Calculate the percent error between an experimental (measured) value and a theoretical (accepted) value. Also shows absolute and relative error.
| Formula | Expression |
|---|---|
| Percent Error | |exp − the| / |the| × 100% |
| Absolute Error | |exp − the| |
| Relative Error | (exp − the) / |the| |
| Signed % | (exp − the) / |the| × 100% |
| Measurement | Value |
|---|---|
| Experimental (measured) | 9.4 |
| Theoretical (accepted) | 9.8 |
| Percent Error | 4.08163265% |
| Signed Percent Error | -4.08163265% |
| Absolute Error | 0.4 |
| Signed Absolute Error | -0.4 |
| Relative Error | -0.04081633 |
| % Error | Interpretation |
|---|---|
| < 1% | Excellent accuracy |
| 1% – 5% | Very good accuracy |
| 5% – 10% | Acceptable accuracy |
| 10% – 20% | Moderate error |
| > 20% | Large error — review method |
Percent error is not the same as percent difference, and you cannot divide by whichever number feels convenient. Percent error always divides by the accepted value, since that is the target you are trying to match. Percent difference, used when neither value is the reference, divides by the average of the two. For a pendulum experiment that measured g as 9.4 m/s² against the accepted 9.8, the two formulas give 4.08% and 4.17%.
Percent error = |experimental − theoretical| ÷ |theoretical| × 100. The page also reports the signed percent error (the same fraction without the absolute value on top, so the sign shows direction), the absolute error |experimental − theoretical|, the signed absolute error, and the relative error (experimental − theoretical) ÷ |theoretical|, which is just the percent error as a decimal. A theoretical value of zero is refused, because the division would be meaningless.
What changes if you alter one input? Swap the roles, with 9.8 measured against an accepted 9.4, and the error rises from 4.0816% to 4.2553%, because the denominator got smaller. Rescale both numbers, 94 against 98, and the answer is the same 4.0816%, so a change of units does not matter as long as both sides use the same one. Move the accepted value to 9.81 and the error becomes 4.1794%. A tiny reference exaggerates things: 0.5 measured against an accepted 0.05 gives 900%. A negative accepted value is handled with its absolute value in the denominator.
Worked example with experimental 9.4 and theoretical 9.8. The absolute error is |9.4 − 9.8| = 0.4. Then 0.4 ÷ 9.8 = 0.0408163, and × 100 gives 4.08163265%, rounded to 8 decimals. The signed percent error is -4.08163265%, meaning the measurement fell below the accepted value, and the relative error is -0.04081633. The page's own table (under 1% excellent, 1% to 5% very good, 5% to 10% acceptable, 10% to 20% moderate, over 20% large) labels this very good. Those bands are a rule of thumb printed on the page, not an official standard.
The headline figure is never negative because it uses an absolute value. The Signed line beneath it keeps the sign: negative means you measured less than the accepted value, positive means more.
Percent error needs a trusted reference. With two measurements of equal standing, use percent difference instead, or report the spread of repeated readings.
The accepted value is probably small. A 0.45 gap against an accepted 0.05 is 900%, while the same gap against 50 is 0.9%. Read it alongside the absolute error.
No more than your measurement supports. 9.4 has two, so 4.08163265% is better quoted as 4.1% or about 4%.
Not by itself. What counts as acceptable depends on the instrument and field, and the page's bands are only a guide. A consistent sign across repeated trials points to a systematic cause, such as calibration.