1-AnalyzingExperimentalData

1-AnalyzingExperimentalData - American University of Beirut...

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Unformatted text preview: American University of Beirut, Physics Department. Phys205L . ___________________________________________________________________________________ 1 ANALYSIS OF EXPERIMENTAL DATA CONTENTS 1. Significant Figures 2. Errors of Observation 3. Theory of Errors 4. Propagation of Errors 5. Least-Squares Curve Fitting 6. References 1. SIGNIFICANT FIGURES , ACCURACY AND PRECISION Physical quantities are measured in the laboratory with the help of various instruments. These instruments have scales with divisions. The digits that are read and estimated on the scale are called significant figures . Suppose the length AB of an object is measured with a meter stick (Fig. 1). The smallest subdivision on the meter stick is 1 cm. Point A is aligned with the 0 graduation mark and point B falls between the 7 and 8 graduation marks. Therefore the length AB is somewhere between 7 and 8 cm. A reasonable estimate is that B is located at 7.6 cm. Of Figure 1. those two digits the first one is certain, the last one is doubtful. The length AB can therefore be given to two significant figures. The digit preceding the doubtful digit represents in general the smallest subdivision on the scale. If the length AB had been recorded as 7.65 cm, the impression would have been given that the scale of the ruler was divided into a tenth of a centimeter and that the point B was located between the 7.6 and the 7.7 graduation mark. Thus the length AB would be closer to the value of 7.65 cm than to 7.64 or 7.66 cm. More information is therefore given and there are now three significant figures. 0 1 2 3 4 5 6 7 8 9 10 11 A B American University of Beirut, Physics Department. Phys205L . ___________________________________________________________________________________ 2 Very often in the process of calculations extra figures are accumulated and sometimes reported in the end result. This is a meaningless procedure . No figures should be included beyond the precision of the original data. In the case of random errors the root-mean-square error in the arithmetic mean (see following sections) determines the number of significant figures. As an example L = (2.56  0.03) m and not L = (2.556  0.034) m or L = (2.56  0.034) m. If a resistance of 105  is written as 0.000105 M  , there are still only three significant figures. The resistance can namely also be written as 105 * 10-6 M  . A figure such as 25600 has five significant figures. It means that the true value is located somewhere between 25601 and 25599. There are only three significant figures if the figure had been reported as 256 * 10 2 . In this case the number has been specified to the nearest hundredth. Some other examples are 30.31 Four significant figures 35 Two significant figures 15.5 Three significant figures 20.000 Five significant figures 0.0001059 Four significant figures 5.0001050 Eight significant figures In publications and reports the results of the measurements in the laboratory are given....
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1-AnalyzingExperimentalData - American University of Beirut...

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