# Attacking Types

## Overview

You’ve all heard that Fire and Ice are great attacking types, while Poison and Normal are bad ones. This article will attempt to justify these claims mathematically.

## What makes an attack type great?

In order to tackle this problem mathematically, it is important to know what makes an attacking type good or bad. What makes Fire a better attacking type than, say, Bug? The answer is that the more common Pokemon in the metagame are dealt more damage by the better attacking types.

This suggests that we need to look into the following aspects of Pokemon:

1. Their usage.
2. Their type effectiveness.
3. Their defensive stats.

This means that all of the above need to be in a hypothetical mathematical formula if we need to solve this problem.

## The damage dealt to a Pokemon

Of course, to quantify the amount of damage dealt to a Pokemon, we need to resort to the damage formula. Here, we are going to use a simplified version, because we can assume that all Pokemon are at level 100 and have 31 IVs in all stats. Let us further assume that the Pokemon that is going to deal damage has 300 Atk (or SpA) stat and is using a move having 80 Base Power. One final assumption is that we will always deal the maximum amount of damage, i.e. the random factor in the damage formula is ignored.

The damage dealt to a Pokemon would then be equal to

`Damage = 0.84 * 300 * 80 * TypeEffectiveness / (OppHP * OppDef/SpD)`

Where TypeEffectiveness is 4, 2, 1, 0.5, 0.25 or 0 depending on how effective that move is on the foe, OppHP is the average HP stat of the foe and OppDef/SpD is the average Def or SpD stat of the foe.

This can be simplified to

`Damage = 20160 * TypeEffectiveness / (OppHP * OppDef/SpD)`

However, since the damage dealt cannot be greater than 100% (or 1), the formula needs to be slightly modified:

`Damage = min(1, 20160 * TypeEffectiveness / (OppHP * OppDef/SpD))`

Notice that we have already used points 2) and 3) above. The only thing that remains is incorporating the usage of the Pokemon. We are simply going to weigh the damage done by the usage percentage of the Pokemon in question:

`DamageUsage = Damage * PercentageUsage`

where PercentageUsage is the percentage usage of the foe. This can be written as

`DamageUsage = min(PercentageUsage, 20160 * PercentageUsage * TypeEffectiveness / (OppHP * OppDef/SpD))`

We thus have a formula that incorporates all three aspects we’re looking for. Note, however, that the formula above is really two formulae in one. One formula looks at the physical aspect of the type, while the other looks at the special aspect of the type. So what we really have are the following two formulae:

`PhysicalDamageUsage = min(PercentageUsage, 20160 * PercentageUsage * TypeEffectiveness / (OppHP * OppDef))`
`SpecialDamageUsage = min(PercentageUsage, 20160 * PercentageUsage * TypeEffectiveness / (OppHP * OppSpD))`

## Attacking Type Effectiveness (ATE)

To get to our final attacking type effectiveness, which we shall call ATE for short, all we need to do is to sum up all the damage done to all the Pokemon in existence. That is:

`ATE = DamageUsage_1 + DamageUsage_2 + … +DamageUsage_n`

where n is the number of Pokemon available (in DPP this is equal to 505).

We can either find the physical ATE only (PATE), the special one only (SATE), or combine them together. If we want to combine them together, we just find the average of the physical and special attacking type effectiveness.

## Attacking Type Modifier Estimate (ATME)

Sometimes, we wish to know an estimate as to how much damage a move would deal to the metagame on average. For this purpose, the Attacking Type Modifier Estimate (ATME) can be used.

We basically want the ATME to replace the part in the damage formula where we divide by the product of the opponent’s HP and Def/SpD, like so:

`Average Damage done by move = 0.84 * Atk/SpA * MovePower * STAB / ATME`

It turns out that the ATME for a particular type can be found easily from the attacking type effectiveness of the move as follows:

`ATME = 24000 / ATE`

Again we can find the physical ATME (PATME), the special ATME (SATME) or a combination of both, by using the appropriate ATE in the formula above.

