PHYSICS 133
ERROR AND UNCERTAINTY
In Physics, like every other experimental science, the numbers we “know” and the ones
we measure have always some degree of uncertainty.
In reporting the results of an
experiment, it is as essential to give the uncertainty, as it is to give the best-measured
value.
Thus it is necessary to learn both the meaning or definition of uncertainty, and the
techniques for estimating this uncertainty.
Although there are powerful formal tools for
this, simple methods will suffice for us.
To large extent, we emphasize a “common
sense” approach based on asking ourselves just how much any measured quantity in our
experiments could be in error.
The experimental error is NOT the difference between your measurement and the
accepted “official” value.
Error means
your experimental estimate of the range of values
within which the “true experimental value” of your measurement is likely to lie.
This
range is determined from what you know, or can figure out experimentally, about your
lab instruments and methods. It is conventional to choose the error range as that which
would comprise about 68% of the results, if you were to repeat the measurement a very
large number of times. In fact, we seldom make the many repeated measurements, so the
error is usually an estimate of this range. Note that the error range is defined so as to
include most of the likely outcomes, but not all. You might think of the process as a
wager: pick the range so that if you bet on the outcome being within your error range,
you will be right about 2/3 of the time. If you underestimate the error, you will lose
money in your betting; if you overestimate it, no one will take your bet!
Error:
If we denote quantities that are measured in an experiment by, say,
X, Y
and
Z
,
then their corresponding errors would be denoted by
Δ
X
,
Y
and
Z
. So if
L
represents
the length of a book measured with a meter stick, then you might say the length
L =
25.1
±
0.1 cm, where the central value (usually the most probable value) for the length is 25.1
cm and the error,
L
is 0.1 cm.
Both central value and error of measurements must be
quoted in your lab writeups.
Note that in this example, the central value is given with just
three significant figures. Do not write significant figures (
e.g. L
= 25.08533
±
0.1 cm)
beyond the first digit of the error on the quantity.
Failure to round off to 25.1 suggests
that you assign additional precision to your number, which is misleading.
Since the