Percent Error Calculator

What is percent error?

Percent error tells you how far a measurement landed from the value it should have had, expressed as a percentage of that correct value. The correct value is called the true, accepted, theoretical or reference value, depending on the field. The measured one is the observed or experimental value.

Because the gap is divided by the true value, the result does not depend on units. A 2 gram miss on a 100 gram sample and a 2 kilometre miss on a 100 kilometre trip are both a 2% error, which is what makes the figure useful for comparing instruments, experiments and methods of very different sizes.

How to use the calculator

1. Enter what you measured in Observed value.
2. Enter the value you expected or the accepted figure in True value.
3. Press Calculate. The answer appears in the Percent error row.

• The result is signed. A minus sign means the observed value was lower than the true value; no sign means it was higher. If you only care about the size of the miss, ignore the sign.
• Whole-number results are shown without decimals (20%). Everything else is rounded to two decimals (133.33%), with thousands separators for large values.
• The true value cannot be zero, because the formula divides by it. In that case the tool reports an error instead of a number.
• Both fields accept negative numbers and decimals. Exponent notation such as 1e3 is not accepted, so type the full number.

The percent error formula

percent error = (observed − true) ÷ true × 100The calculator keeps the sign of the difference.

Written in steps: the absolute error is the plain difference observed − true. Dividing it by the true value gives the relative error as a decimal, and multiplying by 100 turns that into a percentage. So a relative error of 0.015 is a percent error of 1.5%.

Many textbooks and lab manuals put absolute value bars around the difference so that the answer is never negative. The calculator shows the signed version because the direction of the error is usually the first thing you want to know (a scale that reads high, a sensor that drifts low). Drop the sign mentally if your course asks for the absolute form.

Worked examples

Measuring gravity. A pendulum experiment gives g = 9.78 m/s² while the accepted value is 9.81 m/s². The difference is 9.78 − 9.81 = −0.03, and −0.03 ÷ 9.81 = −0.003058. Multiplying by 100 gives −0.31%: the experiment came out a third of a percent low.

A thermometer check. Boiling water at sea level should read 100 °C, but the thermometer shows 101.2 °C. The error is (101.2 − 100) ÷ 100 × 100 = 1.20% too high. Note that percent error on the Celsius scale depends on where zero sits, so for temperatures it is more meaningful to quote the error in degrees.

From an absolute error. A kitchen scale is checked with a 250 g calibration weight and reads 250.5 g, so the absolute error is 0.5 g. The percent error is 0.5 ÷ 250 × 100 = 0.20%. You can enter 250.5 and 250 in the calculator, or divide by hand as shown.

A tape measure. A tape that is supposed to span 100 cm is laid against a certified bar and found to cover only 99.4 cm of real length. Treating the 100 cm marking as the observed value and 99.4 cm as the true value, the error is (100 − 99.4) ÷ 99.4 × 100 = 0.60%. Every length you read from that tape is, on average, 0.6% too long.

Percent error compared with similar measures

• Percent error compares a measurement with a value you trust. The denominator is the trusted (true) value.
• Percentage difference compares two values that have equal standing, such as two measurements of the same thing, and divides by their average. It has no direction. The percentage calculator has a mode for it.
• Percent change compares a later value with an earlier one. See the percent increase and percent decrease calculators.

Percent error is also not the same thing as a statistical standard error or a margin of error, which describe the spread of an estimate across repeated samples rather than a single miss from a known target. For spread, see the standard deviation calculator.

Common mistakes

• Dividing by the wrong value. The denominator must be the true value, not the observed one. With an observed 120 and a true 100 the percent error is 20 ÷ 100 = 20%, but dividing by the observed value would give 16.67%. The two drift further apart as the error grows.
• Mixing units. Convert both numbers to the same unit before entering them.
• Rounding too early. Keep full precision in the difference and round only the final percentage.
• Using a negative true value. The formula divides by the true value as entered, so with a negative reference the sign of the result is flipped compared with "observed above or below". Compare magnitudes instead, or measure from a different baseline.
• Calling a systematic offset random. A consistent +0.4% on a scale is a calibration issue you can correct; scattered errors need repeated measurements to average out.

Percent error is a descriptive figure for a single measurement. It says nothing about whether the true value itself is reliable, or about the uncertainty of the instrument.