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EDIT: It has been hinted by Drysc (http://forums.worldofwarcraft.com/thread.html?topicId=1778587218&pageNo=1&sid=1#16) that the penalty incurred by improved fireball/frostbolt talents will be *REMOVED* in 2.3. Until 2.3 occurs this research stands as the current method for calculating coefficients.
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EDIT 2: And it has been CONFIRMED (http://forums.worldofwarcraft.com/thread.html?topicId=2856224307) in 2.3 this research is no longer valid (or rather, that the improved frostbolt component isn't), see the thread for more details
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Round two! Last time I did this I established that the imp frostbolt/fireball nerf was multiplicative, this time I'm out to prove one way or the other whether emp frostbolt/fireball are additive or multiplicative.

Here's the dump from my data crunching perl script, enjoy folks, full data release available upon request as always:


First Dataset, chaincasting frostbolt rank 13, no gear, no spec, tooltip damage range of 600 - 647 damage, expected average of 623.5. This dataset establishes a baseline average for the spell so we know that we can trust a simple averaging of the tooltip.


Stats for dataset:
Max: (647)
Min: (600)
Mean: (623.973053892216)
Datapoints: (334)


Second Dataset, chaincasting frostbolt rank 13, wearing +399 damage gear, no spec, tooltip damage range of 600 - 647 damage, expected average of (((600 + 647) / 2) + (((3.0 / 3.5) * .95) * 399)) = 948.4 damage. This dataset establishes a baseline coefficient before we modify it with further talents.

Stats for dataset:
Max: (972)
Min: (925)
Mean: (946.822485207101)
Datapoints: (169)


Third dataset, chaincasting frostbolt rank 13, wearing +399 damage gear, specced around imp frostbolt and piercing ice to get 5/5 emp frostbolt, tooltip damage range of 600 - 647.

Assuming Additive Emp Frostbolt, expected average is (((600 + 647) / 2) + ((((3.0 / 3.5) * .95) + .1) * 399)) = 988.3
Assuming Multiplicative Emp Frostbot, expected average is (((600 + 647) / 2) + (((3.0 / 3.5) * .95 * 1.1) * 399)) = 980.89

This dataset establishes whether or not emp frostbolt is an additive or multiplicative modification to the base coefficient.

Stats for dataset:
Max: (1011)
Min: (965)
Mean: (989.425414364641)
Datapoints: (181)




So, for the big crayon explanation, the theorycrafting for empowered Frostbolt/Fireball has been correctly handled by most people as additive, which is good as it nets you slightly more damage than you would have had if it was multiplicative.

Still remaining to be definitively researched is what order things happen if you have both empowered and improved frostbolt, since:

(.8143 * .9 + .1) = .83287
(.8143 + .1) * .9 = .82287

a .01 difference in the coefficient, nothing huge, but if we want to be retentive about theorycrafting (Which we mostly do), then we should know. That I'll do tomorrow, I need to sleep before work now. Enjoy!

Effort

I would love to take a gander at the full data, but your math seems simple enough to follow that I can't really see any possible way for it to be off.

TLDR XD

Interesting though.

Nice work, thanks for crunching the numbers.

And heres that fourth dataset:

Fourth dataset, chaincasting frostbolt rank 13, wearing +399 damage gear, specced 5/5 imp frostbolt 5/5 emp frostbolt, tooltip damage range of 600 - 647.

Assuming we multiply then add, expected average is (((600 + 647) / 2) + ((((3.0 / 3.5) * .95) *.9 + .1) * 399)) = 955.81
Assuming we add then multiply, expected average is (((600 + 647) / 2) + (((((3.0 / 3.5) * .95) + .1) * .9) * 399)) = 951.82
This dataset establishes what order things happen in when you have both imp and emp talents (a 1% difference in coefficient overall)

Stats for dataset:
Max: (975)
Min: (928)
Mean: (950.628676470588)
Datapoints: (272)


So, for a full summary of the important stuff here. Dataset 1 established that the spell range really is 600-647 and that over a reasonable sampling the average will meet theorhetical, so using average baseline tooltip information is acceptable for theorycrafting purposes.

Dataset 2 established that the coefficient assumption of ((3.0 / 3.5) * .95) is an accurate one to use as a baseline for +damage calculations for frostbolt. Solving to a coefficient of 0.814285714286 for untalented frostbolts.

Dataset 3 established that empwowered frostbolt is an additive .1 to that coefficient and that the coefficient assumtpion of (((3.0 / 3.5) * .95) + .1) is accurate, solving to 0.914285714286.

