The Flaw in 1st Pitch Hitting Stats

The other night I'm watching a game on TV and a stat appeared on the screen regarding the batter.  It indicated that he had a higher than normal batting average on the first pitch of his at-bats.  The announcers discussed it for a little bit and, not surprisingly, warned that opposing pitchers better be very careful not to lay a fat first pitch over the plate with this guy.  I would contend that laying a fat pitch over the plate to any major league hitter in any count is not a good idea.  However, I wanted to give another perspective on that notion that a hitter is a "great first pitch hitter."

Still the best pitch in baseball no matter how
good the batter is.

Pitchers of all ages and abilities have a common problem.  The problem is that they forget how hard hitting actually is.  We all have heard numerous times that the best hitters in baseball fail between 60%-70% of the time.  This alone should give pitchers a great deal of confidence to throw strikes but it doesn't always do so.  Especially on the first pitch to a batter who apparently is a "great first pitch hitter."
But there's a different way to look at those first pitch stats.  Unfortunately, the stats that follow are rarely, if ever, considered.  It's a shame because they can add to every pitcher's confidence level if shown to them.  Let's look at what I see to be a flaw in first pitch hitting stats and then do some math to help make the case.
To make it easy, we'll use a team as an example.  Let us say that a certain team had a first pitch batting average of .400 during their last game.  Sounds good, right?  But in reality, what that really means is that they batted .400 "when they put the ball in play."  Most calculations don't take into account all the times a first pitch is not put in play even when the pitch is a strike.
There are five things that can happen on a first pitch strike:

1. It can be taken for a strike.
2. It can be fouled off.
3. It can be put into play resulting in an out.  
4. It can be put into play resulting in an error.
5. It can be put into play for a hit.

The first point for pitchers to take from this is that only one of those five things raises a batting average.  The other four do not.  The next point is that the first two are not even considered in the first-pitch average math at all.  That, in my opinion, is where the flaw exists.
Let us say that the team mentioned above put the first pitch into play ten times over the course of the game, four of them resulting in hits. That's where the .400 average comes from.  But let us also say that the pitcher threw twenty first pitch strikes over the course of the nine inning game.  Four ended up being put into play for hits but the other sixteen were not.  Four hits in twenty at-bats is a .200 average.  Of course, that is vastly different from the .400 average that was initially presented.   Sort of softens the "great first pitch hitting team" description a bit doesn't it?
Obviously, there are, by comparison, good first pitch hitters.  However,  informing a pitching staff that a team or particular hitter is batting .400 on the first pitch may result in a reluctance to throw first pitch strikes.  Analyzing the numbers and presenting them as I described gives a very different - I would say more realistic - picture and may improve the confidence a pitcher needs in order to go after hitters early in the count.