This one is going to seem overly simplistic for some of you, but it's necessary to have a think about the implications of rate stats and counting stats in order to get to where we want to be going, soo...
Prerequisites for understanding: None.
Prerequisites for derivation: N/A; conceptual.
Rates and Numbers
There's a fundamental divide at the very heart of baseball statistics. On one side lie the counting statistics, on the other lurk the rates. What's the difference? The former deal in numbers. Home runs, hits, wins, they all live within the realm of the counting stats. The latter include raw numbers too, of course, but then they divide by some other value: batting average is the number of hits divided by the number of at bats, and so on and so forth. As discussed earlier in the series, teams will be looking at value in terms of wins, which is clearly a counting stat. This means that we'll eventually want to have a counting stat version (in runs) of more or less every aspect of play.
Special Cases
There are special cases for both counting and rate stats that we should be aware of. We'll deal with the rate first, as it's simpler. When we're faced with a rate statistic, it's often nice to know how it relates to average. We convert the pure rate stat into a 'normalised' statistic through a process that essentially involves dividing by league average then multiplying by 100 (it's a little more complicated than that, and different stats do this differently, but this definition is close enough to give you an idea). This is useful because we don't need to know so much about the context of what we're looking at: average is built in; all one needs is to know that above 105 means 5% above league average and 95% means 5% below. Normalising a statistic still doesn't give us information about the typical differences in talent for a given statistic, so a value of 110 in one statistic isn't necessarily more impressive than 105 in another.
The special case for counting stats involves setting a benchmark at a certain point and converting from the rate statistic. Imagine our league's batting average is .270, and we have a batter who's hitting .275 in 400 at-bats. Said batter has managed 110 hits, while the league average hitter would be at 108 hits in the same number of at-bats. We can therefore say that our hitter is two hits above average. This isn't very useful for hits, but when we start converting to runs it becomes a big deal. However, there's a problem. Although comparisons to league average are nice from a mathematical point of view, it distorts information somewhat. A hitter can get to one hit above average by hitting .273 in 400 at-bats, or by hitting .278 in 100 at-bats. Worse, a batter who has one awful at-bat and never sees the majors again might be ranked far higher than a slightly below average hitter who plays out the whole season. These statistics are therefore not correctly weighting playing time, which is a fairly major flaw.
There's a nice little solution to this, but in order to understand it fully we'll have to look at the way the roster works as well.
What Follows
'Plus' statistics; runs above average; runs above replacement.