# Thread: The Numbers Game: Learning to Understand Math in WoW

1. Just a small point here, I've never done any maths on procs before and I'm sure someone can explain it better than me but the 'maths' part wouldn't be the stat gain of the trinket such as +300 str, it would be something like the effect of the proc on dps. Stats don't necessarily increase your dps in a linear fashion, for example in the case of arp, having 0% arp 90% of the time and 100% arp 10% of the time is not equivalent to having 10% arp 100% of the time.
with any stat that does not have increasing or diminishing returns you can simply divide the stats given by the proc by the % uptime. so for something like DMC:G if it gave 400str and was up 25% of the time you could say the proc was effectively 100str. However with stats like arp, and crit, their DPS value varies depending how much of that stat you have, and as such you can't make give those procs an equivalent constant stat value

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Originally Posted by Delmonte
Just a small point here, I've never done any maths on procs before and I'm sure someone can explain it better than me but the 'maths' part wouldn't be the stat gain of the trinket such as +300 str, it would be something like the effect of the proc on dps. Stats don't necessarily increase your dps in a linear fashion, for example in the case of arp, having 0% arp 90% of the time and 100% arp 10% of the time is not equivalent to having 10% arp 100% of the time.
ah, ok, if that's his intention than he's right. (But in this case, i would really appreciate an short explanation in the guide what's meant with math/Y, as it isn't that clear imho.)
My problem with that seems to be, that there's "missing" a coefficient for the proc, it would've maybe helped, if there was something like Y*C(oefficient)*math+(1-Y)*math...
as i write this, i see my real problem :P
that what TGM has written is 'wrong', as it equals to "math". (Y*math+(1-Y)*math = (y+1-y)*math
with a C, it would be something like y*c*math+(1-y)*math = (1-y+yc)*math )

For example say that we got a proc, which grants 30% DIM for 10% of the time and the style does 1000 dmg.
y = 0,1
c = 0,3
m = 1000
TGM: 0,1*1000+0,9*1000 = 1*1000 = 1000
I: 0,1*1,3*1000+0,9*1000 = 1,03*1000 = 1030

Hope that somebody can understand this.

3. (c*d)(a*b(ability math))+(c-(c*d))(a(ability math))+(d-(c*d))(b(ability math)+(1-(c+d))(ability math)

the thing about a and b is that you must account for if the damage modifier is increasing or decreasing as I wanted the formula to work both ways, by either adding a and b to 1 or subtracting them from 1
to give you the proper modifiers, once you do that then the equation is fine. I should probably make a note there to make sure people remember to do that.

I'll carefully go over my formulas to see if I made any mistakes once I am done with my classes for the day.
Last edited by Thegreatme; 08-27-2010 at 08:48 AM.

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damned -.-
i got the wrong variables. should be c instead of a and d instead of b in post #17.
I was talking about the uptime, not the modifier himself.

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Limits in general aren't mentioned in the guide and are a key concept of a lot of World of Warcraft math.

During a Lunar Eclipse, I have 113% chance to crit with Starfire based on stats and buffs. However, there are functional limits. It is impossible to crit more than 100% of the time.

My average damage from Starfire from the crit calculation provided would be...

1.13 * 21,160 + (1 - 1.13) * 9,014 = 22,738 average damage
However, because crit is actually bound to not exceed 100%, my actual would be.
1.00 * 21,160 + (1 - 1.00) * 9,014 = 21,160 average damage
Which makes sense, if I crit 100% of the time, I average what my crit damage is.

Many mechanics have functional limits which actually apply to the equations.

6. A very well written guide.

Too bad when I got to the last part with the huge ass formula, my face went blank like it normally does. =/ I just do not do well with long formulas. It's almost as if my mind goes Ok..ok...OK..TOO MUCH! then shuts off.

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Another concept neglected is ability coefficients, which are the most basic type of damage increase modifiers (DIMs).

Ability Effect = Base Effect + (Base Coefficient * Modifier Stat)

Then expanding on that it is important to understand that DIMs can be added at any point in the calculation like so.

