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
   Measurement
   Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
   
    of the data
   Data
   The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...
   
    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
   Pi
   ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
   
   ).

In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, the numerical stability

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....

 of an algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 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

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

 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


where the vertical bars denote the absolute value

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

.
If : the relative error is


and the percent error is


These definitions can be extended to the case when and are n-dimensional vectors, by replacing the absolute value with an n-norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

.

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.

See also

• Accepted and experimental value
• Percent difference
  Percent difference
  The relative difference, percent difference, relative percent difference, or percentage difference between two quantities is the difference between them, expressed as a comparison to the size of one or both of them. Such measures are unitless numbers...
  
  
• Relative difference
• Uncertainty
  Uncertainty
  Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
  
  
• Experimental uncertainty analysis
  Experimental uncertainty analysis
  The purpose of this introductory article is to discuss the experimental uncertainty analysis of a derived quantity, based on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship to calculate that derived quantity...
  
  
• Propagation of uncertainty
  Propagation of uncertainty
  In statistics, propagation of error is the effect of variables' uncertainties on the uncertainty of a function based on them...
  
  
• Errors and residuals in statistics
  Errors and residuals in statistics
  In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...