There is no way to manipulate this. Stats are affected by 5 things:
- Base Stats
Base stats are always the same for the same breed of Pokemon. Abra will always have a base speed of 90, and Kadabra will always have a base speed of 105. Note that this number is relatively meaningless on its own, but can be used to usefully distinguish relative stats of different Pokemon. The base stat is tossed into an equation to calculate the current stat.
Nature is always determined at the time that the Pokemon is initially created. It either has no effect on stats, or increases one by 10% and decreases another by 10%.
The Level of the Pokemon is obviously included. The higher your level, the higher your stats get.
IVs are set per individual Pokemon. They are randomized at the time that the Pokemon is initially created, and can never be altered. They are an invisible modifier, so there is no convenient way to check them.
EVs are acquired by battling with the individual Pokemon, or by feeding it certain berries. This is the only way to customize stats after the Pokemon has been initially rolled. It is worth noting that you will only see changes to stats based on EVs when you level up. This can make it appear that you are getting a huge boost to your stats if you do a ton of EV training between levels. This may be the effect you are describing.
For more information on how these things are combined into the actual value of the stat, see Bulbapedia's article on the formula.