PokéBase - Pokémon Q&A
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When you use Swords dance or have the ability Huge power/Pure power does it multiply your base attack x2 or your total attack x2. I have commonly heard people say that Scizors attack becomes base 260 but I have noticed in many situation it seems that many times the base stat double is not as high as a total stat double example. Azumarill 50 attack stat becoming 100 attack stat does not raise it nearly as high as total 217 to 434. So which is it? a base stat raise or final stat raise.

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The Ability Huge Power / Pure Power double the attack stat. Ie. if you have a base stat of 50, it becomes 100. Swords dance is a temporary raise, and will go away after you switch out or have it lowered. Swords dance essentially doubles your attack as well, but only temporarily.

Conclusion:

Swords Dance: Doubles total attack.

Huge/Pure Power: Doubles the base attack stat.

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I did some research including damage calculation and Bulbapedia and found that no move or ability increases base stats because base stats are the Pokemon's stats at level 50 with no evs and no ivs. So both increase total attack.
Umm Speed, even the first part of what you said is wrong,

Flygon at level 50 with No Iv's or Ev's and a Neutral nature has

140 Hp
105 Atk
85 Def
85 Sp ATk
85 Sp Def
105 Speed

Although you are right about the fact that it doesn't involve base.
Oh right it also has something to with nature or something.