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ChemLab113-0010

ChemLab113-0010 - Propagation of errors If the ﬁnal...

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Unformatted text preview: Propagation of errors If the ﬁnal result is calculated from measurements of two or more numbers, we must combine (“propagate”) the uncertainties. As an illustration, consider the measurement of the area of a rectangle to ﬁnd a length and width of 14.47 :t 0.02 cm and 5.83 :I: 0.02 cm, respectively. Examine the error propagation by calculating the following: most probable area: 14.47 cm x 5.83 em = 84.360] cm2 largest possible area: 14.49 cm x 5.85 cm : 84.7665 cm2 smallest possible area: 14.45 cm x 5.81 em = 83.9545 an2 After rounding the differences to one signiﬁcant ﬁgure we see that the largest value is 0.4 cm2 larger than the most probable value, and the smallest value is 0.4 cm2 smaller than the most probable value. So the area can be reported as 84.4 :I: 0.4 cm2. The idea just illustrated is the simplest way to make an uncertainty estimate. This is the primitive error propagation method, or the maximum uncertainty method, and is the only method we will learn to use in this course. It is an approximate approach, and your engineering friends will scoff at you for using it rather than doing the elaborate calculations of the more advanced approaches. Feel free to point out that it is very quick and easy, and gives a perfectly reasonable and useful estimate of the uncertainty. For your interest: The primitive method overstates the uncertainty. For example, in an area measurement there is a good chance that the errors in the length and width will be in opposite directions, resulting in some error cancellation in the calculation. The probability of such a cancellation is taken into account when the uncertainty is calculated by substitution into the following more accurate formula. AA = [(41402 + (44.91“: The term AA in the formula is the calculated uncertainty in the area, and the terms AA; and MW are the uncertainties in the area resulting from the uncertainties in the length and width that would result if each of those measured quantities were the only uncertain one. These quantities in general are calculated using calculus. We are not going to use this formula in this course! In this particular example of the rectangle area, the application of the more accurate formula is very easy. For the mm of it, we can look at the result AA = [(0.12 cm2)2 + (0.29 cm2)2]l"2 = 0.3 cm2 so the correct result for the area is 84.4 :t: 0.3. As expected, this more accurate uncertainty esimate is a little smaller than given by the primitive method. 18 ...
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