Ability Effect = (Base Effect * Cumaltive Base Effect DIM) + (Base Coefficient * Cumulative Coefficient DIM * Modifier Stat)

A good example of how this works is Wrath of Cenarius talent only modifies the Coefficient portion of the equation, however Moonfury modifies both the Base Effect and the Coefficient portions of the equation.

8. @kren
if you are afraid of long equations like at the end of the uptime section, break it in to parts.
(c*d)(a*b(ability math))+(c-(c*d))(a(ability math))+(d-(c*d))(b(ability math)+(1-(c+d))(ability math)

is the same as doing the following 4 equations, and adding the answers together:
(c*d)(a*b(ability math))=w
(c-(c*d))(a(ability math))=x
(d-(c*d))(b(ability math)=y
(1-(c+d))(ability math)=z
w+x+y+z= success!

@Quin
I'll see what I can do about explaining those two aspects. Be prepared to fix any dumb mistakes I might make for the coefficients section because I am much more familiar with doing melee-type math as opposed with caster-type math.

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Originally Posted by Thegreatme
with any stat that does not have increasing or diminishing returns you can simply divide the stats given by the proc by the % uptime. so for something like DMC:G if it gave 400str and was up 25% of the time you could say the proc was effectively 100str. However with stats like arp, and crit, their DPS value varies depending how much of that stat you have, and as such you can't make give those procs an equivalent constant stat value
Mmm, I wasn't sure if strength did increase your dps in a perfectly linear fashion, perhaps it does but something in the back of my mind makes me unsure about that. I'd have to look through the maths though, just a hunch I had, can't get landsoul's spreadsheet to work on my mac though -.- The main point was that it doesn't work for all stats so it's a confusing system to use, you might as well use a system that does work for all stats, or perhaps put a reminder that it doesn't work that way for all stats. The problem otherwise is that anyone wanting to use this maths needs a clear understanding of the relationship between every stat and dps so they know which they can use this sort of formula for, your target audience is people who aren't that confident with the basic maths who presumably won't fully understand these relationships.

Edit: sorry just realised you didn't mention stats in your guide, just 'ability math' which is a good way to put it.

Originally Posted by Quietsch
ah, ok, if that's his intention than he's right. (But in this case, i would really appreciate an short explanation in the guide what's meant with math/Y, as it isn't that clear imho.)
My problem with that seems to be, that there's "missing" a coefficient for the proc, it would've maybe helped, if there was something like Y*C(oefficient)*math+(1-Y)*math...
as i write this, i see my real problem :P
that what TGM has written is 'wrong', as it equals to "math". (Y*math+(1-Y)*math = (y+1-y)*math
with a C, it would be something like y*c*math+(1-y)*math = (1-y+yc)*math )

For example say that we got a proc, which grants 30% DIM for 10% of the time and the style does 1000 dmg.
y = 0,1
c = 0,3
m = 1000
TGM: 0,1*1000+0,9*1000 = 1*1000 = 1000
I: 0,1*1,3*1000+0,9*1000 = 1,03*1000 = 1030

Hope that somebody can understand this.
Sorry, I should have been more clear, I didn't actually look at the maths you posted fully and some of what you said seemed to be true, my comment wasn't about the validity of your equations just that you might talk about increased dps from the stat averaging out rather than the stat itself. In terms of the times when neither is procced I agree with you, it should be 1 - [p(a) + p(b) - p(a&b)] where 'p(a)' is the probabilitiy that a will be up and the '- p(a&b)' has been left out.

In other words, step 7 should read:

7) figure out the percentage of time that neither a or b will be up
1-[c+d-(c*d)]

Last edited by Delmonte; 08-27-2010 at 12:23 PM.

10. I think the biggest problem with Math is that it looks intimidating honestly. It's the fact that the formulas can get so long that you're sitting here going..wow, where the heck do I start first. That alone makes it intimidating to no end and it's one of those things that I guess becomes uncomfortable.

