PHY 2048L
Lab 0

Uncertainties

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Updated: 7/28/2004
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Dealing with Uncertainties

The idea that numbers are used to describe imprecise information is probably new to you. As a result, most students find it difficult to deal with the uncertainties inherent in making measurements, particularly when deciding if the data support or reject some hypothesis or model of the world. Suppose you determine that the acceleration of gravity is 8.5 m/s2. Since it is not the same as 9.8 m/s2, you probably think that your result is "wrong" and that you made a "mistake". However, when one includes the known uncertainties in the measurement, it could be that you actually measured (8.5 +/- 0.9) m/s2. This changes the question.

The question really is: Is the difference between my value and the known one, (1.3 +/- 0.9) m/s2, significantly different from zero? The answer to this question is "no". The reason it is "no" is that there is a 1 in 3 chance that your measurement would fall between 1 and 2 standard deviations (that is, between 0.9 and 1.8 m/s2) away from the true value. In our lab, two or three groups out of eight would be expected to be that far off just from the random variation of measured values within the range that is quantitatively specified by the standard deviation.

This is the new idea. Our measured value can only be compared to another value if we also include the measured uncertainty in that quantitative or qualitative comparison.

That is why we make a distinction between accuracy and precision:

A value for the acceleration of gravity such as (8.5 +/- 0.9) m/s2 is not very precise but it is reasonably accurate. A value of (9.5 +/- 1.5) m/s2 is very accurate but extremely imprecise, making it difficult to use it to test the value of g. A value of (9.2 +/- 0.1) m/s2 is very precise but not accurate at all. Why is it inaccurate even though it is closer to 9.8 than the 8.5 result? Because it is six standard deviations away instead of one and a half, a difference that is extremely unlikely to be explained by random variations in your measurements.

A related idea is that, in addition to the "known unknowns" represented by the calculated (measured) standard deviation, there are also "unknown unknowns" due to systematic (hidden) uncertainties and errors in the experiment. These are much harder to deal with and will not be emphasized in this course.

An Exercise with Uncertainties

In the social sciences and medical studies, this situation arises because of what is called "sampling error". That topic is covered in detail in statistics classes such as STA2023. In the physical sciences, the situation arises because it is impossible to measure anything to the infinite precision implied when you use a number to report the result. The effect is similar to sampling error because each measurement will be (slightly) different, as if you were drawing them from a hat that contains the values you might get because of observational or instrumental uncertainties.

We can get a feel for what we can responsibly and reliably deduce about the true value of some quantity from the mean and standard deviation of a small "data set" by comparing what we measure for a small sample to the known properties of the entire population. We will do this in a class exercise that I will not describe here other than to say that each group will randomly select some "data" from a known population and find the mean and standard deviation of that data. Although the population used has nothing to do with physics, it does have some characteristics of real experimental data. [This population has a skewed rather than "normal" distribution, so some samples should deviate significantly (?) from the mean, just as happens when experiments are done.]

We should discover that each group's data is within one or two standard deviations of the known mean, and that the average of the averages (thus including a much larger sample) agrees quite well with the exactly known mean of the full population. I hope this will help you see that some of the deviations you will see can be due to "bad luck" rather than "bad work", and that the standard deviation can be used to quantify whether your results agree or disagree with the expected value because of random variations in measurements or due to significant systematic or personal errors.

 
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