the distribution and the chance of Type 1 or Type 2 errors.

To show that the calculated curves are reasonably close to the simulated curves, I'll start of by showing the situation where you cast Force Pulse a 100 times so 300 data points and your recorded the total damage. If you repeated it a 100 times you'll get something like the green curve below

It doesn't match the calculated curve (blue) that well because repeating it 100 times is a bit low. If you repeated it 10000 times, things get much better

For me this means that I'm better off just calculating the mean and basing my conclusions off that or calculating the damage distributions (now that I've done the maths once and have a few scripts that are easily editable) than spending time in game casting Force Pulse on a boss repeatedly to get determine which set is better. I'll still keep testing in game to ensure the base equations are accurate though.

## Scathing Mage and Law of Julianos

Spoiler:In the Thieves Guild update, the Scathing Mage set was buffed. In this post, I’ll compare Scathing Mage to Law of Julianos.

There are several approaches in determining when Scathing Mage is preferred over Julianos. I suggest looking at the number of attacks required to proc Scathing Mage. From there we can estimate the uptime of Scathing Mage and then it is a straightforward comparison to Julianos. I’ll show some example calculations based on my sorcerer.

The proc chance per attack with Scathing Mage is quite simplistically

Assuming that all attacks are independent we can model the resulting distribution of required attacks to proc Scathing Mage with a geometric distribution. In order to demonstrate that the geometric distribution is a suitable model I did some in-game testing. I counted the number of hits required to proc Scathing Mage. I only did 50 trials but I was reasonably convinced by the results. I trial here is defined as the number of hits needed to proc Scathing Mage. The image bellow summarises my in-game testing as well as the theoretical model. I was reasonably convinced with validity of the model despite the low number of trials.

The median of this distribution is

The median corresponds to the point where 50% of the time you’ll need less than X attacks to proc Scathing Mage and equivalently 50% of the time you’ll need more than X attacks to proc Scathing Mage. From this, it is reasonably straightforward to calculate the median amount of time required to proc Scathing Mage

From this the uptime of Scathing Mage is expected to be

where there is an implicit assumption that the internal cooldown of Scathing Mage is the duration of the proc, which is based on personal testing during the IC PTS. The effective spell damage of Scathing Mage is then

Or to simplify it in a single equation

Let us consider a magicka Sorcerer using the Thief mundus (6 pieces of Legendary divines) and with a precise staff. The spell critical of this character is 71.2% or 74.2% with Minor Prophecy active. I timed myself doing 50 Force Pulse/LA weaves and could do it in 1.16 seconds per weave or equivalently 3.45 attacks per second. Since this is probably close to the upper limit of attacks per second, I would estimate that the upper limit of the Scathing Mage 5 piece bonus for the magicka Sorcerer in question is

A more practical approach to estimating SM SD equivalence is by looking at a parse and determining the number of non-DoT attacks per second. In the example parse below, I estimate the number of non-Dot attacks to be 84 (Force Pulse + Crystal Fragments + Light Attack) which means that the number of non-DoT attacks in this example is 2.37 (parse duration 35.5s).

This is 97 SD greater than the 5 piece bonus of Julianos. Since this character has 43486 Max Magicka and 3189 Spell Damage, the 80 spell damage corresponds to an average tooltip damage increase of ~1.7%

So based on the example parse above, if I were to use Scathing Mage instead of Julianos I would expect a DPS increase of around 425 (1.7%*25000). There is a slight simplification here since Light Attacks depend more strongly on Spell Damage than abilities so in fact my Light Attacks will do more than 1.7% damage with Scathing Mage.

However most Sorcerers rely quite a bit on Overload so the SM SD equivalence during Overload needs to evaluated. The magicka Sorcerer is using Nirnhoned swords (Spell Critical of 64.2%) for Overload and does 0.862 non-Dot attacks per second

This is 38 SD less than the 5 piece of Julianos and corresponds to an average damage loss of ~0.6% during Overload or around 200 DPS (0.6%*32000).

During Overload, I use about 18 Ultimate per second. It takes 6 seconds of non-Overload time to generate 18 Ultimate. Thus on average Scathing Mage provides an increase of 336 DPS.

This is based on the assumption that you Overload all things equally.

To Precise or not?An additional question that arises when using Scathing Mage is whether a Precise or Sharpened/Nirnhoned weapon should be used.