## Attacking Type Effectiveness in the various Metagames

I have endeavored to calculate the ATE and ATME for the standard, underused and uber metagames. Credits go to DougJustDoug’s Smogon server statistics, without which I could not have done this. From Doug’s statistics, I extracted the average HP, average Def and average SpD stats of every Pokemon, taking into account also item bonuses. Then I calculated the type effectiveness of every Pokemon, remembering that a few abilities can also modify it (for example Water Absorb, Thick Fat and Wonder Guard). Of course, I had to do the above separately for each of the three metagames.

Following are two tables for each metagame. The first table lists, in descending order, the physical and special ATEs and ATMEs for all types. I also included a hypothetical type called ‘typeless’, which does neutral damage to every type, for comparison purposes. The second table lists the combined ATEs and ATMEs for all types.

### Standard Metagame:

Type ATE ATME
Special Fire 34.04% 70506
Physical Fire 33.50% 71638
Physical Ice 33.07% 72572
Special Ice 32.64% 73534
Physical Rock 31.60% 75955
Special Rock 30.64% 78330
Physical Flying 30.51% 78651
Physical Electric 30.29% 79227
Physical Fighting 30.12% 79681
Special Water 30.03% 79914
Special Electric 29.90% 80264
Special Flying 29.84% 80417
Physical Dark 29.80% 80550
Special Fighting 29.74% 80696
Physical Water 29.56% 81203
Physical Ghost 28.89% 83063
Physical Typeless 28.83% 83261
Special Dark 28.80% 83326
Physical Dragon 28.75% 83466
Physical Ground 28.74% 83498
Special Typeless 28.52% 84150
Special Ghost 28.40% 84510
Special Ground 28.26% 84931
Special Dragon 27.64% 86821
Physical Psychic 27.51% 87241
Special Psychic 26.90% 89206
Physical Bug 26.05% 92137
Special Grass 25.71% 93357
Physical Grass 25.63% 93632
Special Bug 24.99% 96037
Physical Steel 23.89% 100468
Special Steel 23.05% 104103
Physical Normal 22.84% 105087
Special Normal 22.22% 107999
Physical Poison 21.07% 113896
Special Poison 19.84% 120991
Type ATE ATME
Fire 33.77% 71068
Ice 32.85% 73050
Rock 31.12% 77124
Flying 30.18% 79524
Electric 30.10% 79742
Fighting 29.93% 80185
Water 29.79% 80554
Dark 29.30% 81914
Typeless 28.67% 83703
Ghost 28.65% 83780
Ground 28.50% 84208
Dragon 28.20% 85110
Psychic 27.21% 88212
Grass 25.67% 93494
Bug 25.52% 94046
Steel 23.47% 102253
Normal 22.53% 106523
Poison 20.45% 117337

### Underused Metagame:

Type ATE ATME
Physical Flying 38.60% 62175
Special Flying 36.83% 65157
Physical Ice 36.75% 65311
Physical Rock 36.69% 65421
Special Ice 36.01% 66656
Special Rock 34.76% 69036
Physical Psychic 33.64% 71349
Physical Fire 33.42% 71812
Physical Dark 33.24% 72191
Special Water 32.53% 73782
Physical Water 32.43% 74010
Special Fire 32.43% 74014
Physical Electric 32.28% 74357
Special Psychic 31.82% 75419
Special Dark 31.72% 75656
Special Electric 31.59% 75962
Physical Typeless 31.58% 75997
Physical Dragon 31.23% 76843
Special Fighting 31.14% 77069
Physical Fighting 31.10% 77175
Physical Ground 30.96% 77509
Physical Ghost 30.60% 78427
Special Typeless 30.58% 78473
Special Ground 30.19% 79486
Special Grass 30.11% 79715
Special Dragon 30.06% 79841
Physical Grass 30.02% 79960
Physical Bug 29.71% 80770
Special Ghost 29.08% 82519
Special Bug 28.97% 82848
Physical Poison 28.04% 85580
Physical Normal 27.34% 87793
Physical Steel 27.24% 88110
Special Poison 26.61% 90181
Special Normal 26.55% 90409
Special Steel 26.52% 90509
Type ATE ATME
Flying 37.72% 63631
Ice 36.38% 65977
Rock 35.72% 67180
Fire 32.92% 72896
Psychic 32.73% 73327
Dark 32.48% 73883
Water 32.48% 73896
Electric 31.94% 75151
Fighting 31.12% 77122
Typeless 31.08% 77215
Dragon 30.65% 78313
Ground 30.58% 78485
Grass 30.06% 79837
Ghost 29.84% 80421
Bug 29.34% 81796
Poison 27.33% 87820
Normal 26.94% 89082
Steel 26.88% 89293