But that suggestion somewhat made it a bit more bearable actually. Thanks for that. One of the more annoying problems is the ones on EJ where they have roots and divisions and blah blah blah. The more advanced stuff is just absolutely ridiculous to understand less you're studying a math degree in University. Those just despair me. Also I find some people just write the stuff and expect people to understand, and yet don't really go into detail what a means, or b means or x or the root or whatnot.

It really comes down to communication when it comes to math. Exactly how are you using the formula to explain things, and how are you explaining SAID formula to have average joe understand what it means. It's nice you wrote this guide because it shows the basics and I wish some guides would take more time to actually explain their math formulas instead of spewing.

This is also one of the reasons I try to avoid sending average joe to EJ because of that exact problem: EJ presumes you got a strong math knowledge. That doesn't help at all. So I like seeing sites like Tankspot with the guides we write be a bit less about the math (Although some still is there) and more about explaining the process.

11. Added a Spell power coefficient section. Still need to do a section on limits, and talk about timescales in the Rates section.

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Still looking for fixes
- coefficient in the uptimes section (something like y*DIM*math+(1-y)*math)
- uptimes 7) change to 1-c+d+(c*d)
- uptimes 8) change to (1-c+d+(c*d))(ability math)
- uptimes 9) to
(c*d)(a*b(ability math))+(c-(c*d))(a(ability math))+(d-(c*d))(b(ability math)+(1-(c+d)+(c*d))(ability math)

(Even if i would delete nearly all parenthesis, as most of them aren't needed...

13. you're missing a set of parenthesis.
let's say you have two DIM's that both have a 30% uptime.
down time = 70% for each.
time that you neither of the two DIM's = 0.70*0.70= 0.49

under your proposed method it would look like this:
1-0.3+0.3+(0.3*0.3)=1.09
adding in the parenthesis you missed:
1-(0.3+0.3)+(0.3*0.3)= 0.49

I changed the sections accordingly.

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I should've known it. If i say something about the whole bunch of too many parenthesis, i forget they, where they are needed. :P
(interesting... where they aren't needed i write them, where they are needed i forget they...)

if you write now a little explanation to it, what's Y, z and math, it would be perfect It's even at the moment clear (at least for me), but maybe for someone who see's a "complex" uptime calculation the first time in his life, it isn't.

15. changed the "neither DIM's are up" part in the uptimes section to (1-c)*(1-d) because I think it's a more intuitive way of doing it and will be easier for beginners to understand. oh FOIL you never cease to be everywhere 1^2-c-d+c*d = 1-(c+d)+(c*d)

EDIT: added a discussion of timescales to the rates section. it needs a lot of work, but I am having brainfarts and can't get what I am trying to say in to words the way I want.

Still need to talk about limits, but I need to make diagrams first because I think that will help people understand them much easier. I'm very hesitant about adding this though because people are generally not taught about limits until they take an intro calculus course... and this guide is about basic math, not intro calculus. Also, most limits in the game are pretty much common sense (ex. crit is limited at 100%)
Last edited by Thegreatme; 08-28-2010 at 10:33 AM.

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It's getting quite long so I'd suggest adding an index and numbering the sections.

I think you need to break the 'rates' bit down into a few sections and remove/change the 'here's why' part because it sounds a bit Zen atm . It depends how concise you want it to be but I think if you make it too concise you run the risk of people needing more and more mathematical background to know what the formulae mean. I'm probably going to put a lot more wordiness in than most would but nvm, perhaps someone can condense it without losing any key points. Is there an option to hide stuff and allow people to expand sections for more explanation and examples if they would like?

I'd maybe split it up something like this:

There are different things you might be interested in in terms of how much DPS an ability does, 2 basic things are:

1) How much damage does it do, relative to the amount of time it takes to use/cast?

Let's consider 2 abilities that have no cooldown, i.e. you can cast them as often as you like.

One ability (ability A) takes 5 seconds to cast and does 10,000 damage, another (ability B) takes 2 seconds to cast and does 3,000 damage.

