As far as measurements go, ability/attribute scores are pretty abstract. They’re a great tool for comparing creatures and characters, but that is it. Something is missing if a game leaves it at that. While some systems will use dice pools for a measurement, they will still not fully explain how the numbers work. What it comes down to is this: it is one thing to have a quantitative measure of a character’s ability to interact with the world and another to provide a definition of what that truly means. This is where derived values and modifiers come into play.
Regardless of your familiarity with roleplaying games, it helps for everyone to have the same definitions with which we can work without too much confusion. To start, we must borrow from computer programming terminology. In a program there are two types of variables: global and local. Global values are applied to different routines as needed to keep various functions synchronized with user input, data flow, etc. These values do not change globally unless mandated to do so. The same happens in RPGs, but less often and will be discussed elsewhere. Local variables are more specific as they apply to a specific routine and are internal to it and do not affect anything else.
In the language of the roleplaying game community, global variables are termed “adjustments” and local variables as “derived values/stats.” Both categories function in much the same way. The difference between the two types is often expressed in mathematical terms unique to the value in question. Adjustments are often static for all ability/attribute scores in the game. This allows the core mechanic to work with all of them using the same formulae and language. Hence the global nature of these numbers. The adjustment scores also get applied to skills, attacks, health, and so on. Derived values are calculated for specific purposes such as how much weight one can lift, how well armored one is, the ease at which magic can be learned, etc. These do not affect the core mechanic as much as they do subsystems in a game.
So why do games require these secondary stats? In truth, games can only simulate so much through their core mechanics. The rest has to be handled by subsystems, some of which are used in conjunction with the core mechanic. What is really being provided is a method (or series therein) for describing exactly how a character’s ability/attribute scores translate from abstract measures to how he interacts with the world and vice versa. In effect, these values are more important than the stats they are derived from because of their tangibility. There are fewer instance in which an attribute/ability score is used other than as a generic, catch-all, or raw ability. Such uses boil down to a roll with the stat as the threshold against which success is determined.
A question that may come up in the design process is how many derived values does a game need. There is no simple answer to their inquiry. The complexity of the system should suggest the number needed. The more present, the more subsystems or calculations necessary to define character interactions. Too few and the game may fail to address fundamental situations, leading to player dissatisfaction or arguments impeding play. It is a fine line a designer has to walk in order to simulate reality without ruining the entertainment the system is meant to facilitate. What is important is that a player is armed with enough information to determine what his character can or cannot do. Anything unique that comes up and players will be smart enough to improvise.