Error Propagation Coefficients
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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 error propagation division them. When the variables are the values of experimental measurements they error propagation calculator have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables
Error Propagation Physics
in the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be
Error Propagation Chemistry
defined by the relative error (Δx)/x, which is usually written as 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 error propagation square root statistical probability distribution of the variable is known or can be assumed, it is possible to 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 Furthe
measurands based on more complicated functions can be done using basic propagation of errors principles. For example, suppose
Error Propagation Inverse
we want to compute the uncertainty of the discharge error propagation excel coefficient for fluid flow (Whetstone et al.). The measurement equation is $$ C_d = error propagation average \frac{\dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ where $$ \begin{eqnarray*} C_d &=& \mbox{discharge coefficient} \\ \dot{m} &=& \mbox{mass flow rate} https://en.wikipedia.org/wiki/Propagation_of_uncertainty \\ d &=& \mbox{orifice diameter} \\ D &=& \mbox{pipe diameter} \\ \rho &=& \mbox{fluid density} \\ \Delta P &=& \mbox{differential pressure} \\ K &=& \mbox{constant} \\ F &=& \mbox{thermal expansion factor (constant)} \\ \end{eqnarray*} $$ Assuming the variables in the equation are uncorrelated, the squared uncertainty of the http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc553.htm 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. $$ \frac{\partial C_d}{\partial \dot{m}} = \frac{\sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial d} = \frac{-2\dot{m} d}{K F D^4 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} - \frac{2 \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^3 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial D} = \frac{2 \dot{m} d^2}{K F D^5 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} $$ $$ \frac{\partial C_d}{\partial \rho} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \rho^{\frac{3}{2}} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial \Delta P} = \frac{- \dot{m} \sqrt{1-\left( \f
techniques coupled with type B analyses and propagation of error. It is a continuation of the case study of type A uncertainties. Background - http://www.itl.nist.gov/div898/handbook/mpc/section6/mpc64.htm description of measurements and constraints The measurements in question are volume resistivities (ohm.cm) of silicon wafers which have the following definition: $$ \rho = X_0 \cdot K_a \cdot F_T \cdot t \cdot F_{t/s} $$ with explanations of the quantities and their nominal values shown below: $$ \begin{eqnarray*} \rho & = & \mbox{resistivity} = 0.00128 \,\,\, \mbox{ohm} \cdot \mbox{cm} \\ error propagation X & = & \mbox{voltage/current} \,\, \mbox{(ohm)} \\ t & = & \mbox{thickness}_{wafer} \,\, (\mbox{cm}) \\ K_a & = & \mbox{factor}_{electrical} = 4.50 \,\,\, \mbox{ohm} \cdot \mbox{cm} \\ F_T & = & \mbox{correction}_{temp} \approx 1 ^\circ \mbox{C} \\ F_{t/s} & = & \mbox{factor}_{thickness/separation} \approx 1.0 \end{eqnarray*} $$ Type A evaluations The resistivity measurements, discussed in the case study of error propagation coefficients type A evaluations, were replicated to cover the following sources of uncertainty in the measurement process, and the associated uncertainties are reported in units of resistivity (ohm.cm). Repeatability of measurements at the center of the wafer Day-to-day effects Run-to-run effects Bias due to probe #2362 Bias due to wiring configuration Need for propagation of error Not all factors could be replicated during the gauge experiment. Wafer thickness and measurements required for the scale corrections were measured off-line. Thus, the type B evaluation of uncertainty is computed using propagation of error. The propagation of error formula in units of resistivity is as follows: $$ \large{ s_\rho = \rho \sqrt{\frac{s_X^2}{X^2} + \frac{s_t^2}{t^2} + \frac{s_{K_a}^2}{K_a^2} +\frac{s_{F_T}^2}{F_T^2} + \frac{s_{F_{t/s}}^2}{F_{t/s}^2} } } $$ Standard deviations for type B evaluations Standard deviations for the type B components are summarized here. For a complete explanation, see the publication (Ehrstein and Croarkin). Electrical measurements There are two basic sources of uncertainty for the electrical measurements. The first is the least-count of the digital volt meter in the measurement of \(X\) with a