John Gastineau's post is excellent on the subject of precision, but
what about accuracy? That is, how does one look for the systematic
errors that would produce a value of g, say, of 9.6 even after an
infinite number of repetitions?
I don't mean to suggest that the following is what should be included
in either a high school or beginning elementary course, but as one
who has been involved in "precision measurements" since the 1960s, it
certainly concerns me in my research.
The textbook method of reducing systematics is to repeat the
experiment using a different method. (I.e., try a different falling
body, a different timer or meter stick, or use a different method) Of
course, even then there are the psychological needs of the
experimenter to get the "correct" answer (often the one reported in
his or her last paper).
See Peter Galison's book "How Experiments End" (Univ of Chicago
Press) for a fascinating history of several sets of experiments,
including the Einstein-de Haas measurement of the gyromagnetic ratio
of the electron which gave g = 1.02 +/- 0.10 when the presently
accepted answer is essentially 2.
To quote Vernon Hughes (Yale), "If experiment and theory differ by
three standard deviations, then there is a 50% chance that the
difference is real." That is, systematic errors are often much more
subtle than we could ever imagine.