How do i determine the uncertainty this appendix will

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Unformatted text preview: very careful to estimate or eliminate (by other means) systematic uncertainties well because 243 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY measurement could be off is a fraction of a mm. To be more precise, we can estimate it to be about a third of a mm, so we can say that the length of the key is 5.37 ± 0.03 cm. they cannot be eliminated in this way; they would just shift the distributions in Figure 1 left or right. Roughly speaking, the average or “center” of the distribution is the “measurement,” and the width or “deviation” of the distribution is the random uncertainty. Another time you may need to estimate uncertainty is when you analyze video data. Figures 3 and 4 show a ball rolling off the edge of a table. These are two consecutive frames, separated in time by 1/30 of a second. How do I determine the uncertainty? This Appendix will discuss three basic techniques for determining the uncertainty: estimating the uncertainty, measuring the average deviation, and finding the uncertainty in a linear fit. Which one you choose will depend on your situation, your available means of measurement, and your need for precision. If you need a precise determination of some value, and you are measuring it directly (e.g., with a ruler or thermometer), the best technique is to measure that value several times and use the average deviation as the uncertainty. Examples of finding the average deviation are given below. Figure 3 How do I estimate uncertainties? If time or experimental constraints make repeated measurements impossible, then you will need to estimate the uncertainty. When you estimate uncertainties you are trying to account for anything that might cause the measured value to be different if you were to take the measurement again. For example, suppose you were trying to measure the length of a key, as in Figure 2. Figure 4 Figure 2 The exact moment the ball left the table lies somewhere between these frames. We can estimate that this moment occurs midway between them ( 1 t 10 60 s ). Since it must occur at some point If the true value were not as important as the magnitude of the value, you could say that the key’s length was 5cm, give or take 1cm. This is a crude estimate, but it may be acceptable. A better estimate of the key’s length, as you saw in Appendix A, would be 5.37cm. This tells us that the worst our between them, the worst our estimate could be off by 244 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY is 1 60 s . We can therefore say the time the ball leaves 1 the table is t 10 60 1 60 3.131 average The deviations are: 0.009g, 0.002g, 0.013g, 0.013g, 0.005g, 0.006g Sum of deviations: 0.048g Average deviation: (0.048g)/6= 0.008g Mass of penny B: 3.131 ± 0.008g s. How do I find the average deviation? If estimating the uncertainty is not good enough for your situation, you can experimentally determine the un-certainty by making several measurements and calculating the average deviation of those measurements. To find the average deviation: (1) Find the average of all your measurements; (2) Find the absolute value of the difference of each measurement from the average (its deviation); (3) Find the average of all the deviations by adding them up and dividing by the number of measurements. Of course you need to take enough measurements to get a distribution for which the average has some meaning. Finding the Uncertainty in a Linear Fit Sometimes, you will need to find the uncertainty in a linear fit to a large number of measurements. The most common situation like this that you will encounter is fitting position or velocity with respect to time from MotionLab. When you fit a line to a graph, you will be looking for the “best fit” line that “goes through the middle” of the data; see the appendix about graphs for more about this procedure. To find the uncertainty, draw the lines with the greatest and least slopes that still roughly go through the data. These will be the upper and lower limits of the uncertainty in the slope. These lines should also have lesser and greater yintercepts than the “best fit” line, and they define the lower and upper limits of the uncertainty in the yintercept. In example 1, a class of six students was asked to find the mass of the same penny using the same balance. In example 2, another class measured a different penny using six different balances. Their results are listed below: Class 1: Penny A massed by six different students on the same balance. Mass (grams) 3.110 3.125 3.120 3.126 3.122 3.120 3.121 average. The deviations are: 0.011g, 0.004g, 0.001g, 0.005g, 0.001g, 0.001g Sum of deviations: 0.023g Average deviation: (0.023g)/6 = 0.004g Mass of penny A: 3.121 ± 0.004g Note that when you do this, the uncertainties above and below your “best fit” values will, in general, not be the same; this is different than the other two methods we have presented. For example, in Figure 5, the y-intercept is 4.25 +2.75/-2.00, and the slope is 0.90 +0.20/-0.25. Figure 5a Class 2: Penny B massed by...
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