PokéBase - Pokémon Q&A
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A burn breaks a Pokémon’s attack in half, and a swords dance doubles a Pokémon’s attack, but every time I see something like that happen the attack seems weaker, could someone please explain this to me?

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First lets see about Burn

Generally, if a Pokémon is burned, it will lose a set amount of HP
every turn, and its damage dealt by physical moves will be halved.

Bulbapedia says.It reduces the Base power of the move and not the stat while Swords Dance increase the Attack stat and not the Base Power.

Swords Dance increases the user's Attack stat by two stages.

For Swords Dance.

So I'm not sure using Swords Dance can even out a Burn everytime but still I'll give you some calculation to give a better conclusion.

252+ Atk Arceus Extreme Speed vs. 248 HP / 244+ Def Arceus-Fairy:
87-103 (19.6 - 23.2%) -- possible 5HKO

252+ Atk burned Arceus Extreme Speed vs. 248 HP / 244+ Def
Arceus-Fairy: 43-51 (9.7 - 11.5%) -- possible 9HKO

+2 252+ Atk burned Arceus Extreme Speed vs. 248 HP / 244+ Def Arceus-Fairy: 87-102 (19.6 - 23%) -- possible 5HKO

According to this and some more calculation the damage deal output is that Swords Dance can even out a Burn but not directly but by raising the Attack stat.

Hope I Helped

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Thanks, this helped a lot.
Always happy to help :)
This is true only if the physical attack stat has 0 or -2 stages. If it's at less than -2 or more than 0, swords dance is less effective than a burn. If it's at exactly -1, swords dance is more effective than a burn.