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Heart Gold and Soul Silver: EVs
Introduction - What are EVs?
EV stands for Effort Value. EVs are hidden values that help determine the amount that each stat is raised when a Pokemon gains a level. EVs are the reason why a certain stat might gain 4 points, while another stat may only gain 1.

Explanation - How do EVs work?
When your Pokemon defeats another Pokemon in battle a few things happen. Your Pokemon gains experience and EV points. You can visibly see the amount of experience your Pokemon has gained, but the EV points gained are hidden. Luckily, it is easy to determine how many EV points your Pokemon has received after a battle. Typically, basic Pokemon give out 1 EV point for one stat and evolved Pokemon give out 2 EV points in one stat. However, some Pokemon give 1 EV point in 2 different stats. For example, Parasect gives 1 EV point in Attack and 1 in Defense.

Every 4 EV points means 1 stat point. For example, if your Pokemon gets 4 Attack EV points from fighting 4 Mankey, that equals +1 Attack point the next time it levels up. The total amount of EV points a Pokemon can have is 510. The total number of EV points a single stat can use is 255. Let's use a little bit of simple math to maximize the effect of the EVs: 255 + 255 = 510 Now remember that EV points convert to stat points in divisions of 4. 255 is not evenly divisible by 4, which means spreading EV points between only 2 different would be wasting a few valuable EV points. To remedy this, the spread of EV point is 252 252 4, meaning 2 stats get 252 EV points and another stat get 4 EV points. Because 252 + 252 + 4 = 510, this spread ensures no wasted EV points.

In-Depth - How do I use EVs to my advantage?
Training a Pokemon with certain EVs in mind is called EV training. For example, training a competitive Gengar probably means you want it to have good Speed and Special Attack. This means you would train your Gengar against Poliwag for Speed and Gastly for Special Attack. This will ensure your Gengar has the highest Speed and Special Attack it possibly can.

In-Depth - How do I optimize EV training?
A valuable tool in EV training is the Macho Brace. The Macho Brace doubles the EV points your Pokemon receives at the end of a battle. Let's say you want to EV train a Pokemon in HP, so you go to the Viridian Forest to battle Caterpie. Each Caterpie gives out 1 HP EV point. That means you would normally have to battle 252 Caterpie in order to fill out its HP EV points. When the Pokemon you are EV training is wearing a Macho Brace it would only have to battle 126 Caterpie because your Pokemon receives 2 HP EV points instead of 1.

Along with the Macho Brace, there are the Power Items. Each Power Item gives 4 EV points in its related stat at the end of battle, regardless of the Pokemon you fight, in addition to whatever EV points the opposing Pokemon gives. For example, if your Pokemon is holding a Power Belt and it defeats a Caterpie in battle, your Pokemon will receive 1 HP EV point from Caterpie and 4 Defense EV points from the Power Belt.

Power WeightHP
Power BracerAttack
Power BeltDefense
Power LensSpecial Attack
Power BandSpecial Defense
Power AnkletSpeed

The Macho Brace can be obtained through an in-game trade for a Machop in Goldenrod City. The Power Items can be found in the Battle Frontier and can be bought with Battle Points.

Wrap Up:
If you want an advantage against uninformed Pokemon players, EVs are the best place to start. They allow you to manipulate your Pokemon's stats to a certain extent which gives a competitive edge.