Error Propagation Mathworld
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Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and error propagation formula Statistics>Error Analysis> Interactive Entries>Interactive Demonstrations> Error Propagation Given a formula with an absolute
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error in of , the absolute error is . The relative error is . If , then (1) where denotes
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the mean, so the sample variance is given by (2) (3) The definitions of variance and covariance then give (4) (5) (6) (where ), so (7) If and are uncorrelated, then so
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(8) Now consider addition of quantities with errors. For , and , so (9) For division of quantities with , and , so (10) Dividing through by and rearranging then gives (11) For exponentiation of quantities with (12) and (13) so (14) (15) If , then (16) For logarithms of quantities with , , so (17) (18) For multiplication with , and , so (19) error propagation mathematica (20) (21) For powers, with , , so (22) (23) SEE ALSO: Absolute Error, Accuracy, Covariance, Percentage Error, Precision, Relative Error, Significant Digits, Variance REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p.14, 1972. Bevington, P.R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, pp.58-64, 1969. Referenced on Wolfram|Alpha: Error Propagation CITE THIS AS: Weisstein, Eric W. "Error Propagation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ErrorPropagation.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal» Collection of teaching and learning tools built by Wol
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch relative error calculator Number Theory>Rounding> Interactive Entries>Interactive Demonstrations> Roundoff Error Roundoff error is the difference relative error formula between an approximation of a number used in computation and its exact (correct) value. In certain types of computation, relative error definition roundoff error can be magnified as any initial errors are carried through one or more intermediate steps. An egregious example of roundoff error is provided by a short-lived index devised at http://mathworld.wolfram.com/ErrorPropagation.html the Vancouver stock exchange (McCullough and Vinod 1999). At its inception in 1982, the index was given a value of 1000.000. After 22 months of recomputing the index and truncating to three decimal places at each change in market value, the index stood at 524.881, despite the fact that its "true" value should have been 1009.811. Other sorts of roundoff error can also http://mathworld.wolfram.com/RoundoffError.html occur. A notorious example is the fate of the Ariane rocket launched on June 4, 1996 (European Space Agency 1996). In the 37th second of flight, the inertial reference system attempted to convert a 64-bit floating-point number to a 16-bit number, but instead triggered an overflow error which was interpreted by the guidance system as flight data, causing the rocket to veer off course and be destroyed. The Patriot missile defense system used during the Gulf War was also rendered ineffective due to roundoff error (Skeel 1992, U.S. GAO 1992). The system used an integer timing register which was incremented at intervals of 0.1 s. However, the integers were converted to decimal numbers by multiplying by the binary approximation of 0.1, As a result, after 100 hours ( ticks), an error of had accumulated. This discrepancy caused the Patriot system to continuously recycle itself instead of targeting properly. As a result, an Iraqi Scud missile could not be targeted and was allowed to detonate on a barracks, killing 28 people. SEE ALSO: Absolute Error, Accuracy, Error Propagation, Precision, Relative Error, Rounding, Significant Digits REFERENCES: Chartier, T. "Devast
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 http://mathworld.wolfram.com/AbsoluteError.html entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Error Analysis> History and Terminology>Disciplinary Terminology>Religious Terminology> Absolute Error The http://web.tecnico.ulisboa.pt/~mcasquilho/compute/errtheory/Fx-errors.php difference between the measured or inferred value of a quantity and its actual value , given by (sometimes with the absolute value taken) is called error propagation the absolute error. The absolute error of the sum or difference of a number of quantities is less than or equal to the sum of their absolute errors. SEE ALSO: Error Propagation, Percentage Error, Relative Error REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, error propagation calculator and Mathematical Tables, 9th printing. New York: Dover, p.14, 1972. Referenced on Wolfram|Alpha: Absolute Error CITE THIS AS: Weisstein, Eric W. "Absolute Error." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AbsoluteError.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal» Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, int
“S”) Data: generated; or supplied by user. • μ, σ (for G) Mean and standard deviation — ignored if"S". ntrial, .seed (for G) No. of trials and seed — ignored if"S". • Supplied data, X (forS) 26.1 25.8 26.3 26.8 25.9 26.2 User-supplied data (n must always match them !) — ignored if"G". • Outlier ? smallest largest Outlier (?), smallest or largest, to be assessed. P % Probability for interval for μ. • Show values No Yes Shows the coordinates of the graph. • From a randomly generated (G) or user-supplied (S) sample (set of observations), calculates some common statistics: average (x-bar), standard deviation (s), average deviation (δ). The graph shows: for G, δ simulated for random samples of size 2k trials, with (integer) k = 1..log2(ntrial) (log2100000 = 16.6 → 16); for S, the user data, without the (possible) outlier. An extreme (smallest or largest) value in the sample is assessed to estimate whether it is an outlier (to be rejected). The decision on what is an outlier is a delicate question. See, for example, Barnett and Lewis [1998]. Excel: t = Studinv([(1+prob)/2, ν] becomes TINV[(1-prob, ν] Typically, ν = n − 1. (Here, n had already been reduced, so ν=noriginal − 2 .) Other suggested data: 26.6 26.8 26.2 26.9 27.1 27.0 (smallest) 26.6 26.9 26.7 26.6 26.7 26.6 (largest) References: Plate: Errth08804 •Scuro, Sante R.: Error theory.pdf (Texas A&M Univ.). •Err. analysis (P. C. Eklund, PennState U.). •Petridis, A. (Drake Univ.). •Wolfs, F., see "Error analysis" (Univ. of Rochester). •Weisstein, Eric W., "Error propagation". From MathWorld — a Wolfram Web Resource (accessed 2008-08-04). •Barnett, Vic and Toby Lewis, 1998, "Outliers in statistical data", 3.rd ed. (=1994, 1978), Wiley, New York, NY (USA). (At ISEG Library: 1994.) •Bevington, Philip R. and D. Keith Robinson, 2003, "Data reduction and error analysis for the physical sciences", 3.rd ed., McGraw-Hill, New York, NY (USA). •Google search: simulation "Monte Carlo" "Chemical Engineering"… •Kurtosis, skewness.xls •Descriptive statistics (SAS v7 doc., Oklahoma State Univ., Stillwater). •Student's t (this site). •Taylor, John R., 1997, "An introduction to error analysis", 2.nd ed., Univ. Sci. Books (USB), Sausalito, CA (USA) (intro IE). •[Lyons, 1991] •[J. K. Taylor, 1987] •Weisstein, Eric W., "Student's t-distribution". From Mat