PokéBase - Pokémon Q&A
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What the title says. How are points added to stats every time a Pokemon levels up? Is there a set amount that is affected by EVs, natures, etc?

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There's a formula that determines what the stat is, which uses variables such as IVs, EVs, Nature, base stat and level.

Stat formula for Gen III onward
(btw that last word is nature lol. Nature = 0.9 if it hinders the stat; Nature = 1.1 if it benefits the stat.)

So a Rare Candy would still raise a Pokemon's stats by some amount. However, the EVs gained from battles while on that level grow the stat higher upon level increase.

Those are the only two variables in the formula that really change at any point; the other variables are determined at the moment of generation and remain constant forever unless it is hacked or something.

Hope I helped. :)
Source: Individual Values article from Bulbapedia + Experience

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Yup I agree this is accurate. Basically those two equations constantly update and recalculate each time there is change in EVs or level, which is what produces the addition (or subtraction in some cases) of stats. Note, the changes do apply without a level up in modern games; this is why stats change on non-level up evolutions (due to base stat changes) and why you'll occasionally observe your Pokemon's stats, especially HP, arbitrarily increasing outside of level up -- this is their new EVs taking effect.
Thank you both for the help!
*fizz's approval increases my self-esteem +1*