I have been using take home labs as part of my undergraduate fluid mechanics class for a while now and have often been frustrated with students approach to uncertainty and possible error in their measurements and resulting calculations. All the take home labs require the students to measure some quantity using multiple techniques and then compare their results from the different tests. Ideally, any differences in their values can be explained by analysis of the uncertainty in their underlying measurements. If it cannot be accounted for in this way then there is a potential problem in the way the students did the test or the analysis. Given this frustration I have come to the point where I see a major goal of the take home labs to be to teach the students about basic error analysis. I typically have 4-5 take home labs during each semester and see the teaching process with regard to uncertainty analysis as a 4 step program.
Step 1. Accepting that you have uncertainty – In the first lab of the semester I ask the student to quantify their uncertainty. This very rarely goes well. I get vague statements in their lab reports like “possible sources of error include human error, timing errors, not being able to hold the camera steady, and errors from measuring the area of …” They are generally all lumped together regardless of their relative magnitude or their influence on the final result. However, it does force them to think about possible sources of error in their experiments.
Step 2. Quantifying your measurement uncertainty – When I hand back the first graded lab report I talk to the class about quantifying their uncertainty. I explain that what 95% of them did was present a qualitative discussion of possible sources of error. However, being engineers, they need to actually quantify that uncertainty. I then discuss measurement errors, primarily associated with scale resolution. For the second take home lab they are required to provide a table with every measurement they made and an associated uncertainty (that is XXXX ± YY). I can then provide feedback on this when grading and returning the second lab report.
Step 3. Quantifying your calculation uncertainty – Once they have quantified their measurement uncertainty in the second lab we discuss how those uncertainties propagate through their calculations. I tend to keep this fairly easy and only use basic linear error analysis such as is described here. For the third lab they have to present each of their their experimental results with the associated propagated error and state if each of their different sets of results (from each of the different types of experiments they ran) agree within the margin of error/uncertainty. If they do not then, most likely, one or more of three things has happened. (1) they have made a mistake in either their experimental design or analysis such that the resulting calculation is not representative of the quantity they were supposed to measure. (2) their estimates of their uncertainty in their measurements is too small. This sometimes happens when they report that their times are accurate to a hundredth of a second when actually their uncertainty is mostly dominated by their ability to hit start and stop in a timely manner. (3) their experiments were not repeatable. That is, the experimental conditions changed between tests. For example, they may not always turn the hose on to the same flow rate each test.
Step 4. Putting it all together – For the fourth (and possibly fifth) take home lab they need to look at their underlying assumptions when they make their measurements. For example, one lab is to measure the mass flow rate from a compressed air can using at least two different methods. One problem is that the air decompresses as it leaves the can so if you make an assumption about density there is an uncertainty associated with that. Another problem with that lab is that the flow rate decreases over time as the pressure in the can decreases. Therefore, they need to discuss repeatability of the test and its impact on the final results. This is more complex, and we do not go over this in detail in class but the lab does make them aware of other issues with running lab experiments.
An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (firstname.lastname@example.org). I also welcome comments (through the comments section or via email) on improving the demonstrations.