Dataset 4 re-established that the 10% reduction in coefficient from improved frostbolt is multiplicative, and that the relation between the two is that we add the .1, then multiply by .9, leaving us with a coefficient of ((((3.0 / 3.5) * .95) + .1) * .9) which solves to 0.822857142857

And for just 5/5 improved we have (((3.0 / 3.5) * .95) * .9) which solves to .732857142857

So, using good old signifigant figures, I've got three sig figs, so I have four useful digits in those readings, and so my final table is as follows:

No Spec: 0.8143
5/5 Improved: 0.7329
5/5 Empowered: 0.9143
5/5 Imp/Emp: 0.8229



Someone asked for data earlier, looking for some webspace to post it up to for you, its in a string of numbers seperated by :'s, thats how my addon records them and my perl script parses them, I'll post the full script once I find somewhere to hide it in a few minutes. Any questions?

Edit: D'oh, moved a decimal point and got caught.

Just out of curiosity, what were you chain casting on? And what is your crit chance?

Well I'm assuming crits were thrown out but yes the target would be good to know. Really nice work on this btw.

Troggs and wolves in the dwarven starting area, and base of 18% with no crit talents, I filtered crits from the data since to get base damage on them I'd need to do a calculation, and that introduces another variable.

I would be very surprised if those mobs have any resistance so it sounds good to me.

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Q u o t e:

Dataset 3 established that empwowered frostbolt is an additive .1 to that coefficient and that the coefficient assumtpion of (((3.0 / 3.5) * .95) + .1) is accurate, solving to 0.824285714286.

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This solves to .9142856. You're adding .1 not .01

Thanks Inian, got decimal place happy.

And Brewmaster, yeah if level 1/2 monsters have any resistances / weird stuff with them I'd be very surprised. Also part of why I did the first set of data to show that it falls between the expected range.

A side note, if we are to assume that Imp Fireball and imp Frostbolt behave the same way, and that Emp Fireball and Emp Frostbolt behave the same way (not a bad assumption to make, one worth proving at some point), then we have theoretical coefficients for fireball of the following:

Bare Spec: (3.5 / 3.5) = 1.000
5/5 Imp: ((3.5 / 3.5) * .9) = 0.900
5/5 Emp: ((3.5/3.5) + .15) = 1.150
5/5 Imp/Emp: ((3.5/3.5) + .15) * .9) = 1.035



Edit: Proved here - http://forums.worldofwarcraft.com/thread.html?topicId=108301152

proof of intelligence on mage forums

keep up the good work



PS
while browsing other class forums, i have seen that it is not only the mage forums that contains numerous trolls, QQ posts, and general idiocy, in fact, it exists on every single forum on the internet

----------We've heard that a million monkeys at a million keyboards could produce the complete works of Shakespeare; now, thanks to the Internet, we know that is not true.--------

holy crap, that's cool that you did that but i for one graduated college in the hopes that i would never have to do math again. evar. well, i work for an architecture firm now, so i do end up doing some calcs but it's mostly just degrees and simple fractions, adding subtracting etc.

anyway, it always blows my mind to see people willingly writing up these data sheets when they have nothing better to do, but hey some people really enjoy math and solving things... it's like soldiers, i sure as hell ain't gonna do it but i'm glad someone wants to

props

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Q u o t e:

No Spec: 0.8143
5/5 Improved: 0.7329
5/5 Empowered: 0.9143
5/5 Imp/Emp: 0.8229

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These numbers are +dmg per spell cast. To compare effectively, it's better to look at +dmg per second of casting:

No Spec: 0.8143 / 3.0 = 0.2714
5/5 Improved: 0.7329 / 2.5 = 0.2932
5/5 Empowered: 0.9143 / 3.0 = 0.3048
5/5 Imp/Emp: 0.8229 / 2.5 = 0.3292

For Fireball:
Bare Spec: (3.5 / 3.5) / 3.5 = 1.000 / 3.5 = 0.2857
5/5 Imp: ((3.5 / 3.5) * .9) / 3.0 = 0.900 / 3.0 = 0.3000
5/5 Emp: ((3.5/3.5) + .15) / 3.5 = 1.150 / 3.5 = 0.3286
5/5 Imp/Emp: ((3.5/3.5) + .15) * .9) / 3.0 = 1.035 / 3.0 = 0.3450

Other "normal" spells (eg Scorch): 0.2857

So adding 100 +dmg to your toon will add 27DPS for unspecced frostbolt spam, or 34DPS for fully talented fireball spam, or 28DPS for Scorch. Plus interactions from other talents.