Percentage penetration can be converted into equivalent spell damage with the following equation

where the variables Mit_pen and Mit_Base are defined as

For simplicity,

bis assumed to be 10.5. Using some typical values, 14% penetration (Sharpened) is worth aboutAdditional Spell Critical from Precise can be converted into equivalent spell damage with the following equation

Again using typical values, 7% Spell Critical (Precise) is worth about

This means that for mobs with 13k resistance which is a typical value for bosses debuffed with Major Breach, Sharpened/Nirnhoned is preferred over Precise. However, going with Precise will increase the uptime of Scathing Mage and the question is whether the increased uptime will compensate for the difference. Using my first parse with Force Pulse, the spell damage equivalence of Scathing Mage is expected to be 418 with a Sharpened Staff

While the spell damage equivalence with Precise is

Thus the average Spell Damage gain from using a Precise weapon with Scathing Mage compared to Sharpened is 9 base spell damage which is increased to around 11 by Major Sorcery and Expert Mage. However, the spell damage equivalent difference between Sharpened and Precise on a PvE monster with 13000 resistance is 93 which is larger thus a Sharpened weapon is still preferred even when Scathing Mage is used.

## Molag Kena and Nerien'eth

Spoiler:In this post, I’ll demonstrate how to compare Molag Kena to Nerien’eth on a single target.

The image for Nerien’eth was taken without CP. The tooltip is increased by both Elemental Expert and Thaumathurge.

My Max Magicka and Spell Damage with Nerien’eth is 43486 and 3189, respectively. The spell damage is with Major Sorcery and 3 Sorcerer abilities slotted. Molag Kena provides 129 Spell Damage and an additional 516 Spell Damage when proc’ed. I estimate a maximum uptime of Molag Kena at 86% (6/7 as you cannot re-proc it when it is active). This means that the maximum Spell Damage from Molag Kena is

This is increased to

722 Spell Damageafter taking into account Major Sorcery and 3 Sorcerer abilities (6% increased Spell Damage from Expert Mage passive). Based on my Max Magicka and Spell Damage, this corresponds to an increase of ability tooltips by ~9.8%So we can roughly say that we would expect a DPS increase of ~9.8% with Molag Kena. This is an approximation since Light Attacks depend more on Spell Damage than abilities so will be increased by more than 9.8%.

There are two ways to determine the contribution of Nerien’eth. One way is to look at a parse. In the example parse below, we can estimate the DPS increase of Nerien’eth by dividing the total damage contribution of Lich Crystal by the total damage dealt less the damage done by Lich Crystal. For the example parse below it turns out to be ~9.8%, similar to the estimated upper bound DPS increase with Molag Kena.

The method above does depend on the RNG or a particular parse so a theoretical approach would be beneficial. The DPS contribution of Nerien’eth can be modelled by first modelling the number of attacks required proc Nerien’eth with a geometric distribution and then we will be able to obtain the uptime of Nerien’eth and finally estimate the DPS from the uptime.

I first tested the validity of a geometric distribution with in-game testing. I counted the number of attacks required to proc Nerien’eth and made the following image

The median number of attacks required to proc Nerien’eth is

which means that the median amount of time required to proc Nerien’eth is

Since you cannot have two Lich crystals at once, the internal cooldown of Nerien’eth is estimated to be 3 seconds. Thus, I would expect one Lich crystal every

From my parse above, I did 84 non-DoT attacks over 35.5 seconds which means the number of non-DoT attacks that could proc Nerien’eth is 2.37. So I would expect one Lich crystal every

The average damage for each Lich Crystal is

Let’s say I have 100 in Elemental Exper (25%) and 2 in Thaumathurge (1.6%) and my spell critical is 71.2% and my critical modifier 0.62 then the average damage for each of my Lich Crystal is

So my expected DPS with Nerien’eth is 2319 (13385/5.77) which was reasonably close to the example parse. Based on this, I conclude that Nerien’eth is comparable to Molag Kena for the rotation shown in the parse.

Additional note: I considered making a comparison between Molag Kena, Nerien’eth and Skoria but there is something very peculiar about Skoria. It appears that DoTs must be running for some period of time before it can actually proc. I counted the number of Puncturing Sweep hits required to proc Skoria and found that for the first few hits Skoria will never proc. This is shown in the image below. A similar result can be obtained when using Elemental Blockade (data not shown)

**Cost increase in Dark Brotherhood**

# Character SpreadSheets

Best character building site

http://www.uesp.net/wiki/Special:EsoBuildEditor

Spoiler:Only made them for Sorcerers, Templar and Nightblades at the moment.Feel free to make a copy for your own calculations.

https://docs.google.com/spreadsheets/d/1e2M7ZU9ZKBxCNhDmvfS4GKPs_Nx7YLUny9vBmPaUmgA/edit?usp=sharing

@Beltan3 has made an awesome calculator. The post is at

http://tamrielfoundry.com/topic/stats-calculator/

and the calculator itself can be found at

Other useful spreadsheets (I've listed them at various points in the post but sometimes I forget where they are too :/ )

TG Mage CP Distribution

https://docs.google.com/spreadsheets/d/1Zp9v1Vp4Z7X6zfDfcxTwyAnejv-tEC5LujbXYBiVMDk/edit?usp=sharing

TG Skill Coefficients

https://docs.google.com/spreadsheets/d/1YN8YWDpi1-d4CfoagRy1F9ath2w2nb-TniL4MjdJdz4/edit?usp=sharing