# Absolute and Relative Error Calculation

Absolute error and relative error are two types of experimental error. You'll need to calculate both types of error in science, so it's good to understand the difference between them and how to calculate them.

## Absolute Error

Absolute error is a measure of how far 'off' a measurement is from a true value or an indication of the uncertainty in a measurement. For example, if you measure the width of a book using a ruler with millimeter marks, the best you can do is measure the width of the book to the nearest millimeter. You measure the book and find it to be 75 mm. You report the absolute error in the measurement as 75 mm +/- 1 mm. The absolute error is 1 mm. Note that absolute error is reported in the same units as the measurement.

Alternatively, you may have a known or calculated value and you want to use absolute error to express how close your measurement is to the ideal value. Here absolute error is expressed as the difference between the expected and actual values.

Absolute Error = Actual Value - Measured Value

For example, if you know a procedure is supposed to yield 1.0 liters of solution and you obtain 0.9 liters of solution, your absolute error is 1.0 - 0.9 = 0.1 liters.

## Relative Error

You first need to determine absolute error to calculate relative error. Relative error expresses how large the absolute error is compared with the total size of the object you are measuring. Relative error is expressed as a fraction or is multiplied by 100 and expressed as a percent.

Relative Error = Absolute Error / Known Value

For example, a driver's speedometer says his car is going 60 miles per hour (mph) when it's actually going 62 mph. The absolute error of his speedometer is 62 mph - 60 mph = 2 mph. The relative error of the measurement is 2 mph / 60 mph = 0.033 or 3.3%

## Sources

• Hazewinkel, Michiel, ed. (2001). "Theory of Errors." Encyclopedia of Mathematics. Springer Science+Business Media B.V. / Kluwer Academic Publishers. ISBN 978-1-55608-010-4.
• Steel, Robert G. D.; Torrie, James H. (1960). Principles and Procedures of Statistics, With Special Reference to Biological Sciences. McGraw-Hill.
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