Percent Error Calculator

How to Use This Calculator

1. Enter your experimental (measured) value and the theoretical (accepted) value.
2. Read the percent error, absolute error, and relative error instantly.
3. Use Signed (Bias) tab to see whether you overestimated or underestimated.
4. Use Multiple Measurements tab to enter up to 5 trial values and get the mean percent error.
5. Professional tier adds uncertainty propagation — enter your instrument uncertainty (±) to see the error bounds.

Formula

Example

Frequently Asked Questions

• What's the difference between percent error and percent difference? ▼
  Percent error and percent difference solve different problems and should never be confused. Percent error compares an experimental measurement to a known, accepted theoretical value: percent error = |experimental − theoretical| / |theoretical| × 100%. There is a clear reference point — the theoretical value is treated as correct. Use percent error when one value is a standard (e.g., g = 9.81 m/s², the speed of light, a known density). Percent difference, by contrast, compares two measurements where neither is more correct than the other — both are experimental, measured, or estimated values. Its formula is symmetric: percent difference = |a − b| / ((a + b) / 2) × 100%, using the average as the reference denominator. If you are comparing two students' experimental results, use percent difference. If you are comparing one student's result to the textbook value, use percent error. A third concept, percent change, is directional (A to B), used when comparing before/after values like stock prices or population growth. Mixing these three formulas is one of the most common errors in introductory science courses.
• When is percent error positive vs negative? ▼
  The standard percent error formula takes the absolute value of the difference, so the result is always non-negative: percent error = |experimental − theoretical| / |theoretical| × 100%. This means a 5% error is a 5% error regardless of whether the measurement was too high or too low. However, signed percent error — (experimental − theoretical) / |theoretical| × 100% — preserves the direction. A positive signed error means your measurement overestimated the true value (experimental > theoretical); a negative signed error means you underestimated (experimental < theoretical). Signed error is useful for identifying systematic bias in your measurement method. If you consistently get positive signed errors across many trials, your instrument may be reading high due to a calibration offset, or there may be a systematic source of error such as air resistance increasing measured time, a scale that reads high, or a thermometer with a positive offset. Recognizing the sign of systematic error helps correct it. The unsigned (absolute) percent error tells you the magnitude of discrepancy; the signed version tells you the direction of your systematic bias.
• What is an acceptable percent error in scientific experiments? ▼
  Acceptable percent error depends entirely on the field, the precision of the instruments, and the purpose of the experiment. In introductory physics lab courses, a percent error of 1-5% is generally considered good, reflecting the limitations of typical lab equipment like meter sticks, spring scales, and stopwatches. In analytical chemistry, errors below 1% are expected with modern balances and volumetric glassware, and professional labs may require under 0.1%. In engineering manufacturing, tolerances vary enormously: machined metal parts may need under 0.01% precision while rough construction measurements may accept 5-10%. In economics and social sciences, percent errors of 10-20% in predictions may be considered reasonable given the complexity of the systems. In astronomy, measuring distances to nearby stars with parallax achieves about 0.1-1% error, while distances to distant galaxies carry uncertainties of 5-15%. The key principle is that acceptable error must be less than the error that would change a decision or conclusion. If 5% error changes your engineering safety factor from acceptable to unsafe, 5% is not acceptable regardless of what is typical in the field.
• How does percent error relate to precision and accuracy? ▼
  Accuracy and precision are related but distinct concepts, and percent error primarily measures accuracy. Accuracy describes how close a measurement is to the true or accepted value — a low percent error means high accuracy. Precision describes how repeatable or consistent a set of measurements is, regardless of whether they are close to the true value. You can be precise without being accurate (consistently getting 9.50 m/s² when g = 9.81 m/s² — low spread but high percent error) or accurate without being precise (measurements scattered around the true value — high variation but correct on average). The ideal is both high accuracy and high precision. Percent error alone does not tell you about precision — for that you need standard deviation or coefficient of variation across multiple trials. When reporting scientific measurements, give both: the mean (representing accuracy via comparison to accepted value) and the standard deviation (representing precision). A measurement reported as 9.79 ± 0.08 m/s² tells you the experimental mean (percent error ≈ 0.2% vs 9.81) and the precision (sigma = 0.08 m/s², about 0.8% relative uncertainty).
• How do I calculate percent error when the theoretical value is zero? ▼
  Standard percent error is undefined when the theoretical (accepted) value is zero because you would be dividing by zero: |experimental − 0| / |0| × 100% is undefined. This situation arises more often than expected — in physics measuring a residual signal expected to be zero (noise floor), in chemistry testing for zero concentration of a contaminant, or in engineering checking that displacement returns to exactly zero. Several approaches handle this case. First, use absolute error instead: simply report |experimental − theoretical| = |experimental| with appropriate units. Second, if you have a meaningful scale for the measurement, normalize by that scale rather than the theoretical value — this is called normalized or scale-relative error. Third, use percent deviation relative to the measurement range or instrument full scale, which is standard in metrology for zero-point calibration. The JCGM Guide to the Expression of Uncertainty in Measurement recommends expressing uncertainty as an absolute quantity rather than a percentage when the reference value is zero. Reporting that your measurement gave 0.003 mg/L when the theoretical is 0 mg/L communicates more clearly than an undefined percent error.

Related Calculators