A Saiya-Jin is always very strong, stronger than almost any race out there. They are at their full potential during the full moon.
Strength is obviously a very important thing to the Saiya-Jin as they even base their social system on this. Creatures that are less strong than the Saiya-Jin are seen as inferior.
After every fight a Saiya-Jin fights, their power will increase. Whenever a Saiya-Jin has been hurt badly, their power will increase greatly. This is why some of the lesser liked methods of getting stronger has been to simply shoot one in a vital spot in its body and hope that he’ll recover.