Propagation Error Quotient Two Numbers
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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 propagation of error division can you state your answer for the combined result of these measurements error propagation average and their uncertainties scientifically? The answer to this fairly common question depends on how the individual measurements are error propagation formula physics 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, error propagation square root and dZ, and your final result, R, is the sum or difference of these 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
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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 = 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/
metres long, but I’ve only got a 4 metre tape measure. I’ve also got a 1 metre ruler as well, so what I do is extend the
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tape measure to measure 4 metres, and then I measure the last metre error propagation inverse with the ruler. The measurements I get, with their errors, are: Sponsored Links Now I want to know the adding errors in quadrature entire length of my room, so I need to add these two numbers together – 4 + 1 = 5 m. But what about the errors – how do I add these? Adding http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm and subtracting numbers with errors When you add or subtract two numbers with errors, you just add the errors (you add the errors regardless of whether the numbers are being added or subtracted). So for our room measurement case, we need to add the ‘0.01m’ and ‘0.005m’ errors together, to get ‘0.015 m’ as our final error. We just need to put this on the end http://www.math-mate.com/chapter34_4.shtml of our added measurements: You can show how this works by considering the two extreme cases that could happen. Say the measurement with our tape measure was over by the maximum amount – when we measured 4 m it was actually 3.99 m. Let’s also say that the ruler measurement was over as well by the maximum amount – so when we measured 1.00 m it was really 0.995 m. If we add these two amounts together, we get: This number is exactly the same as the lower limit of our error estimate for our added measurements: You’d find it would also work if you considered the opposite case – if our measurements were less than the actual distances. Adding or subtracting an exact number The error doesn’t change when you do something like this: Multiplication or division by an exact number If you have an exact number multiplying or dividing a number with an error in it, you just multiply/divide both the number and the error by the exact number. For instance: Multiplication of numbers with errors Say we had these two numbers and were multiplying them together: Th
an experiment is calculated from a number of observations taken from different instruments, connected through a formula. Sub Topics Maximum permissible error in different cases is calculated as follows Result involving sum of two observed quantities http://www.tutorvista.com/content/physics/physics-iii/physics-and-measurement/propagation-errors.php Result involving difference of two observed quantities Result involving the product of two observed quantities Result involving quotient of 2 observed quantities Result involving product of powers of observed quantities Maximum permissible error in different cases is calculated as follows Back to Top Result involving sum of two observed quantities Back to Top X is the sum of 2 observed quantities a and b. X = a + b Maximum error propagation absolute error in X = Maximum absolute error in a + Maximum absolute error in b Result involving difference of two observed quantities Back to Top Suppose X = a - b Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be maximum error in X. Maximum absolute error in X = Maximum absolute error in a + Maximum absolute propagation error quotient error in b From equations (1) and (2) it is evident that, when result involves sum or difference of 2 observed quantities, absolute error is the sum of absolute errors in the observed quantities. Result involving the product of two observed quantities Back to Top Suppose X = ab Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be the maximum possible error in X. Dividing both sides by X = ab, we get are relative errors of fractional errors in values of a, b and x. Neglecting as its product is very small. The above result is obtained by logarithmic differentiation. Take log on both sides, Log X = log a + log b Differentiating, we get , Thus, maximum relative error in X = maximum relative error in a x maximum relative error in b Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b Result involving quotient of 2 observed quantities Back to Top Let Da and Db be absolute errors in measurement of quantities a and b and DX be maximum possible error in X. Maximum possible relative error in X, Maximum relative error in X = maximum