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The base damage of Acrobatics is 55 but with no held item it comes 110 and with a flying Pokémon with a held item, Acrobatics does 55 plus 50% stab = 77.5, when considering in a battle with a stab Acrobatics with no held item how much damage will it do? will stab be activated first then doubles (55+50%)*2 = 155 or will the doubling of the Acrobatics damage occurred then stab? (5*2)+50% = 165.

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Just so you know, 55+50% STAB is 82.5, not 77.5. So either way, you'll get 165.

1 Answer

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Acrobatics's no-item mechanic is a power modifier, and STAB is a damage modifier. This means the game will double the base power before applying STAB.
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The damage varies a lot depending on other numbers, most importantly the attacker's level and physical attack stat and the defender's physical defense stat. It will very rarely do exactly 55 * 2 * 1.5 = 165 damage. However, if you change only the base power and keep the other numbers constant, the damage is roughly proportional to the base power, so acrobatics in this situation would do roughly the same damage as a move with 165 base power.

Also this has nothing to do with Pokemon, but this website uses a formatting thing called Markdown. When you type an asterisk, it thinks you want italic text. You can "escape" the asterisk by putting a backslash before it. In other words...
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Thanks for specifying the technicalities I didn't know about. Nice answer :P