Error Propagation Multiple Measurement
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Error Propagation Division
Forgot password Expand/collapse global hierarchy Home Core Analytical Chemistry Quantifying Nature Expand/collapse global location Propagation of Error Last updated 20:33, 14 May 2016 Save
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as PDF Share Share Share Tweet Share IntroductionDerivation of Exact FormulaDerivation of Arithmetic ExampleCaveats and WarningsDisadvantages of Propagation of Error ApproachTreatment of Covariance TermsReferencesContributors Propagation of Error (or Propagation of Uncertainty) is defined as the effects on a function by a variable's uncertainty. It is a calculus derived statistical error propagation calculus calculation designed to combine uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. Introduction Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Therefore, the ability to properly combine uncertainties from different measurements is crucial. Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. Typically, error is given by the standard deviation (\(\sigma_x\)) of a measurement. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. For example, lets say we are using a UV-Vis Spectrophotometer to determine the molar absorptivity of a molecule via Beer's Law: A = ε l c. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must
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
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for the combined result of these measurements and their uncertainties scientifically? The answer error propagation average to this fairly common question depends on how the individual measurements are combined in the result. We will treat each error propagation chemistry 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 http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error 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 position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm 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 [(0.4/5.1)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
measurands based on more complicated functions can be done using basic propagation of errors principles. For http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc553.htm example, suppose we want to compute the uncertainty of https://www.physicsforums.com/threads/error-propagation-with-averages-and-standard-deviation.608932/ the discharge coefficient for fluid flow (Whetstone et al.). The measurement equation is $$ C_d = \frac{\dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ where $$ \begin{eqnarray*} C_d &=& \mbox{discharge coefficient} \\ \dot{m} error propagation &=& \mbox{mass flow rate} \\ d &=& \mbox{orifice diameter} \\ D &=& \mbox{pipe diameter} \\ \rho &=& \mbox{fluid density} \\ \Delta P &=& \mbox{differential pressure} \\ K &=& \mbox{constant} \\ F &=& \mbox{thermal expansion factor (constant)} \\ \end{eqnarray*} $$ Assuming the variables in the equation are error propagation multiple uncorrelated, the squared uncertainty of the discharge coefficient is $$ s^2_{Cd} = \left[ \frac{\partial C_d}{\partial \dot{m}} \right]^2 s^2_{\dot m} + \left[ \frac{\partial C_d}{\partial d} \right]^2 s^2_d + \left[ \frac{\partial C_d}{\partial D} \right]^2 s^2_D + \left[ \frac{\partial C_d}{\partial \rho} \right]^2 s^2_{\rho} + \left[ \frac{\partial C_d}{\partial \Delta P} \right]^2 s^2_{\Delta P} $$ and the partial derivatives are the following. $$ \frac{\partial C_d}{\partial \dot{m}} = \frac{\sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial d} = \frac{-2\dot{m} d}{K F D^4 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} - \frac{2 \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^3 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial D} = \frac{2 \dot{m} d^2}{K F D^5 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} $$ $$ \frac{\partial C_d}{\partial \rho} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \rho^{\frac{3}{2}} \sqrt{\Delta P}}
Community Forums > Mathematics > Set Theory, Logic, Probability, Statistics > Dismiss Notice Join Physics Forums Today! The friendliest, high quality science and math community on the planet! Everyone who loves science is here! Error propagation with averages and standard deviation Page 1 of 2 1 2 Next > May 25, 2012 #1 rano I was wondering if someone could please help me understand a simple problem of error propagation going from multiple measurements with errors to an average incorporating these errors. I have looked on several error propagation webpages (e.g. UC physics or UMaryland physics) but have yet to find exactly what I am looking for. I would like to illustrate my question with some example data. Suppose we want to know the mean ± standard deviation (mean ± SD) of the mass of 3 rocks. We weigh these rocks on a balance and get: Rock 1: 50 g Rock 2: 10 g Rock 3: 5 g So we would say that the mean ± SD of these rocks is: 21.6 ± 24.6 g. But now let's say we weigh each rock 3 times each and now there is some error associated with the mass of each rock. Let's say that the mean ± SD of each rock mass is now: Rock 1: 50 ± 2 g Rock 2: 10 ± 1 g Rock 3: 5 ± 1 g How would we describe the mean ± SD of the three rocks now that there is some uncertainty in their masses? Would it still be 21.6 ± 24.6 g? Some error propagation websites suggest that it would be the square root of the sum of the absolute errors squared, divided by N (N=3 here). But in this case the mean ± SD would only be 21.6 ± 2.45 g, which is clearly too low. I think this should be a simple problem to analyze, but I have yet to find a clear description of the appropriate equations to use. If my question is not clear please let me know. Any insight would be very appreciated. rano, May 25, 2012 Phys.org - latest science and technology news stories on Phys.org •Game over? Computer beats human champ in ancient Chinese game •Simplifying solar cells with a new mix of materials •Imaged 'jets' reveal cerium's post-shock inner strength May 25, 2012 #2 viraltux rano said: ↑ I was wondering if someone could please help me understand a simple problem of error propagation going from multiple measurements with errors to an average incorporating these errors. I have looked on several error propagation webpages (e.g. UC physics or UMaryland