Comparing measured and theoretical values and independent measurements.

 

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 ohm-meter

 

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 instrumentís 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 experimenterís 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 doesnít 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 experimenterís 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

 

ď How likely is it for to lie between ? ď .

 

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 8:00 am without an alarm, then you need to establish a time window.Chances become arbitrarily small if you want to make the time exact. It will never happen that someone will wake up at exactly 8:00 AM down to the nanosecond.]

 

 

 

 

 

 

 

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-1Ť1

68.3%

-2 Ť 2

95.4%

-3 Ť 3

99.7%

How likely for two values to lie outsidewhne both values are correct ?

large neg. value Ť -1and also 1Ť large pos. value

31.7%

outside

large neg. value Ť -2and also 2Ť large pos. value

4.5%

outside

large neg. value Ť -1and also 1Ť 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.