Approximation error

The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because

1. the measurement of the data is not precise due to the instruments. (e.g., the accurate reading of a piece of paper is 4.5cm but since the ruler does not use decimals, you round it to 5cm.) or
2. approximations are used instead of the real data (e.g., 3.14 instead of π).

In the mathematical field of numerical analysis, the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm.

Overview

One commonly distinguishes between the relative error and the absolute error. The absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.

As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002. The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 and in the second it is only 0.000003.

Another example would be if you measured a beaker and read, 5mL. The correct reading would have been 6mL. This means that your % error (Approximate error) would be 16.66666.. % error.

Definitions

Given some value v and its approximation v_approx, the absolute error is

$\epsilon = |v-v_\text{approx}|\ ,$

where the vertical bars denote the absolute value. If :$v\ne 0.$ the relative error is

$\eta = \frac{|v-v_\text{approx}|}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right|,$

and the percent error is

$\delta = \frac{|v-v_\text{approx}|}{|v|}\times 100% = \left| \frac{v-v_\text{approx}}{v} \right|\times 100%.$

These definitions can be extended to the case when v and v_approx are n-dimensional vectors, by replacing the absolute value with an n-norm.^[1]

Instruments

In most indicating instruments, the accuracy is guaranteed to a certain percentage of full-scale reading. The limits of these deviations from the specified values are known as limiting errors or guarantee errors.^[2]

See also

• Accepted and experimental value
• Percent difference
• Relative difference
• Uncertainty
• Experimental uncertainty analysis
• Propagation of uncertainty
• Errors and residuals in statistics

References

1. ^ Golub, Gene; Charles F. Van Loan (1996). Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press. pp. 53. ISBN 0-8018-5413-X. 
2. ^ Albert D. Helfrick, Modern Electronic Instrumentation and Measurement Techniques, pg16, ISBN 81-297-0731-4

External links

• Weisstein, Eric W., "Percentage error" from MathWorld.

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This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Applied mathematics stubs Algebra stubs Numerical analysis Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Elm Park tube station Francis I, Duke of Brittany Look at other dictionaries: approximation error — apvalinimo paklaida statusas T sritis informatika apibrėžtis Skaičiaus tiksliosios reikšmės (a) ir jo suapvalintos reikšmės (a₁) skirtumo modulis: |a – a₁|. atitikmenys: angl. approximation error …   Enciklopedinis kompiuterijos žodynas Error (disambiguation) — Error may refer to: *Error, a mistake. *Error coram nobis, a type of writ. *Postage stamp error *Error coin *Error, a character from cience*Approximation error, error in scientific measurements *Standard error (statistics) *Errors and residuals… …   Wikipedia Approximation — An approximation (represented by the symbol ≈) is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as… …   Wikipedia Error analysis — is the study of kind and quantity of error that occurs, particularly in the fields of applied mathematics (particularly numerical analysis), applied linguistics and statistics. Error analysis in numerical modelling In numerical simulation or… …   Wikipedia Approximation theory — In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by best and simpler will depend on …   Wikipedia Error function — Plot of the error function In mathematics, the error function (also called the Gauss error function) is a special function (non elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined …   Wikipedia Approximation theory/Proofs — Proof that an N th degree polynomial that gives rise to an error function that has N + 2 maxima, of alternating signs and equal magnitudes, is optimal in the sense of approximation theory. Such an optimal polynomial gives the minimum value, over… …   Wikipedia error — errorless, adj. errorlessly, adv. /er euhr/, n. 1. a deviation from accuracy or correctness; a mistake, as in action or speech: His speech contained several factual errors. 2. belief in something untrue; the holding of mistaken opinions. 3. the… …   Universalium Approximation algorithm — In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP hard problems; since it is unlikely that there …   Wikipedia error of approximation — artinio paklaida statusas T sritis fizika atitikmenys: angl. error of approximation vok. Näherungsmethodefehler, m; Verfahrensfehler, m rus. погрешность аппроксимации, f; погрешность приближения, f pranc. erreur d’approximation, f …   Fizikos terminų žodynas 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”