Error Propagation Formula Wiki
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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 values of experimental measurements they have uncertainties due to measurement
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limitations (e.g., instrument precision) which propagate to the combination of variables in the function. error propagation formula excel The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties
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can also be 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 error propagation formula calculator 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 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 error propagation formula for division 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_ ρ 2)\}} be a set of m functions which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_ σ 7,x_ σ 6,\dots ,x_ σ 5} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 … m ) {\displaystyle A_ σ
advice or instruction. (March 2011) (Learn how and when to remove this template message) This article needs more links to other
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articles to help integrate it into the encyclopedia. Please help improve general error propagation formula this article by adding links that are relevant to the context within the existing text. (October
Error Propagation Rules
2013) (Learn how and when to remove this template message) The purpose of this introductory article is to discuss the experimental uncertainty analysis of a derived quantity, based https://en.wikipedia.org/wiki/Propagation_of_uncertainty on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship ("model") to calculate that derived quantity. The model used to convert the measurements into the derived quantity is usually based on fundamental principles of a science or engineering discipline. The uncertainty has two components, namely, bias (related to https://en.wikipedia.org/wiki/Experimental_uncertainty_analysis accuracy) and the unavoidable random variation that occurs when making repeated measurements (related to precision). The measured quantities may have biases, and they certainly have random variation, so what needs to be addressed is how these are "propagated" into the uncertainty of the derived quantity. Uncertainty analysis is often called the "propagation of error." It will be seen that this is a difficult and in fact sometimes intractable problem when handled in detail. Fortunately, approximate solutions are available that provide very useful results, and these approximations will be discussed in the context of a practical experimental example. Contents 1 Introduction 2 Systematic error / bias / sensitivity analysis 2.1 Introduction 2.2 Sensitivity errors 2.3 Direct (exact) calculation of bias 2.4 Linearized approximation; introduction 2.5 Linearized approximation; absolute change example 2.6 Linearized approximation; fractional change example 2.7 Results table 3 Random error / precision 3.1 Introduction 3.2 Derived-quantity PDF 3.3 Linearized approximations for derived-quantity mean and variance 3.4 Matrix format of variance approximation
the https://en.wikipedia.org/wiki/Error_analysis model vary about a mean. Error analysis (linguistics) https://en.wikipedia.org/wiki/Error_analysis_(mathematics) studies the types and causes of language errors. Error analysis for the Global Positioning System This disambiguation page lists articles associated with the title Error analysis. If an internal link error propagation led you here, you may wish to change the link to point directly to the intended article. Retrieved from "https://en.wikipedia.org/w/index.php?title=Error_analysis&oldid=724970265" Categories: Disambiguation pagesHidden categories: All article disambiguation pagesAll disambiguation pages Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog error propagation formula in Namespaces Article Talk Variants Views Read Edit View history More Search Navigation Main pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikipedia store Interaction HelpAbout WikipediaCommunity portalRecent changesContact page Tools What links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationWikidata itemCite this page Print/export Create a bookDownload as PDFPrintable version Languages Español Edit links This page was last modified on 12 June 2016, at 19:18. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view
issue is particularly prominent in applied areas such as numerical analysis and statistics. Contents 1 Error analysis in numerical modeling 1.1 Forward error analysis 1.2 Backward error analysis 2 Applications 2.1 Global positioning system 2.2 Molecular dynamics simulation 2.3 Scientific data verification 3 See also 4 References 5 External links Error analysis in numerical modeling[edit] In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean. For instance, in a system modeled as a function of two variables z = f ( x , y ) {\displaystyle \scriptstyle z\,=\,f(x,y)} . Error analysis deals with the propagation of the numerical errors in x {\displaystyle \scriptstyle x} and y {\displaystyle \scriptstyle y} (around mean values x ¯ {\displaystyle \scriptstyle {\bar {x}}} and y ¯ {\displaystyle \scriptstyle {\bar {y}}} ) to error in z {\displaystyle \scriptstyle z} (around a mean z ¯ {\displaystyle \scriptstyle {\bar {z}}} ).[1] In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. Forward error analysis[edit] Forward error analysis involves the analysis of a function z ′ = f ′ ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z'=f'(a_{0},\,a_{1},\,\dots ,\,a_{n})} which is an approximation (usually a finite polynomial) to a function z = f ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z\,=\,f(a_{0},a_{1},\dots ,a_{n})} to determine the bounds on the error in the approximation; i.e., to find ϵ {\displaystyle \scriptstyle \epsilon } such that 0 ≤ | z − z ′ | ≤ ϵ {\displaystyle \scriptstyle 0\,\leq \,|z-z'|\,\leq \,\epsilon } . Backward error analysis[edit] Backward error analysis involves the analysis of the approximation function z ′ = f ′ ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z'\,=\,f'(a_{0},\,a_{1},\,\dots ,\,a_{n})} , to determine the bounds on the parameters a i = a i ¯ ± ϵ i {\displaystyle \scriptstyle a_{i}\,=\,{\bar {a_{i}}}\,\pm \,\epsilon _{i}} such that the result z ′ = z {\displaystyle \scriptstyle z'\,=\,z} .[2] Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numeri