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So when I breed my Pokemon I try to get the offspring the same nature and characteristic in hopes that it would double max out a certain stat.

Like for example my Electrivire is Proud of its Power (atk) and has Adamant nature (atk). Does grouping the two make any difference or am I just wasting stats and should aim for a different characteristic to its nature?

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And anyways, even if you have perfect IVs and a good Nature, you're not assured to win..

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Best answer

These characteristic is actually telling you how many Individual Values the Pokemon has. THe characteristic doesn't only mean that the IV is for a specific stat, it just indicates which stat is highest (if all IV's are 31, it will randomly indicate a characteristic)

Also, look Here to learn more about IV's and what the characteristics mean.
Anyways, here's how many IV's a characteristic indicates a Pokemon has:

HP
:Loves to eat 0/5/10/15/20/25/30
:Often dozes off 1/6/11/16/21/26/31
:Often Scatters things 2/7/12/17/22/27
:Scatters things often 3/8/13/18/23/28
:Likes to relax 4/9/14/19/24/29

Attack
:Proud of its power 0/5/10/15/20/25/30
:Likes to thrash about 1/6/11/16/21/26/31
:A little quick-tempered 2/7/12/17/22/27
:Likes to fight 3/8/13/18/23/28
:Quick tempered 4/9/14/19/24/29

Defense
:Sturdy body 0/5/10/15/20/25/30
:Capable of taking hits 1/6/11/16/21/26/31
:Highly persistent 2/7/12/17/22/27
:Good endurance 3/8/13/18/23/28
:Good perseverence 4/9/14/19/24/29

Special Attack
:Highly curious 0/5/10/15/20/25/30
:Mischeavious 1/6/11/16/21/26/31
:Thoroughly cunning 2/7/12/17/22/27
:Often lost in thought 3/8/13/18/23/28
:Very finicky 4/9/14/19/24/29

Special Defense
:Strong willed 0/5/10/15/20/25/30
:Somewhat vain 1/6/11/16/21/26/31
:Strongly defiant 2/7/12/17/22/27
:Hates to lose 3/8/13/18/23/28
:Somewhat stubborn 4/9/14/19/24/29

Speed
:Likes to run 0/5/10/15/20/25/30
:Alert to sounds 1/6/11/16/21/26/31
:Impetuous and silly 2/7/12/17/22/27
:Somewhat of a clown 3/8/13/18/23/28
:Quick to flee 4/9/14/19/24/29

Basically, for individuals:

31 HP: Often dozes off
31 Atk: Likes to thrash about
31 Def: Capable of taking hits
31 Sp. Atk: Mischevious
31 Sp. Def: Somewhat vain
31 Spd: Alert to sounds

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