Error Analysis. To become familiar with some principles of error analysis for use in later laboratories.

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2 above is correct, any lack of agreement between theory and experimental results is due to such systematic errors. For example, maybe you are measuring the length of several salamanders, and you are using a meter stick so you can measure lengths with a reading error of 1 mm. Since the salamanders are different, we will expect the lengths to vary by more than this reading error because of random effects. The average of all these measurements would be the average length of this set of salamanders, and our uncertainty in this average value would be determined by random effects rather than the reading error from using the meter stick. But what if we were using a meter stick that had accidentally had the first 4.0 cm cut off and we failed to notice? This would be a systematic error affecting our measurements. Every one of our measurements would be 4.0 cm too small, and the average of all of these measurements would be 4.0 cm too small as well. The only way we would be able to notice this systematic error would be by comparing the average of these measurements with the average of measurements made with a good meter stick. They should not agree. If they did agree, however, that would indicate that the random fluctuation of salamander length was larger than the 4.0 cm systematic error of the shortened meter stick. So, systematic errors are permissible if they are smaller than the random errors affecting our measurements. It is only when the random errors are smaller than the systematic error effects that these systematic effects become noticeable. 3.4 Statistical Analysis Throughout the semester we will be making measurements, and to attempt to deal with the errors we will often make repeated measurements of some aspect of a system. We will then take the mean (average) of these measurements in the hope that that is a better overall value for the particular quantity than any single measurement. The mean xx of a bunch of values xx ii is given by: xx = NN ii=1 xx ii NN = xx 1 + xx 2 + xx 3 xx NN NN Where the symbol NN ii=1 means sum up the terms xx ii from i = 1 to i = N. This is just math-speak for something that we all should already know how to do in our sleep; however, it helps to get used to the fancy math-speak. The mean value of x, whatever x happens to be, then is our best guess for what the most likely value of a measurement of x would give. We aren t done yet, no-no-no! We can do more. In fact, we can not only get the mean by doing the above, but we can estimate an error associated with our mean. We do this because it can give us some measure of confidence in the mean we will quote just how good our mean might be, and consequently how dependable our measurements were. To arrive at the error associated with the mean, we first calculate σ, the standard deviation. σ is calculated the following way: (1) σσ = NN ii=1 (xx xx ii )2 NN 1 = (xx xx 1 )2 + (xx xx 2 ) 2 + (xx xx NN ) 2 NN 1 (2) From this we can calculate the standard error on the mean, ss xx, as: ss xx = σσ NN (3) and from this we can finally get the error associated with the mean, ee xx, as: 2

3 ee xx = 1.96 ss xx (4) This then is the error that we associate with the mean or average value for a measurement. When we quote the mean value of a measurement in a lab report, we write it as: Example vvvvvvvvvv = xx ± ee xx (5) As a quick example, suppose you took data in a lab which looked like this: Trial x (cm) We calculate the mean of x as xx = cm. If we now calculate the standard deviation, we get: σσ xx = NN=5 ii=1 (xx xx ii )2 NN 1 = ( )2 +( ) 2 +( ) 2 +( ) 2 +( ) = cm cm Next we calculate the standard error on the mean: ss xx = σσ xx NN = cm Then the error associated with this mean is: ee xx = 1.96 ss xx = cm 0.5 cm Note that up until the last step we kept many digits (we keep all of them that our calculator or spreadsheet gives us to prevent numerical round off errors), but in the last step above we rounded to 0.5 cm. This is how we want you to do things keep all digits in intermediate steps, and round off in the last step with the answer that is important. Usually this answer is the one we want to compare to some theoretical prediction, and the intermediate steps are just a way to get to it. The rule for rounding is this: look at the number that is the error, and use the first non-zero digit only, regardless of where that digit is with respect to the decimal point. For example, if we were 3

5 in long form. Put the results all in a neat table, with units clearly indicated at the top of each column. Make a graph of the time to fall versus hole diameter. What does this look like? If it is not linear, try to find the mathematical relationship between time to fall and hole diameter using a graphing technique from the first lab. Make and plot this linear graph, do a linear regression, and compare the fit parameters with any theoretical values you can determine. 6. Questions 1. Two data sets, each with the same number of trials for the same time measurement of a system, produce standard deviations of 0.23 s and 0.76 s. What can you say about the data sets that produced each of these? 2. Is a mean value from a data set with a small error necessarily correct? Why/why not? 3. Suppose that the holes in the bottom of the bowls were square shaped instead of circular. How would this have changed your results? What would your first graph have looked like? What would a ln-ln graph have for a slope? 5

6 7. Error Analysis Data Table Cap # Hole Dia. (cm) Time 1 (s) Time 2 (s) Time 3 (s) Time 4(s) Avg (s) 6

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