Editing Approximation error (section)

==Examples==
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To illustrate these concepts with a numerical example, consider an instance where the exact, accepted value is 50, and its corresponding approximation is determined to be 49.9. In this particular scenario, the absolute error is precisely 0.1 (calculated as |50 − 49.9|), and the relative error is calculated as the absolute error 0.1 divided by the true value 50, which equals 0.002. This relative error can also be expressed as 0.2%. In a more practical setting, such as when measuring the volume of liquid in a 6&nbsp;mL beaker, if the instrument reading indicates 5&nbsp;mL while the true volume is actually 6&nbsp;mL, the percent error for this particular measurement situation is, when rounded to one decimal place, approximately 16.7% (calculated as |(6&nbsp;mL − 5&nbsp;mL) / 6&nbsp;mL| × 100%).

The utility of relative error becomes particularly evident when it is employed to compare the quality of approximations for numbers that possess widely differing magnitudes; for example, approximating the number 1,000 with an absolute error of 3 results in a relative error of 0.003 (or 0.3%). This is, within the context of most scientific or engineering applications, considered a significantly less accurate approximation than approximating the much larger number 1,000,000 with an identical absolute error of 3. In the latter case, the relative error is a mere 0.000003 (or 0.0003%). In the first case, the relative error is 0.003, whereas in the second, more favorable scenario, it is a substantially smaller value of only 0.000003. This comparison clearly highlights how relative error provides a more meaningful and contextually appropriate assessment of precision, especially when dealing with values across different orders of magnitude.

There are two crucial features or caveats associated with the interpretation and application of relative error that should always be kept in mind. Firstly, relative error becomes mathematically undefined whenever the true value (''v'') is zero, because this true value appears in the denominator of its calculation (as detailed in the formal definition provided above), and division by zero is an undefined operation. Secondly, the concept of relative error is most truly meaningful and consistently interpretable only when the measurements under consideration are performed on a [[Level of measurement#Ratio scale|ratio scale]]. This type of scale is characterized by possessing a true, non-arbitrary zero point, which signifies the complete absence of the quantity being measured. If this condition of a ratio scale is not met (e.g., when using interval scales like Celsius temperature), the calculated relative error can become highly sensitive to the choice of measurement units, potentially leading to misleading interpretations. For example, when an absolute error in a [[temperature]] measurement given in the [[Celsius scale]] is 1&nbsp;°C, and the true value is 2&nbsp;°C, the relative error is 0.5 (or 50%, calculated as |1°C / 2°C|). However, if this exact same approximation, representing the same physical temperature difference, is made using the [[Kelvin scale]] (which is a ratio scale where 0&nbsp;K represents absolute zero), a 1&nbsp;K absolute error (equivalent in magnitude to a 1&nbsp;°C error) with the same true value of 275.15&nbsp;K (which is equivalent to 2&nbsp;°C) gives a markedly different relative error of approximately 0.00363, or about 3.63{{e|-3}} (calculated as |1&nbsp;K / 275.15&nbsp;K|). This disparity underscores the importance of the underlying measurement scale.