CHAPTER 1: Preliminary Description of Errors Experiment Methodology and Errors To introduce the concept of error analysis, let s take a real world

Transcription

1 CHAPTER 1: Preliminary Description of Errors Experiment Methodology and Errors To introduce the concept of error analysis, let s take a real world experiment. Suppose you wanted to forecast the results of the presidential election. You must first devise a method. In this case the method part seems easy enough. Just ask a simple question: If the election were to be held today, for which candidate would you vote? Now you must decide whom should you ask this question. Clearly you have to ask only people of voting age. Moreover, if you want the result to be for the whole country, then you must ask a representative cross section of voting age people across the whole country. You will randomly call people in each state, according to how many electoral votes there are in each state. Now the big question is how many total people should you call? If you have to call all 50 states, that s at least 50 people. You might decide to call people in the states according to the actual number of electoral votes that the state has, or a total of 535 for all the 50 states. If you do call exactly that number, then the important question becomes: How much can you trust your result? If there is a 1% difference between the two top candidates, does that really mean that the top candidate in your poll would win the election? Suppose there was a 10% difference between the two top candidates. Intuitively you would believe the top candidate has more of chance of winning the election, but how much more of a chance according to your poll? The answers to these questions are contained in the subject of this course: Error Analysis. By the end of this semester you should be able to give quantitative answers to the questions. The Definition of Error In scientific methodology an experimental error does not mean a blunder or a mistake. It simply means that any experimental measurement always carries with it some uncertainty. The uncertainty may be relatively large or relatively small compared to what is being measured. It is always there. Good methodology reduces the errors, not necessarily to be as small as possible, but rather to be as small as need be to make the measurement worthwhile. In the above example of a political poll, then you want to forecast the election to some accuracy, say a few percent. In turn that desired accuracy dictates the number of people whom you should question. 1

2 Lecture 1: Preliminary Description of Errors 2 Practical Example of Experimental Uncertainty Measuring the Height of Doorway: first method Suppose you are a carpenter assigned to make a new door for the classroom. You need to know how high the doorway is in order to cut the wood for the door. What you could do first is simply estimate by eye how high the doorway is. You know for example that you yourself are 6 feet tall and easily walk through the doorway without worrying about bumping your head. So you say the doorway is 7 feet high (210 cm). However, you tell yourself that this is not good enough. In other words, you are not just going to cut a door of height 210 cm. Very simply, you don t trust your estimate to be accurate to within a few centimeters. A door cut too large by even a few millimeters will simply not fit, while a door cut too small by even a centimeter will look pretty shoddy. Measuring the Height of Doorway: second method Since a visual estimate was not good enough, you change your methodology to use a meter stick. But that involves moving the meter stick at least two times which is a little sloppy. So instead you buy a tape measure, say 12 feet long ( 360 cm) and have your partner hold one end while you get pull on the other end. Now you have got the length to about 1 mm accuracy. Of course, instead of a tape measure costing a few dollars, you might decide to spend a few hundred dollars and buy a laser interferometer. That would get you the door height to a few microns instead of a millimeter. However, you don t do that because for the purposes of this experiment, micron accuracy is not necessary. The Problem of Definition Another reason not to buy a laser interferometer is the fact that the doorway height may not be a well defined quantity. One side of the door may be slightly different, by a few millimeters, than the other side of the door. So there is not much point in getting each side to a few microns. The same problem of definition, as the book says, would apply to a political poll. Because of voters changing their opinion as a campaign wears on over several weeks, a poll result taken earlier may become inaccurate at a later time.

3 Lecture 1: Preliminary Description of Errors 3 Importance of Knowing Uncertainties Archimedes and the Gold Crown The mythological story of Archimedes and the gold crown serves as another example of the importance of quoting uncertainties. Suppose you know that gold has a density of ρ gold = 15.5 gm/cm 3 and that a cheap alloy has a density of ρ alloy = 13.8 gm/cm 3 You consult two experts A and B who tell you that they can measure the density of the crown ρcrown. Expert A works quick and cheap, but B works slowly and expensively. Their results are then given to you as in the following table: Measurement reported Expert A Expert B Best estimate for ρcrown Estimated range for ρcrown By just looking at the best estimate row, you might conclude that expert A is saying that the crown is real gold and expert B is saying that the crown is a fake. However, when you take into account the larger uncertainty in the A expert s measurement, then you realize that expert A has not really told you anything of interest. His estimated range overlaps both the densities of gold and of the alloy. However, B s range of measurement is much smaller and appears to rule out the possibility that the density is consistent with gold. Importance of Error Quoting The above example should convince you that error quoting is not just a meaningless exercise. In fact error analysis is probably the most important part of any real experiment. Unless an error is quoted accurately for a measurement, then the measurement itself becomes meaningless. In the case of some physics lab measurements, you are not trying to find out unknown quantities. Rather you are obtaining measurements of well known quantities, and then are ask to determine whether your measurement is in good agreement, according to your error analysis, of the accepted value.

4 Lecture 1: Preliminary Description of Errors 4 Estimating Uncertainties in Measurements Reading Scales Often in experiments you will have to read a scale such as a meter scale or a voltmeter scale. Such a measurement will always have an error associated with it according to the smallest scale division. Examples are shown in Figs. 1.2 and 1.3 of the text (page 8). In the case of the meter stick, with 1 mm gradations, it would be fair to quote a measurement to ±1 mm. On the other hand, with a voltmeter having unit divisions, then it might be a little harder to be as accurate as 0.1 volts in spite of what the book says (page 9). There you might be a little more conservative and quote the error as ±0.15 volts. Repeated Measurements Often times you will have a measuring apparatus which is extremely accurate, like a digital stopwatch calibrated in hundredths of a second. However, you have to time some occurrence by starting and stopping the watch with your thumb. What you will probably find, if you repeat the measurement of the occurrence often, is that the measured times differ by much greater than a few hundredths of a second. Perhaps the differ by a few tenths of a second. This is probably caused by human reaction time. The best estimate of the measurement in this case is the average of all the measurements. As for the error estimate in such a case, you might take the difference between the smallest and the largest estimate. Actually, if you have made a lot of measurements instead of only three as the book gives (page 10), taking the smallest to largest range is probably too conservative an estimate of the measurement error. In these cases of many repeated measurement, it is mathematically possible to get a much improved and probably smaller error estimate.

5 Lecture 1: Preliminary Description of Errors 5 Systematic vs Random Errors All the errors we have talked about so far are examples of random errors. From one measurement to the next the values are likely to jump around some average value without any predictability. And the average value will be the one mostly likely to be nearest the correct value. However, there is another class of error called systematic error. In that case, it does not matter how many repeated measurements you make. Those will jump around randomly too, but their average will always be too low or too high compared to the accepted or true value. For example, if your stop watch was too slow or too fast. It would not matter how good was your reaction time, or how consistently you pushed the start and stop buttons. In the end your average value would be wrong. Systematic errors always involve some defect in the methodology. Either the apparatus has some constant defect, or the experimenter has some undetected bias while making the measurement. Systematic errors are difficult to track down. One must calibrate one s apparatus against know better devices, or else devise means of having the systematic errors cancel out by doing the measurements in different ways.