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Individual Values, or IVs can be compared to the Pokémon's "Genes".
Individual values
They range from 0 to 31 for every stat and are added to the pokémon's stat for their value at level 100. (example: a level 100 Tyranitar with no Effort Values and 0 IVs has 310 HP, while if it had 31 IVs, it would have 341 HP)
These stats are provided randomly for every pokémon you catch or breed, and although it may seem that 31 points do not make much of a difference, they are indeed important for Ace Trainers to put into account when breeding pokémon with perfect natures/stats.
An example of how IVs can be important: Let's say you have a Terrakion with 0 Attack IVs. At level 100, he will have an attack of 358 (with an attack improving nature), while a Terrakion with perfect Attack IVs would have 392 Attack. This small difference can mean the difference between a OHKO and a survival with 1 hp (it is more common than you may think).
Fortunately for us, Ace Trainers and Pokémon Breeders specially, these "IVs" can be bred so the quest for the perfect pokémon can begin.
The process of breeding IVs is as follows, the example displayed below is to breed Nidorans:
1. The kid's IV's are generated randomly, for example: 7/27/31/14/19/2, in HP/ATK/DEF/SPA/SPD/SPE format 2. Three stats are inherited from his parents, and these can stack. The game makes three checks to the pokémon's stats:
a) First check: a random stat is chosen from either the Mother or the Father and passed upon the kid (HP/ATK/DEF/SPA/SPD/SPE)
b) Second check: a random stat but HP is chosen from either the Mother or the Father and passed upon the kid (ATK/DEF/SPA/SPD/SPE)
c) Third check: a random stat but HP and Defense is chosen from either the Mother or the Father and passed upon the kid (ATK/SPA/SPD/SPE)
This means that HP and Defense are harder to pass upon the son, but do not give up hope! There are ways to make sure the IVs are passed on; Power Items!
By letting any of the parents hold a Power Item, you can make sure that that stat will be passed upon the offspring. Say you put a Power Weight (Increases the HP gain of the holder on battle) on the Mother, who has a perfect IV of 31 on HP
If the first check (a) selects the Mother, you are sure to have a kid with perfect IVs on HP. The other checks will be random, but if you're either lucky or patient you'll eventually get the stats you are looking for. Remember: only three stats are inherited per Pokémon and these can stack: you can get the Defense IV from both the Mother and then the Father, clearly wasting IVs...
How to check your IVs:
There has always been an NPC that allows you to check your Pokémon's IVs since third generation Pokémon games.
Pokémon HeartGold: In the Battle Tower, the scientist-looking NPC closest to the PC, on the top right side of the room. He will only enlighten you in your Pokémon's higher IV everytime you speak to him, so you should speak to him more than once to be sure you know all the stats your pokémon has.
Pokémon Black: In the Battle Subway, the closest NPC to the exit. He will enlighten you on the highest IVs your pokémon has. If it has more than one, he will tell you right away, according to the following list:
Sum of the IV's:
-"This Pokémon's potential is decent all around" (0-90).
-"This Pokémon's potential is above average overall" (91-120).
-"This Pokémon has relatively superior potential overall" (121-150).
-"This Pokémon has outstanding potential overall" (150-186).
Highest IV:
-"It's rather decent in that regard" (0-15).
-"It's very good in that regard" (16-25).
-"It's fantastic in that regard" (26-30).
-"It can't be better in that regard" (31).