What is it about the game, field and players in major league baseball
that tends to cause a distribution of batting averages in the range
from 0.000 to approximately 0.400, with a peak at approx. 0.250 ? A
response with valid analyis or reference to a source of such an
analysis, will be acceptable

HI Moedegue,

I think that you have outlined the answer yourself:  a bell curve  - a
Gauss distribution -  with its peak at 0.250 and a maximum of 0.400.
This would be the expected distribution of batting averages if we just
looked at the curve without reference to its position on the X axis,
so we need to explain that.

Nobody  - just nobody! -  in baseball hits over 0.500 or even seldom over 0.400.
The way the game is played precludes this:  nine men against one, the experience 
gained with a batter improving the team's ability to field against
him, at worse or best resulting in the pitcher's walking him, which
allows him on base but does not  (I believe)  count towards his
batting average.  Thus a team facing a potential 1.000 batter (from
his skill at hitting any ball within the strike range) could walk him
and avoid his achieving such a record of hits per times at bat.
This does happen to some extent (choosing intentionally walk a Ted
Williams with two on base rather than risk his hitting and maybe
letting both score).

This explains why the upper half  - 6/10 -  of the theoretical range
of 0.0 to 1.0 for batting averages is not just in right field
somewhere, it is out of bounds.  The bell curve maxes at 0.400.

At the lower end, if just the batting coach had his way, there would
be no players with batting averages below 0.200  - to pick a number. 
But since some players  - especially pitchers -  are of much greater
value to the team on the field, their weak batting is accepted.

I hope this explanation helps you.  It is not an answer, which a
Researcher can still post.

Myoarin

Stephen Jay Gould (renowned, but recently deceased, evolutionary
biologist and baseball fan) wrote an interesting essay on the decline
of the .400 hitter and discusses some of the factors governing the
distribution of batting averages. The essay was published as his
monthy contribution to Natural History magazine and also appears in
one of his collections of essays. From doing a search at Amazon I
think the essay was called "Losing the edge", which is in the
collection "The Flamingo's Smile". I don't think it's available
online.

Some people disagree with Dr. Gould:
http://www.geocities.com/cyrilmorong@sbcglobal.net/Mismeasure.htm

A bit of a nit....

Despite what myoarin wrote, we know that the distribution of batting
averages cannot follow a true gaussian distribution.  The gaussian has
tails that extend to plus and minus infinity, whereas batting averages
cannot be less than 0 or greater than 1.

Of course, in theory, but not in reality. Sure, the curve based on the
batting averages will not hit O,O (all those poor hitting pitchers),
but there will not be any negative results.