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Stats

Maxiumum Stats

Maximum Stats, generally refered to as Max stats, are the highest attainable stats for a certain species of Pokémon. Finding a Pokémon in the wild that has max stats has a VERY low chance of occurance(infact, a 1/65536 chance). However, wild max stats and trained max stats are different. For example, two level 7 Sandshrew, one max stat trained, the other caught in wild with max stats:

Wild
Trained

HP - 26
HP - 30
Attack - 17
Attack - 22
Defense - 19
Defense - 23
Speed - 12
Speed - 17
Spc.Att - 9
Spc.Att - 14
Spc.Def - 11
Spc.Def - 15

If a species of Pokémon can be male, than the female's of that species will NEVER be max stat'd. This of course, changes if the species is female-only(So its possible for a Nidoqueen to be max stat'd, but not a female Rhydon). Finally, Max Stats have a DV ranking of 15/15/15/15/15.

Shiny Stats

For a Pokémon to be shiny, it must have 1 of 8 DV possibilities. Take a look at the chart:

HP Att Def Spd Spc
8 15 10 10 10
0 14 10 10 10
8 11 10 10 10
0 10 10 10 10
8 7 10 10 10
0 6 10 10 10
8 3 10 10 10
0 2 10 10 10

*Note that the Special stat means both Special Attack and Special Defense

For your reference, the following Pokémon CANNOT ever be a shiny when they are female:

BulbasaurIvysaurVenusaurCharmander
CharmeleonCharizardSquirtleWartortle
BlastoiseEeveeVaporeonJolteon
FlareonOmanyteOmastarKabuto
KabutopsAerodactylSnorlaxChikorita
BayleefMeganiumCyndaquilQuilava
TyphlosionTotodileCroconawFeraligatr
TogepiTogechicEspeonUmbreon

Using Stats to find DV
DVs, short for diversification value/deter value, AKA gene/bits/whatever, affect many different aspects of the game. A main reason to find a Pokémon's DVs is because its the first step in finding its Hidden Power type.

How do we find the DV? Its not as hard as you may think. Using this formula, we can find a Pokémon's attack, defense, speed, special att/def DV:

15-((M-A)/2) = D
M = Maximum Stat
A = Stat of your Pokémon
D = DV

To find HP's DV, it becomes slightly more complicated:

A + B + C + D = H
If your attack DV is odd, then A = 8. If the attack DV is even, then A = 0.
If your defense DV is odd, then B = 4. If the defense DV is even, then B = 0.
If your speed DV is odd, then C = 2. If the speed DV is even, then C = 0.
If your special DV is even, then D = 1. If its odd, then D = 0.
The final product is H, which is your HP DV.

Some help from Charmeleon42's Pokémon Justice page at http://come.to/pokejust. Pokémon who cannot be shiny when female was posted by Dark Lugia on Pokémon Forever's Messageboard. Max stats in wild and trained were from Greycloak's Skill/Stat Finder. Props to Atma for giving me the fraction of finding wild Pokémon in the wild.