These are terms used to describe the accuracy or precision of a measurement or calculation when compared to a true or accepted value. They are commonly used in scientific and engineering contexts to assess the quality of data or results.
Absolute Error:
- Absolute error measures the difference between a measured or calculated value and the true or accepted value. It is expressed in the same units as the quantity being measured.
- The formula for absolute error:
\begin{aligned}\text{Absolute Error} &= |\text{Measured Value} – \text{True Value}|\\\end{aligned}
Relative Error:
- Relative error quantifies the accuracy of a measurement or calculation by comparing the absolute error to the magnitude of the true or accepted value. It is often expressed as a fraction or a decimal.
- The formula for relative error:
\begin{aligned}\text{Relative Error} &= \dfrac{|\text{Measured Value} – \text{True Value}|}{|\text{True Value}|}\\\end{aligned}
Percentage Error:
- Percentage error is a way to express the relative error as a percentage of the true value. It provides a more intuitive sense of the error.
- The formula for percentage error:
\begin{aligned}\text{Percentage Error} &= \text{(Relative Error)}\times 100\ \%\\\end{aligned}
To illustrate these concepts, consider an example where we will measure the length of an object, whose true or accepted value is 100\ \rm cm. If the measured length is 98.5\ \rm cm, what will be the error in this measurement?
- Absolute Error = |98.5\ \rm cm – 100\ \rm cm| = 1.5\ \rm cm
- Relative Error = \dfrac{1.5\ \rm cm }{100\ \rm cm }= 0.015 (or 1.5% when expressed as a percentage)
- Percentage Error = 1.5%
In the above example, the absolute error is 1.5 cm, indicating how far off the measurement is from the true value. The relative error is 0.015, meaning the measurement is 1.5% off from the true value. The percentage error, also 1.5%, provides a simple way to communicate the degree of error in a more understandable format.
These error metrics are valuable for evaluating the quality and reliability of measurements, especially when comparing results to established standards or expectations. They help scientists and engineers assess the accuracy and precision of their data.