Approximation error

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

${\displaystyle \epsilon =|v-v_{\text{approx}}|\ ,}$ [2][3]

where the vertical bars denote the absolute value. If ${\displaystyle v\neq 0,}$ the relative error is

${\displaystyle \eta ={\frac {\epsilon }{|v|}}=\left|{\frac {v-v_{\text{approx}}}{v}}\right|=\left|1-{\frac {v_{\text{approx}}}{v}}\right|,}$

and the percent error (an expression of the relative error) is [3]

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

An error bound is an upper limit on the relative or absolute size of an approximation error.[4]

Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)  

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 = 0.2%. As a practical example, when measuring a 6 mL beaker, the value read was 5 mL. The correct reading being 6 mL, this means the percent error in that particular situation is, rounded, 16.7%.

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 while in the second it is only 0.000003.

There are two features of relative error that should be kept in mind. First, relative error is undefined when the true value is zero as it appears in the denominator (see below). Second, relative error only makes sense when measured on a ratio scale, (i.e. a scale which has a true meaningful zero), otherwise it is sensitive to the measurement units. For example, when an absolute error in a temperature measurement given in Celsius scale is 1 °C, and the true value is 2 °C, the relative error is 0.5. But if the exact same approximation is made with the Kelvin scale, a 1 K absolute error with the same true value of 275.15 K = 2 °C gives a relative error of 3.63×10⁻³.

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.[6]

• Accepted and experimental value
• Condition number
• Errors and residuals in statistics
• Experimental uncertainty analysis
• Machine epsilon
• Measurement error
• Measurement uncertainty
• Propagation of uncertainty
• Quantization error
• Relative difference
• Round-off error
• Uncertainty

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