Calculate percent error between experimental and theoretical values. Shows absolute error, relative error, signed/unsigned result, and step-by-step working.
Percent error (also written as percentage error) is a measurement of the accuracy of an experimental result compared to a known or theoretical value. It is one of the most frequently calculated quantities in school and university science — from chemistry density experiments to physics measurements of gravitational acceleration. Our free percent error calculator provides the result instantly with clear step-by-step working, making it ideal for students in the US taking AP Physics or AP Chemistry, or UK students studying A-Level Physics and A-Level Chemistry.
The standard formula for percent error is:
% Error = |Experimental Value − Theoretical Value| / |Theoretical Value| × 100%
The vertical bars | | denote absolute value — meaning we take the positive version of the quantity regardless of sign.
Breaking it down step by step:
Some applications require knowing the direction of the error, not just its magnitude. Signed percent error removes the absolute value from the numerator:
Signed % Error = (Experimental − Theoretical) / |Theoretical| × 100%
A positive signed percent error means the experimental value is above the theoretical value. A negative signed percent error means it is below. For example, if you measure gravity as 9.65 m/s² when the accepted value is 9.81 m/s², the signed error is −1.63% (measuring lower than expected). This is useful for identifying systematic biases in measurement.
These are related but distinct quantities:
| Type | Formula | Unit | Use |
|---|---|---|---|
| Absolute error | |Experimental − Theoretical| | Same as measurement | Shows magnitude of error in original units |
| Relative error | |Experimental − Theoretical| / |Theoretical| | Dimensionless ratio | Comparison across different scales |
| Percent error | Relative error × 100% | % | Most intuitive for reporting and grading |
One of the classic school physics experiments is measuring the acceleration due to gravity using a pendulum or free-fall apparatus.
The accepted value of the speed of light in a vacuum is c = 2.998 × 10⁸ m/s. If an experimental measurement gives 3.012 × 10⁸ m/s, the percent error is |3.012 − 2.998| / 2.998 × 100% ≈ 0.47%. In precision physics experiments, percent errors below 1% are generally considered excellent.
A student measures the density of copper experimentally as 8.78 g/cm³. The accepted value is 8.96 g/cm³.
% Error = |8.78 − 8.96| / 8.96 × 100% = 0.18/8.96 × 100% ≈ 2.01%
For a density experiment using basic school equipment, 2% error is quite acceptable.
If a student determines the molar mass of NaCl experimentally as 57.8 g/mol vs the accepted 58.44 g/mol:
% Error = |57.8 − 58.44| / 58.44 × 100% ≈ 1.10%
Random errors are unpredictable fluctuations that cause measurements to scatter around the true value. Taking multiple readings and calculating the mean reduces random error. Examples include reaction time variation when using a stopwatch, or electrical noise in a measuring instrument. Random errors affect precision.
Systematic errors consistently shift all measurements in the same direction — always too high or always too low. They cannot be reduced by averaging. Examples include a miscalibrated balance, a zero error on a voltmeter, or parallax error when reading a scale. Systematic errors affect accuracy. A large signed percent error that is consistently positive or negative suggests a systematic error.
Accuracy describes how close a measurement is to the true or accepted value. Percent error directly measures accuracy.
Precision describes how reproducible or consistent measurements are — how close repeated measurements are to each other, regardless of the true value.
A measurement can be precise but inaccurate (all readings close together but far from the true value — indicating systematic error). It can also be accurate but imprecise (average close to true value but individual readings scattered). The ideal is both high accuracy and high precision.
| Field / Context | Typical Acceptable % Error | Notes |
|---|---|---|
| GCSE Physics/Chemistry (UK) | < 5% | Basic lab equipment |
| A-Level Physics/Chemistry (UK) | < 3% | More precise apparatus |
| AP Physics/Chemistry (US) | < 5% | School laboratory conditions |
| University Physics Lab | < 1-2% | Precision instruments |
| Clinical Chemistry | < 1% | Medical testing standards |
| Metrology / Standards Lab | < 0.01% | National measurement standards |
| Nuclear Density (Rutherford) | < 10% | Theoretical model uncertainties |
In A-Level Physics (all major UK exam boards), students are expected to calculate percentage uncertainty and percentage error in required practicals. The A-Level Physics practical endorsement requires evidence of dealing with measurement uncertainty. Percentage difference and percentage error are common question types in Paper 3 (practical) sections of AQA, OCR, and Edexcel A-Level Physics.
In A-Level Chemistry, percentage error is particularly relevant for volumetric analysis (titrations), where the accuracy of burette readings (±0.05 cm³) must be related to the total volume used. A 25 cm³ titre has a percentage uncertainty from burette readings of 2 × 0.05/25 × 100% = 0.4%.
AP Physics 1, 2, and C all include laboratory investigations where percent error calculation is expected. The College Board's AP Science practices include "Representing data and phenomena" and "Analysis and evaluation of evidence" — percent error sits within these practices. AP Chemistry labs similarly require calculation and interpretation of percent error to evaluate experimental techniques and identify sources of systematic error.
% Error = |Experimental Value − Theoretical Value| / |Theoretical Value| × 100%. Take the absolute difference, divide by the absolute theoretical value, and multiply by 100.
Using the standard absolute value formula, percent error is always positive. However, signed percent error (without absolute values) can be negative, indicating the experimental value is below the theoretical value. Some teachers and textbooks use signed error to reveal the direction of systematic bias.
For GCSE and A-Level science in the UK, less than 5% is generally acceptable, and less than 3% is good. For AP Science in the US, less than 5% is typically acceptable. The acceptable threshold depends on the experiment type and available equipment.
Absolute error = |Experimental − Theoretical|, expressed in the original units of measurement. Percent error = absolute error / |theoretical| × 100%, expressed as a percentage. Percent error is more useful for comparing accuracy across measurements with different scales.
The standard accepted value of gravitational acceleration used in UK A-Level Physics is g = 9.81 m/s². This is the value given in AQA, OCR, and Edexcel data booklets. The actual value varies slightly by location: it is ~9.832 m/s² at the poles and ~9.780 m/s² at the equator.
Random errors cause unpredictable scatter around the true value — reducing precision. They can be reduced by taking more readings. Systematic errors consistently shift all readings in the same direction — reducing accuracy. They cannot be fixed by averaging; the source of the systematic bias must be identified and corrected.
Use the same formula: % Error = |Measured − Accepted| / |Accepted| × 100%. For a density experiment measuring copper at 8.78 g/cm³ vs accepted 8.96 g/cm³: % Error = |8.78−8.96|/8.96 × 100% = 0.18/8.96 × 100% ≈ 2.0%.
No. Percent error compares an experimental result to a known accepted value. Percentage uncertainty describes the range of doubt in a single measurement due to instrument limitations (e.g., a ruler reading of 12.5 ± 0.5 mm has a percentage uncertainty of 4%). Both use similar mathematics but serve different purposes.
Results are for educational purposes only. Percent error calculations depend on the accepted theoretical value used. Always use your course's specified accepted values for exam work.