Percent Error Calculator

How to Calculate Percent Error

Percent error measures how far a measured (experimental) value is from the known or accepted (theoretical) value, expressed as a percentage of that accepted value. It's used constantly in science labs to judge how accurate a measurement or experiment was.

$$\%\text{Error} = \frac{|E - T|}{|T|} \times 100$$

Because the numerator uses an absolute value, percent error is always reported as a positive number — it tells you the size of the gap, not whether the measurement came in high or low. For a worked example: a student measures the acceleration due to gravity as 9.80 meters per second squared, against the accepted value of 9.81. The raw difference is 0.01, and the percent error is 0.01 divided by 9.81, times 100, which comes out to about 0.10 percent — a very accurate result.

Why Theoretical Value of 0 Breaks the Formula

The percent error formula divides by the theoretical value, so if that value is exactly 0, the calculation is undefined — the same reason any division by zero is undefined in math. This actually comes up in real experiments, for example when the accepted result of a measurement is genuinely zero (a net force that should cancel out, or a net charge that should balance to nothing). In that case, percent error simply isn't a meaningful figure — report the raw absolute difference instead, since there's no nonzero baseline to compare it against.

Percent Error vs. Percent Difference

These sound alike but answer different questions. Percent error compares a measurement against a known correct answer, so one of the two numbers is treated as the "truth." Percent difference compares two values that are both just measurements, with neither one accepted as more correct than the other — it divides by the average of the two instead of picking one as the reference.

A Brief History of Error Analysis in Science

Formal treatment of measurement error is a comparatively recent addition to the scientific method — early experimental scientists reported raw results with little systematic sense of how much to trust them. That began to change in the late 18th and early 19th centuries, largely driven by astronomy and surveying, where repeated observations of the same star or landmark rarely agreed exactly. Carl Friedrich Gauss and Adrien-Marie Legendre independently developed the method of least squares around the turn of the 19th century as a way to combine many imperfect observations into a single best estimate, laying much of the mathematical groundwork that modern error analysis still builds on.

The specific distinction this calculator relies on — a known "theoretical" value versus a measured "experimental" one — reflects a broader split scientists eventually formalized between accuracy (how close a measurement is to the true value) and precision (how consistent repeated measurements are with each other), two related but genuinely different properties that are easy to conflate. Percent error itself is a comparatively simple, informal metric next to the fuller statistical treatment of uncertainty developed over the same period, but it remains a standard first tool taught in introductory science labs precisely because it's quick to compute and easy to interpret.

Common Percent Error Mistakes

A frequent mistake is dividing by the experimental value instead of the theoretical value — the accepted value always belongs in the denominator, not the measured one. Another is forgetting the absolute value and reporting a negative percent error, which can be a useful signed figure in some contexts but isn't the standard "percent error" most instructions ask for. Rounding intermediate steps too early (rounding the difference before dividing) can also introduce a small but avoidable error into the final percentage.

Percent Error Terms You Should Know

Experimental Value — the value actually measured or observed during an experiment, sometimes called the observed or measured value.

Theoretical Value — the known, calculated, or accepted correct value, sometimes called the true or accepted value.

Absolute Error — the plain difference between the experimental and theoretical values, before it's turned into a percentage.

Accuracy — how close a measurement is to the true value; percent error is one common way to quantify it.