### Uber Metagame

Type ATE ATME
Physical Ice 36.87% 65092
Special Ice 35.79% 67063
Physical Dragon 32.35% 74192
Physical Dark 31.06% 77276
Physical Bug 30.59% 78455
Special Dragon 30.22% 79414
Physical Ghost 30.20% 79458
Physical Rock 29.63% 80987
Special Fire 29.11% 82434
Special Dark 29.09% 82494
Special Ghost 28.80% 83329
Physical Fire 28.65% 83775
Special Bug 28.55% 84062
Physical Flying 28.41% 84475
Special Rock 27.90% 86031
Physical Typeless 27.66% 86752
Physical Electric 27.33% 87800
Special Flying 27.12% 88504
Special Typeless 26.52% 90498
Physical Fighting 26.14% 91828
Special Electric 25.24% 95087
Physical Ground 25.21% 95196
Physical Grass 25.12% 95541
Special Fighting 24.77% 96876
Special Ground 24.57% 97670
Physical Normal 24.07% 99727
Special Grass 23.67% 101404
Special Water 23.23% 103330
Physical Water 23.17% 103584
Physical Steel 22.96% 104515
Physical Poison 22.73% 105606
Special Normal 22.66% 105923
Special Steel 22.03% 108965
Special Poison 20.68% 116073
Physical Psychic 20.36% 117885
Special Psychic 19.21% 124966
Type ATE ATME
Ice 36.33% 66063
Dragon 31.28% 76715
Dark 30.08% 79800
Bug 29.57% 81162
Ghost 29.50% 81347
Fire 28.88% 83099
Rock 28.77% 83433
Flying 27.76% 86443
Typeless 27.09% 88585
Electric 26.29% 91298
Fighting 25.45% 94285
Ground 24.89% 96418
Grass 24.39% 98385
Normal 23.36% 102731
Water 23.20% 103457
Steel 22.49% 106693
Poison 21.70% 110593
Psychic 19.78% 121322

## Examples of how to use ATME

For our first example, say we want to see the difference in damage between Thunderbolt and Sludge Bomb on Gengar in the Standard metagame.

Sludge Bomb has 90 base power and gets STAB. Its ATME is 120991. Hence its estimated power in the Standard metagame would be 90 * 1.5 / 120991 = 0.00112.

Thunderbolt has 95 base power. Its ATME is 80264. Hence its estimated power in the Standard metagame would be 95 / 80264 = 0.00118. Hence the use of Thunderbolt over Sludge Bomb on Gengar is justified, even though Gengar gets STAB on the move.

For a second example, suppose we have a mixed Blaziken in UU and we’re pondering whether to use ThunderPunch or Hidden Power Electric. We’ll assume that it has 324 Atk stat and 350 SpA stat.

ThunderPunch has 75 base power. Its ATME is 74357. Hence its estimated power in the UU metagame would be 324 * 75 / 74357 = 0.3268 (32.68%).

Hidden Power Electric has 70 base power. Its ATME is 75962. Hence its estimated power in the UU metagame would be 350 * 70 / 75962 = 0.3225 (32.25%).

Hence they are basically equivalent as far as average damage is concerned. The choice of the moves would then go down to what you’re going to use your Electric move against, whether against physically-defensive Pokemon or specially-defensive ones. Note also that the percentages produced by the above calculations are the estimated average damage that the moves would deal in the UU metagame by that particular Blaziken.