Figure 5c what are r2 x2 and p it sometimes happens in

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Unformatted text preview: six different students on six different balances Mass (grams) 3.140 3.133 3.144 3.118 3.126 3.125 245 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY Figure 5b Figure 6 Mass of pennies (in grams) with uncertainties An important application of this is determining agreement between experimental and theoretical values. If you use a formula to generate a theoretical value of some quantity and use the method below to generate the uncertainty in the calculation, and if you generate an experimental value of the same quantity by measuring it and use the method above to generate the uncertainty in the measurement, you can compare the two values in this way. If the ranges overlap, then the theoretical and experimental values agree. If the ranges do not overlap, then the theoretical and experimental values do not agree. Figure 5c What are R2, X2, and p? It sometimes happens in statistical analysis that instead of determining whether two numbers agree, you need to determine whether a function (“theoretical value”) and some data (“experimental value”) agree. Our method of comparing two numbers with uncertainties is too primitive for this task. R2 (the Pearson correlation), X2 (Greek letter “Chi,” not Roman X), and p are numbers that describe how well these things agree. They are too sophisticated for this appendix, but you may see them from time to time. If you feel comfortable with some basic statistics, you can look them up. You should never need to calculate them by hand; let your fitting software do it for you if your analysis gets that sophisticated. The most you might encounter in this class is that spreadsheet programs will give you R2 if you use them to fit data; for your purposes, you can consider your fit “good” if R2≥0.95. However you choose to determine the uncertainty, you should always state your method clearly in your report. How do I know if two values are the same? Go back to the pennies. If we compare only the average masses of the two pennies we see that they are different. But now include the uncertainty in the masses. For penny A, the most likely mass is somewhere between 3.117g and 3.125g. For penny B, the most likely mass is somewhere between 3.123g and 3.139g. If you compare the ranges of the masses for the two pennies, as shown in Figure 6, they just overlap. Given the uncertainty in the masses, we are able to conclude that the masses of the two pennies could be the same. If the range of the masses did not overlap, then we ought to conclude that the masses are probably different. Which result is more precise? Suppose you use a meter stick to measure the length of a table and the width of a hair, each with an uncertainty of 1 mm. Clearly you know more about the length of the table than the width of the hair. Your measurement of the table is very precise but your measurement of the width of the hair is rather 246 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY of penny B is different, since the range of the new value does not overlap the range of the previous value. However, that conclusion would be wrong since our uncertainty has not taken into account the inaccuracy of the balance. To determine the accuracy of the measurement, we should check by measuring something that is known. This procedure is called calibration, and it is absolutely necessary for making accurate measurements. crude. To express this sense of precision, you need to calculate the percentage uncertainty. To do this, divide the uncertainty in the measurement by the value of the measurement itself, and then multiply by 100%. For example, we can calculate the precision in the measurements made by class 1 and class 2 as follows: Precision of Class 1's value: (0.004 g ÷ 3.121 g) x 100% = 0.1 % Precision of Class 2's value: (0.008 g ÷ 3.131 g) x 100% = 0.3 % Be cautious! It is possible to make measurements that are extremely precise and, at the same time, grossly inaccurate. Class 1's results are more precise. This should not be surprising since class 2 introduced more uncertainty in their results by using six different balances instead of only one. How can I do calculations with values that have uncertainty? When you do calculations with values that have uncertainties, you will need to estimate (by calculation) the uncertainty in the result. There are mathematical techniques for doing this, which depend on the statistical properties of your measurements. A very simple way to estimate uncertainties is to find the largest possible uncertainty the calculation could yield. This will always overestimate the uncertainty of your calculation, but an overestimate is better than no estimate or an underestimate. The method for performing arithmetic operations on quantities with uncertainties is illustrated in the following examples: Which result is more accurate? Accuracy is a measure of how your measured value compares with the real value. Imagine that class 2 made the measurement again using only one balance. Unfortunately, they chose a balanc...
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