There are no Pokémon that keep every stat from its previous form once it evolves.
The only Pokémon that keeps five stats after evolving is Type: Null into Silvally (keeping everything except speed).
The only Pokémon that keeps four stats after evolving is Cosmog into Cosmoem (keeping everything except physical and special defenses).
There are 5 Pokémon that keep three stats after evolving. They are:
Scyther (70, 110, 80, 55, 80, 105) into Scizor (70, 130, 100, 55, 80, 65)
Seadra (55, 65, 95, 95, 45, 85) into Kingdra (75, 95, 95, 95, 95, 85)
Nincada (31, 45, 90, 30, 30, 40) into Shedinja (1, 90, 45, 30, 30, 40)
Misdreavus (60, 60, 60, 85, 85, 85) into Mismagius (60, 60, 60, 105, 105, 105)
Klang (60, 80, 95, 70, 85, 50) into Klinklang (60, 100, 115, 70, 85, 90)
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