Cant get my head around this one, so, to the group........

Made up a test bar at school for my home lathe tailstock alignment.
Took a piece of silver steel 20mm in diameter. (at least, thats what
its called here - its shiny steel rod, ground (I think) so its
straight and parallel. Set it up in a 3 jaw chuck, centre drilled each
end.

Question - if I then set it up between centers of tailstock and
headstock, will it be square to the bed of the lathe. Will the centre
drills be in the CENTRE of the bar, so any rotational error as show on
a dial gauge indicates the lathe bed being out of true (as shown by
running the carriage, with the dial indicator on it, down the bars
length..). It does show deviation at each end if I rotate it...

Its probably simple, its not a search for world peace, but I cant werk
it out....

OR - do I need to set up the bar in a 4 jaw, centre drill, then put it
in a cylindrical grinder to make sure it is 100% accurate?......

Andrew VK3BFA.