Error Propagation Differential Equations
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propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the error propagation formula physics values of experimental measurements they have uncertainties due to measurement limitations (e.g., propagation of error division instrument precision) which propagate to the combination of variables in the function. The uncertainty u can be expressed
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in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as
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a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, the positive square root of variance, σ2. The value of a quantity and its error are then expressed as an interval x ± u. If the statistical probability distribution of the variable is known or can be assumed, it is possible to error propagation calculus derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability that the true value lies in the region x ± σ. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Resistance measurement 5 See also 6 References 7 Further reading 8 External links Linear combinations[edit] Let { f k ( x 1 , x 2 , … , x n ) } {\displaystyle \ ρ 5(x_ ρ 4,x_ ρ 3,\dots ,x_
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 combined result of
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these measurements and their uncertainties scientifically? The answer to this fairly common question depends error propagation average on how the individual measurements are combined in the result. We will treat each case separately: Addition of measured quantities If error propagation square root 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 quantities, then the uncertainty dR is: https://en.wikipedia.org/wiki/Propagation_of_uncertainty 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 = 14.4 m - 9.3 m = 5.1 http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm 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 multiply the error in X with the absolute value of the constant, and this will give you t
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm of summations in algebraic context. At this mathematical level our presentation can be briefer. We can dispense with the tedious explanations and elaborations of previous chapters. 6.2 http://www.chem.hope.edu/~polik/Chem345-2000/errorpropagation.htm THE CHAIN RULE AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the error propagation relation: [6-1] ∂R ∂R ∂R dR = —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and error propagation differential dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations.x, y, or z leads to an error in the determination of u. This is simply the multi-dimensional definition of slope. It describes how changes in u depend on changes in x, y, and z. Example: A miscalibrated ruler results in a systematic error in length measurements. The values of r and h must be changed by +0.1 cm. 3. Random Errors Random errors in the measurement of x, y, or z also lead to error in the determination of u. However, since random errors can be both positive and negative, one should examine (du)2 rather than du. If the measured variables are independent (non-correlated), then the cross-terms average to zero as dx, dy, and dz each take on both positive and negative values. Thus, Equating standard deviation with differential, i.e., results in the famous error propagation formula This expression will be used in the Uncertainty Analysis section of every Physical Chemistry laboratory report! Example: There is 0.1 cm uncertainty in the ruler used to measure r and h. Thus, the expected uncertainty in V is 39 cm3. 4. Purpose of Error Propagation Quantifies precision of results Example: V = 1131 39 cm3 Identifies principle source of error and suggests improvement Example: Determine r better (not h!) Justifies observed standard deviation If sobserved scalculated then the observed standard deviation is accounted for If sobserved differs significantly from scalculated then perhaps unrealistic values were chosen for sx, sy, and sz. Identifies type of error If uobserrved - uliterature scalculated then error is random error If uobserrved - uliterature >> scalculated then error is systematic error 5. Calculating and Reporting Values when using Error Propagation Use full precision (keep extra significant figures and do not round) until the end of a calculation. Then