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The Basics of Error Analysis
-or-
How Much Can I Trust My Data, Anyway?
If you want to measure the distance between your apartment and this classroom, how would you do it? You could drive there, and look at the change in your car's odometer. This would give you an answer in tenths of a mile, but you would probably agree that the real distance could be several hundred feet different. If you really wanted to know the answer badly, you could hire a surveyor to triangulate between the buildings. She might tell you that the distance is 0.723 miles, but warn you that the answer might be off by a few feet in either direction. Frustrated that no one will give you a straight answer, you decide to measure the distance directly with a ruler, foot by foot. After a day's worth of tedious labor, you get a result of 0.7236 miles (3288 feet and 9 inches). Would you believe that you were off by an inch? How about a tenth of an inch?
The point is that it's not possible to measure things completely accurately. We are limited by our capacity for error, by the quality of our tools, and ultimately by our inability to control the state of what we're measuring. Since the scientific method demands that we compare our theories and models to what's really happening, we need to at least guess what the size of the errors in our measurements are.
These guesses are called uncertainties. The first step in taking data is always to estimate the uncertainty of your measurements. This process is an even mixture of common sense and technical knowledge. The idea is to write down a range of values which you're reasonably sure the true value lies inside, rather than a single number.
Using a ruler is the most common example. Let's say you're measuring the distance between these two dots:
I would say the distance is about 6.3 ± 0.1 cm. How well do you think you can estimate the distance between these two marks? Would your estimate be better or worse if the marks on the ruler were closer together?
Now try this ruler, which also has millimeter markings on it:
Now you might think that 6.35 ± 0.05 cm is a reasonable estimate, because the dot is clearly between the 6.3 and 6.4 cm marks. You might even want to say 6.33 ± 0.02 cm. Before writing this down, however, you should stop and think a bit about all the things which make this measurement uncertain. How confident are you that the ruler is placed correctly? For that matter, the marks on the ruler look like they're about half a millimeter wide. Is it their edges or their centers which are lined up with the correct distance? With a real ruler, if you move your head a little or close one eye would your estimate change? How confident are you now of your measurement?
Any measurement that you make has all sorts of little details like this whose effects are hard to estimate and harder to control. To account for this, most people use the following rule of thumb with simple measuring devices:
The uncertainty on a measurement is at least ± 1 of the smallest unit on the device used.
This means a ruler marked in mm has a precision of ± 1mm, a voltmeter which displays 2.34 V has a precision of ± 0.01 V, and so on. The rule above can be bent if you have enough room to easily estimate between marks, as in the case of the first ruler shown above. In such cases it's a good idea to have more than one person make the estimate to get some objective idea of how well this can be done.
Complicated measuring devices will often have their uncertainties explained in the owners manual; their precision might even vary depending on what you're measuring!
Sometimes the value you're trying to measure will fluctuate, the needle or display jumping up and down as you try to get a fix on it. This, of course, increases the uncertainty even more. In these cases try to estimate a range of values which encompasses most of the motion.
Practice this skill! You can't compare a theory to the results of your experiment without the uncertainties on your data. It is this kind of formal estimate which allows us to figure out the probability that the experiment agrees with the theory, rather than just saying "It looks close".
Another important habit to get into is keeping track of significant figures. Every digit in a number is significant except for leading and trailing zeroes. The numbers 0.0000355, 71200, and 101 each have three significant digits. If you don't have any uncertainties for your data, then you should at least follow this rule:
The result of any calculation should have as many significant digits at the least precise number used.
In other words, if you perform an elaborate calculation in which all of the numbers have five significant digits except for one which has only two, then your answer should also have only two significant digits. On the other hand, if you know that one of your numbers isn't very accurate then there's no need to measure or look up the others to high significance, either. It's OK to carry extra digits through the calculations so long as you round off properly at the end. Of course, final zeroes in your result might end up being significant; this is indicated by drawing a horizontal line over the proper zeroes.
Ideally, you should use both significant digits and uncertainties in every experimental calculation. You never need to write down any digits of your uncertainty which are smaller than the last significant figure of your result. For example, 15 ± 12 is OK, and 15 ± 2 is fine, but 15 ± 0.1 is a mismatch. If you come out with an uncertainty that much smaller than your main result, then you either underestimated your uncertainties or some of your data had more significant figures than you thought.
Created by Ben Mathiesen
Last Updated by Andy Pawl
Sept. 2003
apawl@umich.edu