Relative and absolute error

Measurement errors

Absolute error
The absolute error is the difference between the absolute value of an exact number x and the measured value of a number x0, calculated using the following formula:
Δx=|x-x0|
The absolute error provides information on how much the measured value differs from the exact value.

Take the following example:
We know that some work should be done in 4 hours, this is our number x. After doing a certain amount of work, we take a measurement and get the following result: 4.5 hours. This is our measured value of x0.
To calculate the absolute error, we substitute our data into the formula:
Δx=|4-4.5|
Δx=|-0.5|
Δx=0.5

The calculation shows that the absolute error is 0.5 hours (i.e., the time taken to do the work has been underestimated by 0.5 hours). Note that the absolute error will be the same in cases where the length of the work measured is 4.5 hours or 3.5 hours. This is as a result of determining the absolute value.

Relative error

The relative error is the quotient of the absolute error ΔX to the exact value of the number X. It is expressed by the following formula:
δ=Δx / x=|x-x0| / x

The relative error is often expressed as a percentage. For this purpose, the action must be multiplied by 100%; therefore:
δ= (Δx / x)*100%

Going back to our example of work completed, to calculate the relative error (δ), divide the absolute error (Δx= 0.5 hours) by the value of the time spent doing the work (x=4 hours). By substituting these values into the formula, we get:
δ= 0.5 / 4= 0.125
The percentage relative error is:
δ= 0.125*100% = 12.5%
The work was done in 12.5% more time than usual.

In summary, the relative error – therefore – shows how much of a given number is the measured value (i.e., the value by which we have decreased or increased the number).