Here is a chart showing the 95% limits of a batting average by at bats. To read the chart, find expected BA* on the leftside of the chart; scan the top for the number of AB (not PA!) the player has had. The number read will indicate the limit of a one-sided 95% probability distribution. In general, randomness accounts for more variation than most people expect.
For example if you expect a player to bat .320 over 600AB, there is a 95% chance they will bat from .282 (.320-.038) to .358 (.320 +.038).
*Check performance histories for expected BA, or use the following approximation:
For a switch hitter exBA~=avg#
For a righty exBA~=avg#-0.020