Lab 0

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Updated: 7/28/2004

__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/s^{2}. Since it is not
the same as 9.8 m/s^{2}, 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/s^{2}.
This changes the question.

The question really is: Is the difference between my value and
the known one, (1.3 +/- 0.9) m/s^{2}, __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/s^{2}) 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__:

**accurate**measurements are ones that agree with the true value of the quantity to within one or two standard deviations.

**precise**measurements have small (less than 5 or 10% in this lab) uncertainties.

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.

Contact me if you have any questions.