I find splits almost fascinating (they're good writing material!), but it's very very easy to misuse them.
Prerequisites for understanding: Sample sizes
Prerequisites for derivation: Data
People like patterns. Love them, in fact. We're also very, very good at seeking them out, spotting patterns in data no computer could ever hope to analyse. Even better; we're able to generalise. Look around for a chair (there's likely one directly under you). Note its inherent chair-ness. People are able to recognise chairs with no problem at all. Swivelling chairs, comfortable chairs, even broken chairs: none of these match your platonic ideal of a chair, but you can identify them without a problem anyway. A computer can't process optical information nearly as efficiently as the human brain.
Then again, computers don't have our false-positive problem either. We see rabbits in clouds, faces on toast, and the Batman logo on Rorschach tests (unless that last one is just me). Clearly, this is silly. There are no giant sky rabbits, toast isn't anthropomorphic, and Batman isn't drawing all over my inkblot tests. Our pattern recognition ability is so strong that it leads us to find patterns where there are none. Obviously, the examples I just gave are silly, but sometimes silliness is not so clear cut.
Take, for example, our habit of slicing up baseball statistics. Want to know a player's batting average on Tuesdays? How many stolen bases did Pitcher A give up in May? How about a player's performance in the clutch? We can cut up our data in pretty much any way we like, and non-whole-season numbers are typically referred to as 'splits'. Splits are everywhere, especially ubiquitous in television broadcasts. But what do they mean? Why do we care? How much of our fascination with split statistics is due to false-positive pattern recognition and how much helps us to further our understanding of the game?
Filtering Our Information
Our first filter on interpreting splits must be an old friend - the ever reliable "does this make sense?". Does it make sense for a player to do especially well on Thursdays in June, or are we merely seeing an artifact of randomness? A player will always do especially well on some day of the week in some month, after all. Why should we care about which specific day, and which specific month? We probably shouldn't. How about a player's offensive ability against same-handed pitchers? Clearly, this makes sense to look at. We know that it's harder for left handed bats to hit left handed pitching than right handed. Hitting in the clutch? This also passes the 'makes sense' test. We all know people who perform in high pressure situations, and those who wilt, and so we apply that to baseball.
Our second filter is appropriate regression. Would you accept a batter going 4 for his first 10 as a .400 hitter? No, so there's no reason to accept a batter going 4-10 against a certain pitcher as being indicative of very much at all. Regress everything. Handedness splits, clutch splits, day-night splits, everything. Doing so will help to haul you out of the false-pattern trap.
We can also look at the predictive value of splits (admittedly, this is an offshoot of regression). Can we predict future performance in a given situation from past performance? In most cases, the answer is "not very well," with some statistics faring worse than others. Let's take clutch statistics. Superficially, we can predict clutch performance. Hitters who have performed well in the clutch will likely to continue to perform well. However, what we can't predict is the delta between clutch hitting and regular hitting: analysis shows that this has a habit of collapsing to zero - meaning that the best clutch hitters are the best hitters. Left/right splits do much better, but still require significant regression before we can use them to make statements about ability.
Playing with splits is a dangerous game. Ensure you filter the information appropriately, however, and you can encounter some rewarding results. A pitcher does badly with the bases empty, even with his numbers heavily regressed? Maybe from that we can hypothesise that there's some flaw in his windup, and then we can look for it. Splits (team-based home and away numbers in particular) can also help immensely in determining park factors. There are multiple applications here, and we're finding more and more ways to refine our data. Ultimately, splits are a very useful tool - so long as you don't buy into the false positives that will invariably pop up.