PokéBase - Pokémon Q&A
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I wanted to know if anyone knew the best stats a eevee can have straight out of the egg


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You can't tell; most starting stats are 5 to 8 (except HP which is usually 10) for nearly all species (with some exception). Stats at level 1 is no indication to how good a Pokemon can be later down the road.

Best way to check in Gen V is go to the Nimbasa Battle Subway. There's a guy dressed as an Ace Trainer just ahead of you as you walk down the steps, and he'll tell you your Pokemon's potential (i.e. IVs).

You can only tell how good your Pokemon's IVs are at a higher level, and usually level 50 or 100 is the most accurate in showing how good your Pokemon's stats are.

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Well, it will have the largest stat total if it has a neutral nature with perfect IVs. It's stats are too low at level 1 to have the nature boost anything, yet it will still lower stats. So these are the highest it can have:

12 HP
6 Atk
6 Def
6 SAtk
6 SDef
6 Spd

Source: I experimented on Showdown.