Propagating Error Through An Equation
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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 combined result of these measurements and their uncertainties scientifically? The answer to this error propagation calculator fairly common question depends on how the individual measurements are combined in the result. We will error propagation physics 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 chemistry 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 error propagation definition 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 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
Error Propagation Excel
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 the error in R: If you compare this to the above rule for multiplication of two quantities, you see that this is just the special case of that rule for the uncertainty in c, dc = 0. Example: If an object is realeased from rest and is in free fall, and if you measure the velocity of this object at some
The approach to uncertainty analysis that has been followed up to this point in the discussion has been what is called a top-down approach. Uncertainty components are estimated from direct repetitions of the measurement result. To contrast this
Error Propagation Average
with a propagation of error approach, consider the simple example where we estimate error propagation square root the area of a rectangle from replicate measurements of length and width. The area $$ area = length \cdot width error propagation inverse $$ can be computed from each replicate. The standard deviation of the reported area is estimated directly from the replicates of area. Advantages of top-down approach This approach has the following advantages: http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm proper treatment of covariances between measurements of length and width proper treatment of unsuspected sources of error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model Propagation of error approach combines estimates from individual auxiliary measurements The formal propagation of error approach is to compute: standard deviation from the length measurements http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm standard deviation from the width measurements and combine the two into a standard deviation for area using the approximation for products of two variables (ignoring a possible covariance between length and width), $$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$ Exact formula Goodman (1960) derived an exact formula for the variance between two products. Given two random variables, \(x\) and \(y\) (correspond to width and length in the above approximate formula), the exact formula for the variance is: $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$ with \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula) \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to \(s^2\) for width and length, respectively, in the approximate formula) \( E_{ij} = {(\Delta x)^i, (\Delta y)^j}\) where \( \Delta x = x - X \) and \( \Delta y = y - Y \) \( Cov((\Delta x)^2, (\Delta y)^2) = E_{22} - V(x) V(y) \) To obtain the standard d
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the http://math.stackexchange.com/questions/1291341/how-does-uncertainty-propagate-through-an-equation-with-complex-variables workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; error propagation it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top How does uncertainty propagate through an equation with complex variables? up vote 2 down vote favorite I am trying to understand how uncertainty propagates through systems with complex propagating error through variables. Given the general error propagation formula $$ \sigma^2_u = \left(\frac{\partial u}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial u}{\partial y}\right)^2 \sigma_y^2 + \ldots $$ so that if x is uncertain then in the case of multiplication by some constant, if $$ u = Ax $$ then simply $$ \sigma_u = A\sigma_x. $$ I understand that variance can never be complex so what would happens in the case that A is complex? so for example: $$ u = xe^{(-2\pi i f)}. $$ I am assuming that x is drawn from a normal distribution with known parameters. complex-numbers share|cite|improve this question edited May 21 '15 at 8:30 asked May 20 '15 at 15:59 Will 204 add a comment| 1 Answer 1 active oldest votes up vote 0 down vote accepted When variable is complex, you use the product by conjugate complex instead of the square of the value (remember that for a complex number, $\vert z \vert^2 = zz^\dagger$). \begin{eqnarray} \sigma_u & = & \sqrt{E \left[(u-E[u])(u-E[u])^\dagger \right]} \\ & = & \sqrt{E \left[uu^\dagger - uE[u]^\dagger - E[u]u^\dagg
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