# Absolute and Relative Error and How to Calculate Them Absolute, relative, and percent error are ways to measure the error of a measurement or calculation.

Absolute, relative, and percent error are the most common experimental error calculations in science. Grouped together, they are types of approximation error. Basically, the premise is that no matter how carefully you measure something, you’ll always be off a bit due to the limitations of the measuring instrument. For example, you may be only able to measure to the nearest millimeter on a ruler or the nearest milliliter on a graduated cylinder. Here are the definitions, equations, and examples of how to use these types of error calculations.

### Absolute Error

Absolute error is the magnitude (size) of the difference between a measured value and a true or exact value.

Absolute Error = |True Value – Measured Value|

Absolute Error Example:
A measurement is 24.54 mm and the true or known value is 26.00 mm. Find the absolute error.
Absolute Error = |26.00 mm – 25.54 mm|= 0.46 mm
Note absolute error retains its units of measurement.

The vertical bars indicate absolute value. In other words, you drop any negative sign you may get. For this reason, it doesn’t actually matter whether you subtract the measured value from the true value or the other way around. You’ll see the formula written both ways in textbooks and both forms are correct.

What matters is that you interpret the error correctly. If you graph error bars, half of the error is higher than the measured value and half is lower. For example, if your error is 0.2 cm, it is the same as saying ±0.1 cm.

The absolute error tells you how big a difference there is between the measured and true values, but this information isn’t very helpful when you want to know if the measured value is close to the real value or not. For example, an absolute error of 0.1 grams is more significant if the true value is 1.4 grams than if the true value is 114 kilograms! This is where relative error and percent error help.

### Relative Error

Relative error puts absolute error into perspective because it compares the size of absolute error to the size of the true value. Note that the units drop off in this calculation, so relative error is dimensionless (unitless).

Relative Error = |True Value – Measured Value| / True Value
Relative Error = Absolute Error / True Value

Relative Error Example:
A measurement is 53 and the true or known value is 55. Find the relative error.
Relative Error = |55 – 53| / 55 = 0.034
Note this value maintains two significant digits.

Note: Relative error is undefined when the true value is zero. Also, relative error only makes sense when a measurement scale starts at a true zero. So, it makes sense for the Kelvin temperature scale, but not for Fahrenheit or Celsius!

### Percent Error

Percent error is just relative error multiplied by 100%. It tells what percent of a measurement is questionable.

Percent Error = |True Value – Measured Value| / True Value x 100%
Percent Error = Absolute Error / True Value x 100%
Percent Error = Relative Error x 100%

Percent Error Example:
A speedometer says a car is going 70 mph but its real speed is 72 mph. Find the percent error.
Percent Error = |72 – 70| / 72 x 100% = 2.8%

### Mean Absolute Error

Absolute error is fine if you’re only taking one measurement, but what about when you collect more data? Then, mean absolute error is useful. Mean absolute error or MAE is the sum of all the absolute errors divided by the number of errors (data points). In other words, it’s the average of the errors. Mean absolute error, like absolute error, retains its units.

Mean Absolute Error Example:
You weigh yourself three times and get values of 126 lbs, 129 lbs, 127 lbs. Your true weight is 127 lbs. What is the mean absolute error of the measurements.
Mean Absolute Error = [|126-127 lbs|+|129-127 lbs|+|127-127 lbs|]/3 = 1 lb

### References

• 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.
• Helfrick, Albert D. (2005). Modern Electronic Instrumentation and Measurement Techniques. ISBN 81-297-0731-4.
• Steel, Robert G. D.; Torrie, James H. (1960). Principles and Procedures of Statistics, With Special Reference to Biological Sciences. McGraw-Hill.