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Currently I only have the 5th Edition Starter Set and I'm new to D&D. I understand that there is the Proficiency Bonus (+2 for the Starter Set's Cleric for example) which would be applied to the proficiencies the character has (the Cleric has a Procifiency in "playing cards" so I would expect the character to receive a +2 bonus when playing cards).

With this in mind, I also see that there are also positive and negative modifiers listed in the Skills and Saving Throws section of each character sheet. Are these Skill-/saving throw-specific proficiencies that take precedence over the more general Proficiency Bonus, or are they something else entirely?

I'm pretty certain that they are, in fact, a more specific proficiency but I didn't see anything in the starter set text that clarifies and, upon searching Stack Exchange, I mostly see questions regarding whether proficiency bonuses stack. In other words, I hope this question is specific enough and not duplicating an exiting question.

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Those numbers are your overall modifier for that skill or saving throw. They include your proficiency bonus if you are proficient in them, otherwise they are just your ability modifier for that skill or saving throw. They're pre-calculated and listed on the sheet for convenience during the game so that you don't have to add up all your modifiers and Proficiency bonus every time that roll is made.

So if your example Cleric is proficient in Wisdom saving throws and has a Wisdom score of 15, their Wisdom saving throw modifier = proficiency bonus (+2) + Wisdom modifier (+2) = +4, which is the number that will be listed under Wisdom saving throws.

On the other hand, your Cleric probably doesn't have proficiency with Dexterity saving throws, and let's say their Dexterity score is 8. Their Dexterity saving throw modifier = no proficiency bonus (+0) + Dexterity modifier (-1) = -1.

There are other factors that can affect this calculation, such as the Rogue's Expertise feature, but this is the basic calculation that will be used in the vast majority of cases.

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