Understanding error and uncertainty for experimental measurements
requires careful thought about what one knows and what one is trying to
establish. Let us consider two
measurements
1.
Improving our knowledge of the resitivity
of copper
2.
Calibrating an ohmmeter
The first experiment involves determining a value that is unknown (at
least at the level for which the experiment is designed.). The second will
compare an instrument’s performance when measuring values that are taken to be the True Values (the experimenter might
procure, for example, some standards that are believed to be known well beyond
the level of precision of the instrument, standard resistors).
Similarly, in performing a measurement (type 1) for the lifetime of an
unstable particle a calibrated oscillator (type 2) is used for the measurement
of time intervals. Measurements of time
intervals are used to determine a lifetime that is unknown. To perform these
measurements the frequency of the experimenter’s oscillator is compared to a NIST
broadcast of a frequency standard based on atomic clocks. The NIST standard is
known to better that 1 pbb. For the experimenter this value can be taken
as completely known, i.e. the True Value.
What students need to realize is that most of the experiments performed
in introductory laboratory are type 2 measurements. To become acquainted with the measurement
process one must conduct measurements as if the result is unknown. In keeping with this principle when asked to
compare a result with a known value the student is NOT allowed to establish the quality of the measurement by
comparing to this accepted value. If the
result is not consistent with the accepted value, the student is expected to
review the procedure and analysis to see if there was a critical mistake and if
none is found declare that there is a disagreement. Although we know that it is very unlikely
that an accepted value is wrong, the student should adopt the stature of a real
experimenter and state the disagreement.
In real life the experimenter is faced with the decision whether or not
to publish the result. Similarly the student must decide whether or not to
submit the report for grading. If the
student feels that there is a major oversight he/she can repeat the
measurement, or not submit the work (grade of zero). Laboratory instructors realize that some measurements
that are carefully performed without serious oversights or problems may still
produce results that are in disagreement with accepted values. Therefore, a disagreement with an accepted
value should force the student to carefully review the experiment,
it does not necessarily imply a bad experiment. Also a result that agrees
doesn’t guarantee that the experiment was correctly performed. Some calibrations are known to be dependent
on conditions such as temperature that is beyond the control of the student and
these effects are not expected to be included in the analysis. If conditions cause a shift in
instrumentation performance then the introductory laboratory experiments should
disagree with accepted values.
Beyond the measured result, the experimenter must establish a level of
confidence in his/her measurement. We
use the standard approach and describe this confidence in terms of single
number the uncertainty. However,
establishing a single precise value is clearly impossible except perhaps in the
type 1 measurement (calibration), where the instrument in question might reveal
a constant shift from the calibration standard. So what does one mean by the uncertainty? For a well behaved measurement the field of
statistics can be used to develop a statement in terms of probability. The number recorded as the uncertainty is
converted into a statement of likelihood.
One can take the approach that comparing results is done by stating how
likely the two results are to being correct.
Comparing results
is done by stating how likely the two results are to being correct.
In evaluating a single measurement the uncertainty can be interpreted in
terms of how likely the true value differs from the measurement and therefore a
statement about how likely any value is to being the True Value.
Uncertainty can be
interpreted in terms of how likely the true value differs from the measurement.
Assume that an experimenter has established a result and an uncertainty. In this discussion we will use
· _{}to represent the uncertainty that the experimenter has
decided characterizes the measurement and
· _{} result
1 to be the final measured value in the experiment.
·
_{} result
2 to be another measurement or a theoretical value.
One can write down an expression that describes
how likely it is that true value, TV, falls a certain distance from the
measured value.
_{}
Pr is a type of probability. Ignoring mathematical details the curve
indicates by its height those values that are likely and those that are not. This
function is plotted below. The most
likely value corresponds to experimenter’s value and the TV being the same. This is the peak in the plot below.
For our purposes we can establish a similar
analysis by taking one result as correct and ask about the second result.
_{}
One very important aspect of the function is
that it depends on the difference divided by the uncertainty. It is not simply the distance between results
but the distance divided by the uncertainty. To evaluate it is therefore meaningful
to construct differences divided by uncertainties. _{}
Another detail of the evaluation is that the
results represent a continuous set of possible outcomes. Therefore to evaluate
this function one must choose a range of values. The question you actually ask is
“ How likely is it for _{} to lie between _{}? “ .
Choose two choices for one of the results to be
compared a
and another possibility b
and then ask how likely it is that the value _{} is somewhere in
between a and b.
[Using a range of values is just a common sense
approach to comparisons. If you were to
ask some how likely is it for them to wake up at






If the experiment is guided by the above
expression Pr then we compare results as follows:
How likely is it for two
values to lie within _{} when both results
are correct? 
_{} overlap from 1è1 
68.3% 
_{} 
2 è 2 
95.4% 
_{} 
3 è 3 
99.7% 
How likely for two values to
lie outside_{}whne both values are correct ? 
large neg. value è 1 and also 1è large pos. value 
31.7% 
outside _{} 
large neg. value è 2 and also 2è large pos. value 
4.5% 
outside _{} 
large neg. value è 1 and also 1è large pos. value 
0.3% 
The main lesson to learn is that results from an experiment can be compared with other results or with a theoretical value. Most people perform this comparison by examining how far apart the values being compared lie. Since the uncertainty establishes a window. It sets the scale for what far apart means. If you know that your uncertainty is 10 ft then a value that differs by 5 ft from your answer is not far away. However if the uncertainty is 1 inch then 5 ft is a dramatic separation. The expression
_{}
brings together these ideas. It finds out how far you are away in terms of step size or scale at the level of the uncertainty in the experiments. The above table then can provide reasonable statements as to how likely this difference could be if one assumes that both quantities are correct. If for example a comparison indicates that the two quantities differ by _{} one would say that this could happen fewer than 1 out of 100 measurements. So most likely something is wrong. If on the other hand the difference is _{} then this could happen in about 1 out of 20 measurements. In this semester each experiment is performed by about 50 groups so some of the groups will have measurements that far apart even when the groups perform the measurement correctly.