Percent error calculator

How do I calculate percent error?

Percent error = |measured − accepted| ÷ |accepted| × 100. Measuring g as 9.6 m/s² when the accepted value is 9.81 gives |9.6 − 9.81| ÷ 9.81 × 100 = 2.14%.

Why the absolute value?

Percent error usually reports the size of the miss, not its direction. The calculator also shows the signed error, which tells you whether you were high or low.

What is a good percent error?

It depends on the experiment — a few percent is typical in school labs. Compare it with the precision of your instruments.

What is the difference between percent error and percent difference?

Percent error compares with a known true value; percent difference compares two measurements with each other, dividing by their average.

What if the accepted value is 0?

Percent error is undefined — you cannot divide by zero. Report the absolute error instead.

Which value goes in the denominator?

The accepted value, because the question is how far you are from the truth. For a measured 10.2 against an accepted 9.81, the error is 0.39 and 0.39 ÷ 9.81 gives 3.98%. Dividing by your own measurement instead gives 3.82%, which is a different number and not the percent error.

How is percent error different from percent uncertainty?

Percent error needs an accepted value to compare with and measures accuracy. Percent uncertainty needs none: it is the uncertainty of the measurement divided by the measurement, and it describes precision. A reading of 10.2 ± 0.04 m/s² has a percent uncertainty of 0.39% whatever the true value is.

How the formula works

The absolute error is the gap between the measured and the accepted value, |measured − accepted|. Dividing by the accepted value turns the gap into a fraction of what you were trying to measure, and multiplying by 100 makes it a percent. The calculator also shows the signed version, (measured − accepted) ÷ |accepted| × 100, where a plus sign means the reading was high and a minus sign that it was low (Wikipedia, “Relative change”). The formula needs a trusted reference. If you have two measurements and neither is the truth, use the percent difference, |a − b| ÷ ((a + b) ÷ 2) × 100, which is symmetric in the two readings.

Worked example: measuring g

A lab handout from NC State compares a measured acceleration of gravity of 10.2 m/s² with the accepted 9.81 m/s². The absolute error is |10.2 − 9.81| = 0.39 m/s², and 0.39 ÷ 9.81 = 0.0398, so the percent error is 3.98% (the handout rounds it to 4%). It is +3.98% signed: the reading was high. The 9.81 is itself rounded; NIST defines standard gravity as exactly 9.80665 m/s², which gives 4.01%. Two readings from different methods, 10.2 and 9.95, have no true value to compare with, so their percent difference is |10.2 − 9.95| ÷ 10.075 × 100 = 2.5%.

Common mistakes

Dividing by the measured value instead of the accepted one gives 3.82% here instead of 3.98%. Reporting every digit shown overstates the data: 10.2 has three significant figures, so 4.0% is all it supports. And when the accepted value is near zero a tiny denominator inflates the percentage, so quote the absolute error instead.

Sources: NC State University physics lab course (mechanics), Appendix B, “Percent Error and Percent Difference”; Wikipedia, “Relative change” (percent error and its signed form); NIST CODATA, standard acceleration of gravity gₙ = 9.80665 m/s². Related tools: percentage calculator, significant figures calculator, rounding calculator and standard deviation calculator. How we check our calculators.