Of course the best scaling spell int he game is stll fully talented Shaman Lightning Bolt:
(3.0 / 3.5) / 2.0 = 0.4286

Fully talented Lightning Bolt also gets 5% bonus damage and a 5% chance to cast a 2nd Lightning Bolt for free at the target.

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Q u o t e:
Fully talented Lightning Bolt also gets 5% bonus damage and a 5% chance to cast a 2nd Lightning Bolt for free at the target.

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Thanks for letting the Mage community know this important information.

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Q u o t e:


These numbers are +dmg per spell cast. To compare effectively, it's better to look at +dmg per second of casting:

No Spec: 0.8143 / 3.0 = 0.2714
5/5 Improved: 0.7329 / 2.5 = 0.2932
5/5 Empowered: 0.9143 / 3.0 = 0.3048
5/5 Imp/Emp: 0.8229 / 2.5 = 0.3292

For Fireball:
Bare Spec: (3.5 / 3.5) / 3.5 = 1.000 / 3.5 = 0.2857
5/5 Imp: ((3.5 / 3.5) * .9) / 3.0 = 0.900 / 3.0 = 0.3000
5/5 Emp: ((3.5/3.5) + .15) / 3.5 = 1.150 / 3.5 = 0.3286
5/5 Imp/Emp: ((3.5/3.5) + .15) * .9) / 3.0 = 1.035 / 3.0 = 0.3450

Other "normal" spells (eg Scorch): 0.2857

So adding 100 +dmg to your toon will add 27DPS for unspecced frostbolt spam, or 34DPS for fully talented fireball spam, or 28DPS for Scorch. Plus interactions from other talents.

Of course the best scaling spell int he game is stll fully talented Shaman Lightning Bolt:
(3.0 / 3.5) / 2.0 = 0.4286

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+damage benefit/second is not a good measure of "best scaling." +damage benefit / base damage is a much better measurement.

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Q u o t e:

These numbers are +dmg per spell cast. To compare effectively, it's better to look at +dmg per second of casting:

No Spec: 0.8143 / 3.0 = 0.2714
5/5 Improved: 0.7329 / 2.5 = 0.2932
5/5 Empowered: 0.9143 / 3.0 = 0.3048
5/5 Imp/Emp: 0.8229 / 2.5 = 0.3292

For Fireball:
Bare Spec: (3.5 / 3.5) / 3.5 = 1.000 / 3.5 = 0.2857
5/5 Imp: ((3.5 / 3.5) * .9) / 3.0 = 0.900 / 3.0 = 0.3000
5/5 Emp: ((3.5/3.5) + .15) / 3.5 = 1.150 / 3.5 = 0.3286
5/5 Imp/Emp: ((3.5/3.5) + .15) * .9) / 3.0 = 1.035 / 3.0 = 0.3450

Other "normal" spells (eg Scorch): 0.2857

So adding 100 +dmg to your toon will add 27DPS for unspecced frostbolt spam, or 34DPS for fully talented fireball spam, or 28DPS for Scorch. Plus interactions from other talents.

Of course the best scaling spell int he game is stll fully talented Shaman Lightning Bolt:
(3.0 / 3.5) / 2.0 = 0.4286

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While you are generally correct, you're missing a lot in there for a full spell compare, which isn't what this research is for. I'm just establishing proven coefficients and mechanics that relate to coefficients. The rest will come, Ice Shards / Firepower / Playing With Fire are my next targets.

When you're talking about scaling of spells you need to make allowances for the %age increase talents too, it can't be done off of coefficients alone. For a quick example:

Fireball rank x does 100 base damage, gets 100% of + damage
Scorch rank y does 50 base damage, gets 42.86% of +damage

Cast fireball with 100 +damage, you get 200 base hit, factor in firepower, you're at 220 damage.

Cast scorch with 100 +damage, you get 142.86 base hit, 157.146 with firepower. One netted 15ish damage from firepower, the other 20. So since you get two scorches per 1 fireball, you're getting 30 from firepower on scorch, and 20 on fireball, or roughly 6.66 DPS fireball 10DPS scorch. Scalability computations are a bit more complex. Don't worry, I'll get to them at some point once I finish getting a working model of how all the talents really effect things instead of just tooltip theorycrafting them.