Propagation Of Error On Averages
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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 Formula Physics
and their uncertainties scientifically? The answer to this fairly common question depends on how the individual measurements are error propagation average standard deviation 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 calculator 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
Error Propagation Square Root
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 [(
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Error Propagation Inverse
it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Error propagation on weighted mean up vote 1 down vote favorite I understand that, if errors are random and independent, http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the error of $x + y$ is given by $\sqrt{e_1^2 + e_2^2}$, where $e_1$ and $e_2$ and the errors of $x$ and $y$, respectively. However, I have not yet been able to find how to calculate the error of both the arithmetic mean and the weighted mean of the two measured quantities. How do errors propagate in these cases? http://math.stackexchange.com/questions/123276/error-propagation-on-weighted-mean statistics error-propagation share|cite|improve this question edited Mar 22 '12 at 17:02 Michael Hardy 158k16145350 asked Mar 22 '12 at 13:46 plok 10815 add a comment| 2 Answers 2 active oldest votes up vote 3 down vote accepted The first assertion assumes one takes mean squared errors, which in probabilistic terms translates into standard deviations. Now, probability says that the variance of two independent variables is the sum of the variances. Hence, if $z = x + y$ , $\sigma_z^2 = \sigma_x^2 + \sigma_y^2 $ and $$e_z = \sigma_z = \sqrt{\sigma_x^2 + \sigma_y^2} = \sqrt{e_x^2 + e_y^2} $$ Knowing this, and knowing that $Var(a X) = a^2 Var(X)$, if $z = a x + (1-a) y$ (weighted mean, if $ 0\le a \le1$) we get: $$\sigma_z^2 = a^2\sigma_x^2 + (1-a)^2\sigma_y^2 $$ $$e_z = \sqrt{a^2 e_x^2 + (1-a)^2 e_y^2} = a \sqrt{ e_x^2 + \left(\frac{1-a}{a}\right)^2 e_y^2} $$ In particular, if $a=1/2$ , then $e_z = \frac{1}{2}\sqrt{ e_x^2 + e_y^2} $ share|cite|improve this answer answered Mar 22 '12 at 15:05 leonbloy 29.9k63384 If I understand it correctly, and in a more general form, if we had $n$ independent variables with their corresponding weights, $c_{1}, c_{2}...c_{n}$, the error of their weighted mean (when $\sum_{i=1}^n{c_{i}} = 1$) would be $\sqrt{c_{1}^2 e_1^2 + c_{2}^2 e_2^2 + ... + c_{n}^2 e_n^2}$. Right? –plok Mar 23 '12 at 10:56 @plok that's right –leonbloy Mar 23 '12 at 12:12 Thanks so much. –plok Mar 23 '12 at 12:50 add a commen
"change" in the value of that quantity. Results are is obtained by mathematical operations on the data, and small changes in any data quantity can affect the value of a result. We say that https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm "errors in the data propagate through the calculations to produce error in the result." 3.2 MAXIMUM ERROR We first consider how data errors propagate through calculations to affect error limits (or maximum error) of results. It's easiest to first consider determinate errors, which have explicit sign. This leads to useful rules for error propagation. Then we'll modify and extend the rules to other error measures and also to indeterminate errors. The underlying mathematics is that of "finite differences," an error propagation algebra for dealing with numbers which have relatively small variations imposed upon them. The finite differences we are interested in are variations from "true values" caused by experimental errors. Consider a result, R, calculated from the sum of two data quantities A and B. For this discussion we'll use ΔA and ΔB to represent the errors in A and B respectively. The data quantities are written to show the errors explicitly: [3-1] A + ΔA and propagation of error B + ΔB We allow the possibility that ΔA and ΔB may be either positive or negative, the signs being "in" the symbols "ΔA" and "ΔB." The result of adding A and B is expressed by the equation: R = A + B. When errors are explicitly included, it is written: (A + ΔA) + (B + ΔB) = (A + B) + (Δa + δb) So the result, with its error ΔR explicitly shown in the form R + ΔR, is: R + ΔR = (A + B) + (Δa + Δb) [3-2] The error in R is: ΔR = ΔA + ΔB. We conclude that the error in the sum of two quantities is the sum of the errors in those quantities. You can easily work out the case where the result is calculated from the difference of two quantities. In that case the error in the result is the difference in the errors. Summarizing: Sum and difference rule. When two quantities are added (or subtracted), their determinate errors add (or subtract). Now consider multiplication: R = AB. With errors explicitly included: R + ΔR = (A + ΔA)(B + ΔB) = AB + (ΔA)B + A(ΔB) + (ΔA)(ΔB) [3-3] or : ΔR = (ΔA)B + A(ΔB) + (ΔA)(ΔB) This doesn't look like a simple rule. However, when we express the errors in relative form, things look better. When the error a
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