What is the Difference Between Absolute Error and Relative Error?

The difference between absolute error and relative error lies in the way they express the error in a measured value. Here are the key differences between the two:

• Absolute Error: This is the difference between the actual value and the measured value of a quantity. It gives the magnitude of the error without considering the magnitude of the actual value. The formula for absolute error is: $$|x0 - x|$$, where $$x0$$ is the actual value and $$x$$ is the measured value.
• Relative Error: This is the ratio of the absolute error of a measurement and the actual value of the quantity. It gives the magnitude of the error relative to the actual value. The formula for relative error is: $$\frac{|x0 - x|}{x0}$$, where $$x_0$$ is the actual value and $$x$$ is the measured value. Relative error is usually expressed as a percentage by multiplying the fractional error by 100.

In summary, absolute error represents the actual difference between the measured and true values, while relative error represents the difference between the measured and true values relative to the true value itself. Absolute error is a measure of the accuracy of a measurement, while relative error is a measure of the precision of a measurement.

Comparative Table: Absolute Error vs Relative Error

The main difference between absolute error and relative error lies in how they are calculated and the information they provide about the measurement. Here is a table comparing the two:

For example, consider a carpenter measuring wooden floorboards with a tape measure that has an absolute error of 0.5 inches. The carpenter measures one board with the tape and finds that it is 36 inches long. The absolute error of this measurement is 0.5 inches. The relative error of the measurement is $$\frac{0.5}{36} = 0.014$$ or 1.4%.