The CMC Pokèmon League

Pokémon's stats are determined by a few factors:

1) What pokémon it is. Pikachu has different stats to Charizard. This is both obvious and boring.

2) The pokémon's level. A multiplier, again somewhat obvious.

3) How the pokémon has been trained (Effort Values or EVs). Each pokémon defeated, as well as things like Carbos and Proteins, add an amount of EVs that get translated into extra stat points (up to a certain limit).

4) A random pre-determined factor (Individual Values or IVs). Each pokémon has randomised values for each stat between 0-31. These translate to additional stats as well - basically a pokémon with a Defence IV of 31 will have 31 more Defence than a pokémon with a Defence IV of 0 by the time it's level 100, all other factors being the same.

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It's number 4, the IVs, where "legal" netplay pokémon differ from legit pokémon. If you automatically have IVs of 31 to your six stats that's a bonus 186 statpoints compared to a mean of 93, so a sizeable difference. And the odds of naturally getting such a perfect pokémon are 1/32⁶ (1/32 for each stat, ^6 for six stats), which is approximately one in a billion chance.

Or lower chance if you want a beneficial nature (that gives +10% to one stat and -10% to another stat), and lower chance again if you want a specific gender (for evolutions, charm abilities, etc).