How To Calculate Percent Error

Percent error expresses how far a measurement sits from an accepted value, as a percentage of that accepted value. The percentage is what makes it useful: an error of 0.1 grams means something very different when the target is 1 gram than when it is 1 kilogram, and the percentage normalises for that.

The calculation has four steps. Subtract the accepted value from the measured one, take the absolute value, divide by the accepted value, and multiply by 100. Reading 105 where the true figure is 100 gives 5 divided by 100, which is 5 percent. The absolute value matters because a reading that is too high and one that is too low are equally wrong in this measure.

The denominator is the accepted value, not the measured one, and that choice is deliberate. Using the accepted value makes the error a property of the measurement against the truth, which is what you want when comparing results from different experiments against the same standard. It also means every experiment reporting against the same accepted value uses the same denominator.

Percent difference is a different quantity that is often confused with percent error. Percent error compares a measurement against a known truth. Percent difference compares two measurements against each other when neither is authoritative, and it divides by the average of the two rather than by one of them. Reading 105 against 100 gives a percent difference of 0.04878048780487805, slightly smaller than the percent error, because the denominator is the midpoint of 102.5 rather than 100.

A sign convention worth knowing: many textbooks keep the sign so that a positive result means the measurement was high and a negative one means it was low. Others report only the absolute value. Both are correct as long as you say which you are using, and reporting the signed difference alongside the percent error removes the ambiguity entirely.

Percent error is not the same as accuracy or precision. Accuracy is how close a reading is to the true value, which is roughly what percent error measures. Precision is how close repeated readings are to each other, which percent error says nothing about. A set of readings can be tightly clustered and still have a large percent error if they share the same systematic bias.

The magnitude of an acceptable percent error depends entirely on the context. In a school titration, 5 percent might be perfectly reasonable. In analytical chemistry, anything above 1 percent would prompt a re-run. In physics constants, agreement to parts per million is the norm. The number alone means nothing without knowing what the field expects.

Error comes in two flavours. Random error scatters readings around the true value and shrinks when you average repeats. Systematic error pushes every reading in the same direction and does not shrink with repetition. Percent error cannot tell the two apart on its own, but plotting the residuals usually can, and that is why a single percent error figure is a summary rather than a diagnosis.

Percent error is closely related to relative error, which is the same quantity before the multiplication by 100. A relative error of 0.05 is exactly a percent error of 5 percent. Working in relative error for intermediate steps avoids carrying a factor of 100 through the algebra, which is a small but real simplification in longer calculations.

One practical warning: percent error explodes when the accepted value is near zero. Dividing by a tiny number produces a huge percentage even when the absolute error is negligible. When the true value is small or zero, report the absolute error and the units instead, because the percentage has stopped being informative.