Ability A does more damage per use, however, in the same amount of time you could cast ability B more times. In order to compare them more easily you might want to know how much damage they do per second it takes to use them. In order to find this out you divide the amount of damage the ability does by the amount of time it takes to use:

Ability A: 10,000(damage) ÷ 5(seconds to cast) = 2,000 damage per second
Ability B: 3,000 ÷ 2 = 1,500 DPS

So if you repeatedly use A it will do 2,000 DPS over the course of the fight, repeatedly using B will do 1,500 DPS. Another way of looking at this is that every time you use A, for those 5 seconds A is doing 2,000 DPS and every time you use B, for those 2 seconds B is doing 1,500 DPS.

Although many abilities, particularly those of melee classes do not have a cast time ('instant spells') you are still limited in terms of how often you can use them by the global cooldown and in terms of DPS the time to use/cast is effectively equal to the GCD which is normally 1 or 1.5 seconds (as a baseline, it can be affected in certain cases by talents etc).

A quick note: DoTs have a few considerations such as will the target die before the full duration of the DoT? Whilst they make take a long time to do their damage you are really concerned with the time it takes to apply the DoT (i.e. cast the spell) as you are free to use other damaging abilities once it is up.

2) Ability C has a cooldown of 'x' seconds, how much DPS does it do over the course of the fight?

Ability C does 6,000 damage every time you use it, we'll assume it's instant and that the global cooldown for it is 1.5 seconds.
It also has a cooldown of 4 seconds i.e. once you use it, you must wait 4 seconds before you can use the same ability again.

If you could spam C you could use it every 1.5 seconds and it would do 6,000 ÷ 1.5 = 4,000 DPS, however, the most you can use it is once every 4 seconds because it has a cooldown. To know how much DPS this is over the course of a fight, you need to divide the damage by how often you use it. For example, if you use C every 4 seconds i.e. as soon as it comes off cooldown it will do:

6,000 ÷ 4 = 1,500 DPS over the course of the fight.

However, perhaps you can't use it every 4 seconds for some reason e.g. ability D is a higher priority and you only use C every 5.5 seconds. Over the course of the fight it now does:

6,000 ÷ 5.5 = 1090.91 DPS

One small consideration on shorter fights is that the DPS from abilities is less likely to average out. It is easier to demonstrate this than explain it in words. An ability (E) does 30,000 dmg and has a 30 second cooldown, if I use it every 30 seconds it's predicted DPS is:

30,000 ÷ 30 = 1,000 DPS

Over the course of a long fight, this will be quite close to the true value. However, in a short fight, say 35 seconds, I might use it at the start and once more 30 seconds later, but then the fight will end after 5 more seconds. Although I cannot use E for 25 more seconds, I did 35 seconds of damage dealing and used it twice for a total of 60,000 dmg. In this fight E did:

60,000 ÷ 35 = 1714.29 DPS which is a lot more than our predicted value.

If the fight lasted 59 seconds, I'd still only get to use it twice (as it would still be on cooldown for one second when the fight ended) for:

60,000 ÷ 59 = 1016.95 DPS which is close to our predicted value.

Perhaps in a 35 second fight I need to do something else for 10 seconds before I can use E, now I can only use it once in 35 seconds (as it will not come off cooldown in time for a second use now) of damage dealing for 30,000 damage giving:

30,000 ÷ 35 = 857.14 DPS

For comparison, in a fight that lasts 10 minutes and 5 seconds, where I use E at the start and then every 30 seconds, I still end up using it 5 seconds before the fight ends but now I use it 21 times for 630,000 damage in 605 seconds of damage dealing for:

630,000 ÷ 605 = 1041.32 DPS which is close to our predicted value.

