Meta Rate My Team PokéBase - Pokémon Q&A
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Thanks to a variety of posts and websites, I understand (in concept at least) the benefits of natures and EV training.

What I want to better understand is the role of personalities in the game. If I'm correct, the personality tells you what their highest IV is. What would the stats of a Pokémon with a Gentle nature with the Highly Persistent personality look like?

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Knowing just the Nature and the Personality of a Pokemon doesn't give you enough information to determine any real specifics about a Pokemon. You can really only make general guesses.

In this case, a Gentle nature will reduce Defense by 10%, but will also increase Special Defense by 10%. Competitively speaking, this is one of those "never use ever" Natures, because it reduces Defense. Ideally, you always want to reduce the offense stat you're not going to use, or in a Trick Room team to reduce Speed (which effectively increases it instead).

The Personality Trait tells you which stat the Pokemon's highest IV is in, or is tied for. "Highly Persistent" corresponds to Defense. But that's all the information that it gives you. It doesn't give you any indication of how high, just that it is the highest or tied for highest. Meaning it could be a 31. Or it could be a 12. It's just generally not a very useful piece of information.

We do kind of get lucky here, though. A Gentle nature and a Highly Persistent personality both affect the Defense stat. This means that this particular Pokemon is probably not a good choice to use for battling, because it's stat that has it's highest IV is also reduced by 10%.

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The Pokémon in question is a Kangaskhan. Thanks to Serebii, I have been able to see the little differences the Natures can make on Pokémon. I actually did not know that the personality trait would list a tied IV. Thank you for that.

Now let's say for the moment that we took two identical Pokémon with neutral natures, but only one had a perfect IV in any given stat. How much of a difference would a perfect IV make compared to another Pokémon with a zero in the same stat?
Each IV adds one actual stat, so 31.
I just did the math. A detrimental nature plus a perfect IV equals only a slight advantage compared to what it could've been. I don't profess to understand the mindset behind obsessively trying to breed the perfect team, but with this I can appreciate the advantages of paying attention to these variables.

Thanks guys!
In competetive battling, people do calculations to make sure their pokemon will survive a certain number of hits as long as they have full IVs and the right nature and against another specific pokemon with full IVs and the right nature. And if they don't have full IVs or they have the wrong nature, it makes a big difference when going against the standard EV spreads that have been perfected for these things. So overall, it gives you a huge disadavntage if you haven't perfected your pokemon.
Perhaps my 'obsessive' remark was a bit hasty. How do you go about figuring this out?
Go to Smogon, where you will find the stratergy pokedex, the damage calculator and the battle simulator.
Thanks for letting me know. I appreciate your help.