Basic Error Analysis
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these quantities should be presented in terms of Significant Figures. For example, the location of the arrow is to be determined in the figure below. It is obvious that the location is measurement and error analysis lab report between 1 cm and 2 cm. The correct way to express this location
Error Analysis Definition
is to make one more estimate based on your intuition. That is, in this case, a reading of 1.3 cm examples of error analysis is estimated. This measurement is said to contain two significant figures. Note that there should only be one estimated place in any measurement. If data are to contain, say, three significant figures,
Error Analysis Physics
two must be known, and the third estimated. Do not try to locate the position of the arrow in fig. 1 as 1.351 cm. The following rules dictate the handling of significant figures. a Specify the measured value to the same accuracy as the error. For example, we report that a physical quantity is x = 3.45 ± 0.05, not 3.4 ± 0.05 and not 3.452 error analysis linguistics ± 0.05. b When adding or subtracting numbers, the answer is only good to the least accurate number present. For example, 50.3 + 2.555 = 53.9 and not 52.855. c When multiplying or dividing, keep the same number of significant figures as the factor with the fewest number of significant figures. For example, 5.0 · 1.2345 = 6.2 and not 6.1725. Types of Errors Every measurement has its error. In general, there are three types of errors that will be explained below. Random Errors This type of error is usually referred to as a statistical error. This class of error is produced by unpredictable or unknown variations in the measuring process. It always exists even though one does the experiment as carefully as is humanly possible. One example of these uncontrollable variations is an observer's inability to estimate the last significant digit for a given measurement the same way every time. Systematic Errors This class of error is commonly caused by a flaw in the experimental apparatus. They tend to produce values either consistently above the true value, or consistently below the true value. One example of such a flaw is a bad cali
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or more quantities, each with their individual uncertainties, and then combine the information from these quantities in order to come up with a final result of our experiment. How can you state your answer for the http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm combined result of these measurements and their uncertainties scientifically? The answer to this fairly common question depends on how the individual measurements are combined in the result. We will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y, and Z, with uncertainties dX, dY, and dZ, and your final result, R, is the sum or difference of these error analysis quantities, then the uncertainty dR is: Here the upper equation is an approximation that can also serve as an upper bound for the error. Please note that the rule is the same for addition and subtraction of quantities. Example: Suppose we have measured the starting position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. Then the displacement is: Dx = x2-x1 basic error analysis = 14.4 m - 9.3 m = 5.1 m and the error in the displacement is: (0.22 + 0.32)1/2 m = 0.36 m Multiplication of measured quantities In the same way as for sums and differences, we can also state the result for the case of multiplication and division: Again the upper line is an approximation and the lower line is the exact result for independent random uncertainties in the individual variables. And again please note that for the purpose of error calculation there is no difference between multiplication and division. Example: We have measured a displacement of x = 5.1+-0.4 m during a time of t = 0.4+-0.1 s. What is the average velocity and the error in the average velocity? v = x / t = 5.1 m / 0.4 s = 12.75 m/s and the uncertainty in the velocity is: dv = |v| [ (dx/x)2 + (dt/t)2 ]1/2 = 12.75 m/s [(0.4/5.1)2 + (0.1/0.4)2]1/2 = 3.34 m/s Multiplication with a constant What if you have measured the uncertainty in an observable X, and you need to multiply it with a constant that is known exactly? What is the error then? This is easy: just mul
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