Pokemon Go Appraisal System Explained

The latest update of Pokémon Go rolled out yesterday and a new appraisal system was implemented.

There is a new option on the menu of your Pokémon that is called appraise. Your team leader will analyze your Pokémon and will give their feedback if your Pokémon is good or not, this is a way for you to determine which Pokémon is worth evolving and boosting power.

These dialogues of the team leader are equivalent to numbers that basically rank your Pokémon in HP, attack and defense.  These numbers are what is called IVs or Individual Values. These stat bonuses will be added to either attack, defense or stamina ranging from 0-15. So that means, your Pokémon, when powered up, can either increase to a maximum of additional 15 points to every stat that will total to 45 if the dialogue falls under a 37/45 - 45/45 stat bonus

Here are the explanation of every stats:
Increased Attack = Pokémon will deal more Damage
Increased Defense = Pokémon will take Reduced Damage from enemy Moves
Increased Stamina = Pokémon will have More HP

Here is a table of the dialogues of every team leader and the equivalent stat bonuses they represent:

Team Mystic Appraisal Analysis
Phase #1: Overall Stat Analysis	Range of Stat Bonuses
Overall, your [Pokemon Name] is a wonder! What a breathtaking Pokemon!	(37/45 - 45/45)
Overall, your [Pokemon Name] has certainly caught my attention.	(30/45 - 36/45)
Overall, your [Pokemon Name] is above average.	(23/45 - 29/45)
Overall, your [Pokemon Name] is not likely to make much headway in battle.	(0/45 - 22/45)
Phase #2: Individual Stat Analysis	Named Stat Bonus
Its stats exceed my calculations. It's incredible!	Stat bonus of 15
I am certainly impressed by its stats, I must say.	Stat bonus of 13 - 14
Its stats are noticeably trending to the positive.	Stat bonus of 8 - 12
Its stats are not out of the norm, in my opinion.	Stat bonus of 0 - 7
Phase #3: Pokemon Size	Size
Your [Pokemon Name] is tinier than any we have on record. Astounding.	XS
Your [Pokemon Name] is below average in size.	Small
Your [Pokemon Name] is above average in size.	Large
The size of your [Pokemon Name] is... colossal. This is indeed exceptional. Fascinating!	XL
Team Instinct Appraisal Analysis
Phase #1: Overall Stat Analysis	Range of Stat Bonuses
Overall, your [Pokemon Name] looks like it can really battle with the best of them!	(37/45 - 45/45)
Overall, your [Pokemon Name] is really strong!	(30/45 - 36/45)
Overall, your [Pokemon Name] is pretty decent!	(23/45 - 29/45)
Overall, your [Pokemon Name] has room for improvement as far as battling goes.	(0/45 - 22/45)
Phase #2: Individual Stat Analysis	Named Stat Bonus
Its stats are the best I've ever seen! No doubt about it!	Stat bonus of 15
Its stats are really strong! Impressive.	Stat bonus of 13 - 14
It's definitely got some good stats. Definitely!	Stat bonus of 8 - 12
Its stats are all right, but kinda basic, as far as I can see.	Stat bonus of 0 - 7
Phase #3: Pokemon Size	Size
Wh-whoa. That's the tiniest [Pokemon Name] I've ever seen!	XS
Your [Pokemon Name] is a little small for its kind, don't you think?	Small
Your [Pokemon Name] is a BIG one!	Large
Your [Pokemon Name] is just HUGE!	XL
Team Valor Appraisal Analysis
Phase #1: Overall Stat Analysis	Range of Stat Bonuses
Overall, your (Pokemon Name) simply amazes me. It can accomplish anything!	(37/45 - 45/45)
Overall, your [Pokemon Name] is a strong Pokemon. You should be proud!	(30/45 - 36/45)
Overall, your [Pokemon Name] is a decent Pokemon	(23/45 - 29/45)
Overall, your [Pokemon Name] may not be great in battle, but I still like it!	(0/45 - 22/45)
Phase #2: Individual Stat Analysis	Named Stat Bonus
I'm blown away by its stats. WOW!	Stat bonus of 15
It's got excellent stats! How exciting!	Stat bonus of 13 - 14
Its stats indicate that in battle, it'll get the job done.	Stat bonus of 8 - 12
Its stats don't point to greatness in battle.	Stat bonus of 0 - 7
Phase #3: Pokemon Size	Size
Your [Pokemon Name] is so tiny, I almost didn't notice it!	XS
Aww, what a small [Pokemon Name]! It's rather cute, I'd say.	Small
Your [Pokemon Name] is rather sizable, that's for sure!	Large
Your [Pokemon Name] is gigantic—the largest I've ever seen!	XL

BTW, the internet went crazy with the dialogues and produces some "dick jokes" by naming their Pokémon into genitals. Here are a few examples:

Source: PokemonDB

Update: The appraisal system was tested and verified over the subreddit of Pokemon Go
Update 2: Reddit user BoredMai made a calculator for all the dialogues to narrow down the results