A 10 minute and 29 second fight means 21 uses in 629 seconds (our ability is still not coming off cd in time for another use before the end of the fight) giving:

630,000 ÷ 629 = 1001.59 DPS

It might also be worth explaining how to look at the overall picture of various abilities combined together and comparing different choices but I'm not sure how to do that properly (i.e. a la Landsoul, properly taking into account cds overlapping, pushback, proc chances etc. I just use really rough and ready methods that would be counterproductive to include in a tutorial) so I'll leave that up to someone clever
Last edited by Delmonte; 08-28-2010 at 03:00 PM.

17. Originally Posted by Krenian
I think the biggest problem with Math is that it looks intimidating honestly.
At the risk of stretching "on topic" that psychology is a major part of the challenge for you (and folks like you). If you look at it and are intimidated/confused/daunted even without digging into it, that is just one more step of removal that your brain allows to block out the information. It is also a separate issue to work through and break down. It is not helpful, of course, in any way other than satisfying a weird self-deprecating ego issue. =) So feel free to notice yourself doing it, and say, "hey! You aren't helping, I'mma go read some math!"

Originally Posted by Krenian
It's the fact that the formulas can get so long that you're sitting here going..wow, where the heck do I start first.
And *that* is why TGM's guide is cool, and a handy thing to have around.

If you ever come up against a formula and you are unsure of where to start, mention it here and I bet we can help you identify the parts to make it seem less overwhelming or chaotic.

But that suggestion somewhat made it a bit more bearable actually. Thanks for that. One of the more annoying problems is the ones on EJ where they have roots and divisions and blah blah blah. The more advanced stuff is just absolutely ridiculous to understand less you're studying a math degree in University. Those just despair me. Also I find some people just write the stuff and expect people to understand, and yet don't really go into detail what a means, or b means or x or the root or whatnot.

It really comes down to communication when it comes to math. Exactly how are you using the formula to explain things, and how are you explaining SAID formula to have average joe understand what it means. It's nice you wrote this guide because it shows the basics and I wish some guides would take more time to actually explain their math formulas instead of spewing.

This is also one of the reasons I try to avoid sending average joe to EJ because of that exact problem: EJ presumes you got a strong math knowledge. That doesn't help at all. So I like seeing sites like Tankspot with the guides we write be a bit less about the math (Although some still is there) and more about explaining the process.[/QUOTE]

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DPET = the value of an ability in your priority system, relative to ability use time
DPS = the actual contribution of an ability towards your rate of damage, relative to actual encounter time

Generic rules... (this is a general pattern, but not always the case)
Cooldowns and Damage over Time abilities will always have significantly higher DPET than DPS. The average rate of damage being dealt isn't significant relative to how often the ability is used, however the damage per use is higher than other abilities typically resulting in you wanting to use those abilities when they are off cooldown or in the case of Damage over Time abilities not already active and the target will live for the duration (or close to depending on how high the value) of the debuff. Direct damage abilities not subject to a cooldown will have these two values equivalent and will have higher DPS in but lower DPET relative to Cooldowns and Damage over Time counterparts. Some Cooldowns are also Damage over Time such as the summoning of combat pets.

The key difference between a Cooldown and a Damage over Time ability is the "last use".
As far as the math is concerned, they are almost identical. The only real difference is in the choice to cast it that last time. In the case of a Damage over Time ability, if the debuff lasts 20 seconds, and the target will die in 10 seconds, you're not likely to use it at half of its effective value. In the case of a cooldown however if there are 10 seconds left in a fight and you have a 30 second cooldown ability that deals instant damage, you would choose to use it. When you can use the cooldown again doesn't enter into your consideration of whether or not you should cast it that last time (the cooldown being available again in 30 seconds is irrellevant to me wanting to use it 10 seconds before the boss dies). Some abilities fall under both catagories, such as Starfall is a cooldown ability that deals damage over 10 seconds. In cases like that you would combine both principles to determine is it worth casting that last time.

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Odd question, but having just come back to WoW again. I'd like to be able to really sit down and understand the mechanics of "Why" the game responds how it does.

That being said, has the recent patch changed any of these equations?

20. Simple and straightforward, and surprisingly with no math in it that I couldn't do in 7th grade. Good guide, TGM!