To paraphrase Robert Cohen, he has asked if averaging many imprecise
trials will yield an accurate average (assuming no biases).
I believe the answer is yes. The reasoning is as follows.
The precision associated with an experiment is equivalent to the
standard deviation. The precision does not improve with more trials.
When we calculate the standard deviation by the usual formula, we are
getting an estimate of the "width" of the Gaussian distribution of
results for that particular experiment. Presumably that width is fixed
by the technique and instrumentation. As we increase the number of
trials, we improve our estimate of the standard deviation. Hence, with
more trials the standard deviation does not necessarily become smaller
or larger, but it becomes closer to the true value of the "width" of
the distribution. That is, we become more sure of our precision.
However, the average/mean value of the data (the centroid of the
gaussian distribution) also becomes better known. So more trials will
improve our knowledge of the average. The "standard error of the mean"
is defined as the standard deviation of the data divided by the
square-root of the number of trials.
(Note: Here I am assuming the "sample" standard deviation, i.e. we
divided by (N-1) when we calculated the standard deviation, so we
divide the S.D. by sqrt(N) to find the standard error of the mean. If
we use the "population" standard deviation, then we divide that S.D. by
sqrt(N-1) to find the standard error of the mean. This may be a
trivial distinction, but if I don't say it, someone else probably
will.)
Thus, if the standard error of the mean is (S.D.)/sqrt(N), we see that
quadrupling the number of trials will cut the uncertainty of our
average by a factor of two.
Assuming that the average value is what we seek, reducing the
uncertainty in the average implies we have obtained a better result.
That definitely means we have obtained a better estimate of the
average value of the measurements. If, as Cohen stipulated, there are
no biases (there is no systematic error) this also means we have
improved the accuracy. Stated another way, if our
distribution of measurements is centered on the true value, then
improving our knowledge of the average value of our measurements
implies we have improved our knowledge of the true value. That's
exactly what happens when we perform more trials.
To thoroughly beat this to death, let's state it one more way. We
assume our measurements are part of a gaussian distribution. The
parameters of this gaussian distribution (width and centroid) are
initially unknown to us. As we begin to take data we can calculate
estimates of the width (precision) and centroid (mean value) of this
distribution. The more data points we have, the more sure we become of
these parameters. If the centroid is the true value, as we reduce the
uncertainty in the location of the centroid, we also reduce the
uncertainty of our determination of the true value; i.e. we increase
the accuracy.
Of course, assuring that our distribution is centered on the true value
(i.e. ruling out biases/systematic-error) is the tough part.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817