[Coda]

Pokemon have three stats, which aren't directly exposed on the UI: Stamina, Attack, and Defense. (CP is a weighted value derived from these stats that tries to give a straightforward way to compare two Pokemon to see which is better without exposing the stats directly to the player.) Each Pokemon, when it's generated, is assigned a random value from 0 to 15 for each of these stats. These values are called IVs and they're a bonus on top of the stats the Pokemon would have from its species and level.

Let's take Articuno as an example. It has a base Attack stat of 192. If it had 15 Attack IV (the appraiser would claim that its attack is the highest possible) then it would be treated as 207, but a 0 Attack IV would still be 192. To determine the Pokemon's actual stat, this base value is then multiplied by a fraction based on its level. At level 20 (that is, if your trainer level is 20 and the Articuno's power up bar is maxed out) that multiplier is approximately 0.6. So the 0 IV Articuno would have an actual Attack stat of 115, while the 15 IV one would have an actual Attack stat of 124. The 15 IV Articuno would therefore be dealing ~8% more damage with every attack than the 0 IV one. This isn't an overwhelming advantage, but it's not insignificant either.

However, because of the way the damage formula works and where the rounding takes place, that extra little bump could end up making a HUGE difference in actual damage output: If a fast attack would deal 1.95 damage, it would get rounded down to 1, but if it would deal 2.05 damage, it would get rounded down to 2. That tiny little bump is technically only a 5% increase before rounding, but AFTER rounding that tiny little bump means you're dealing TWICE AS MUCH damage with your fast attack. This is most significant for high-speed low-power fast attacks like Fury Cutter, and when you've only got 80 HP, dealing 2 HP per fast attack instead of 1 HP means you need 40 fewer attacks to win the flight.

On the flip side, consider a Pokemon like Wailord, which actually has surprisingly low defense because it relies on its HP for survivability. A level 20 0 IV Wailord has 204 HP and 52 Defense, while a level 20 45 IV Wailord has 213 HP and 61 Defense. This means that the tougher Wailord can survive 22% more damage than the weaker one -- that's like being able to just ignore one out of every 5-6 incoming hits!