Absolute Error vs Relative Error

What Absolute Error Means

Absolute error is the magnitude of the gap between the experimental and accepted values, ignoring direction. It is measured in the same units as the quantity itself. A 3 gram error on a 50 gram sample is a very different situation from a 3 gram error on a 5 kilogram batch, and absolute error on its own cannot tell you which.

What Relative Error Means

Relative error is the absolute error divided by the accepted value, giving a unitless proportion. An absolute error of 3 against an accepted value of 50 gives a relative error of 0.06. The same 3 gram error against an accepted value of 5000 gives 0.0006. Both describe a 3 gram gap, and relative error shows immediately that the second is far more precise.

How Relative Error Becomes Percent Error

Percent error is simply relative error expressed as a percentage, which means multiplying by 100. Relative error 0.06 becomes 6 percent. This is why the terms are sometimes used loosely in conversation, though strictly percent error carries the absolute value while a signed relative error can be negative.

Precision Versus Accuracy

Relative error measures accuracy, how close you are to the truth. Precision is a different property, how close repeated measurements are to each other. A very precise instrument can still be badly inaccurate if it is miscalibrated, and the way to reveal that is a large percent error across trials that all agree closely with each other.

Why Precision Matters in Reporting

Significant figures in an error figure communicate precision. Reporting 6 percent tells the reader the error is known to about one part in ten. Reporting 6.0003 percent claims far more precision than any realistic measurement supports and undermines the credibility of the whole result.

Which to Report

Report the absolute error when the units matter and the reader needs to know the physical size of the gap, such as in a machining tolerance. Report the percent or relative error when comparing across experiments, when the values are quoted in different units, or when a single headline figure is wanted.

Percentage Uncertainty Follows the Same Shape

An uncertainty interval has both an absolute half width in units and a relative half width as a proportion. When results are combined, absolute uncertainties add by adding their absolute widths, while relative uncertainties add by adding in quadrature. This is why a quantity with a large relative uncertainty dominates the total uncertainty of any calculation containing it.

Handling a Zero Accepted Value

Both relative error and percent error divide by the accepted value, so both are undefined when it is zero. This is a genuine limitation of relative measures rather than a calculational mistake. Absolute error still works, and percent difference is the correct substitute when comparing against another measurement.