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They are really confusing to me. What exactly are they?

(A question like this didn't show up when I was typing the title, but if it is a double, please hide it. :)

Note that this is in Black 2.

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Individual Values.
They are numbers for every stat that are determined when you first meet the Pokemon.

Each stat's IVs can go to a maximum of 31, with each of those 31 numbers being worth one point in the stat it is for. So a Pokemon could have 6 Attack IVs, 15 Defence IVs, 31 SAtk IVs, etc.

This isn't the only thing IVs do. They also determine the characteristic your Pokemon has depending on which stat has the highest IVs (the things like "Proud of it's power" and "Loves to eat"), but this does not mean that the stat that the characteristic matches has 31 IVs. It could have as low as two with the other stats having just one IV. They will also determine which type of Hidden Power the Pokemon has if it learnt the move and how much Base Power it'd have, but there's a load of math behind that.

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IVs are individual values. Each Pokemon is assigned 0-31 IVs in each stat. IVs increase the stats like EVs. There are complicated ways to breed your Pokemon to have perfect IVs, but it takes lots of patience. IVs also affect what type and power your Pokemon's Hidden Power is. That's about it I think. Source: Experience and research done a while ago.

IVs aren't shown ingame except for a characteristic such as loves to eat being HP or likes to thrash about being attack. To find the IVs of a Pokemon, here is a link to a good IV calculator. It is pretty straight forward and easy to use. Psypokes IV Calculator

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