Percent Error Calculator with Formula and Steps
Compare an observed, measured or experimental value with an accepted reference. See signed error, absolute error, absolute relative error, exact percent error and a clear calculation trace.
Last Updated: July 26, 2026
Online Percentage Error Calculator
Enter the observed value first and the accepted or theoretical value second. Whole numbers, decimals, fractions, mixed numbers, negative values and scientific notation are supported.
Your error percentage, direction and exact calculation will appear here.
Calculation steps
- Enter an observed value and a nonzero accepted value, then calculate.
Recent calculations
- Your last six results will appear here.
How to Use This Percent Error Calculator
- Enter the observed value. This is the result you measured, estimated or obtained experimentally.
- Enter the accepted value. This is the theoretical, expected, certified or reference result used as the baseline.
- Choose the decimal places for the rounded display. The exact reduced fraction stays available.
- Select Calculate Percent Error. Review the signed error, absolute error, absolute relative error, percentage and steps.
- Keep both inputs in the same unit. Editing either input clears the old result and blocks stale copying.
Input order matters. The accepted value forms the denominator, so switching the two entries usually changes the answer. If neither number is an accepted baseline, use a percentage difference method instead.
Percent Error Formula
Percent error expresses the size of an observation’s error relative to the magnitude of the accepted value. The absolute value signs make the reported percent error nonnegative.
- Signed error = Observed − Accepted
- Absolute error = |Observed − Accepted|
- Absolute relative error = Absolute error ÷ |Accepted|
- Percent error = Absolute relative error × 100
The accepted value must be nonzero. A zero reference makes the standard relative-error formula undefined because division by zero has no finite result.
Worked Example: Observed 98, Accepted 100
Suppose a test gives an observed value of 98 and the accepted value is 100. First subtract the accepted value from the observation. The signed error is −2, so the observation is below the reference. Its absolute error is 2.
The percent error is 2%. Reporting the absolute percentage alone removes direction, so this calculator also shows whether the observation is numerically above, below or equal to the accepted value.
Observed Value Versus Accepted Value
The observed value comes from a measurement, experiment, estimate, forecast or calculation. The accepted value is the benchmark. Depending on the task, it might be a theoretical constant, certified calibration value, published reference, specification target or agreed expected value.
The accepted value is not automatically perfect. It may have uncertainty, rounding or a limited scope. Record the source, unit, conditions and precision of the benchmark when the result supports laboratory, engineering or quality work.
| Observed | Accepted | Absolute error | Percent error | Direction |
|---|---|---|---|---|
| 98 | 100 | 2 | 2% | Below accepted |
| 102 | 100 | 2 | 2% | Above accepted |
| 9.5 | 10 | 0.5 | 5% | Below accepted |
| 0 | 25 | 25 | 100% | Below accepted |
| 125 | 100 | 25 | 25% | Above accepted |
Percent Error Versus Percentage Difference
Percent error is directional in setup even though its main answer is absolute. One input is the observation and the other is the accepted baseline. Percentage difference treats both inputs as peers and divides their absolute gap by their average magnitude.
| Method | Denominator | Input roles | Best use |
|---|---|---|---|
| Percent error | Magnitude of accepted value | Observed and accepted | Judge an observation against a reference |
| Percentage difference | Average magnitude of both values | Two peer values | Compare values with equal standing |
| Percentage change | Original value, normally positive | Original and new | Measure change over time or state |
| Absolute error | No denominator | Observed and accepted | Report the raw gap in the original unit |
For observed 80 and accepted 100, percent error is 20%. Treating 80 and 100 as peers gives a percentage difference of about 22.222222%. The formulas answer different questions.
Zero and Negative Values
Accepted value is zero
The standard percent-error formula is undefined because its denominator is zero. Use absolute error, a domain-specific normalization value or another approved method. Do not silently replace zero with a small number.
Observed value is zero
If the accepted value is nonzero, the absolute error equals the magnitude of the accepted value. The percent error is therefore 100%.
Negative accepted value
The calculator divides by the magnitude of the accepted value. For observed −90 and accepted −100, the absolute error is 10 and percent error is 10%. Confirm that absolute relative error is meaningful for the scale involved. Celsius and Fahrenheit values, for example, have arbitrary zero points.
Values with opposite signs
The calculation remains defined when the accepted value is nonzero, but the error may exceed 100%. A sign reversal often carries practical meaning that an absolute percentage cannot communicate alone.
Can Percent Error Be More Than 100%?
Yes. Percent error exceeds 100% when the absolute error is larger than the magnitude of the accepted value. If the observed value is 250 and the accepted value is 100, the absolute error is 150 and the percent error is 150%.
Unlike the average-magnitude percentage-difference formula, percent error has no general 200% upper limit. An observation far from a small nonzero reference can produce any large percentage. Check input order, units and the suitability of the reference before interpreting such a result.
Rounding, Significant Figures and Exact Results
A repeating decimal may need rounding. Observed 1/3 and accepted 1/2 produce an exact percent error of 100/3%, or approximately 33.333333% at six places. This tool keeps the exact rational result and marks a rounded decimal with an approximation sign.
Keep full precision during intermediate steps. Round once at the reporting stage according to your course, laboratory method or organization’s rule. Decimal places and significant figures are different. A result of 0.004872 may have six decimal places but only four significant figures. The selector rounds by decimal places; it does not round by significant figures.
Input Rules and Calculation Method
You may enter signed integers, decimals, simple fractions, mixed numbers and scientific notation. Valid examples include 98, −12.5, 3/8, 1 1/2 and 2.5e3. Do not enter commas, currency symbols, unit labels or percent signs. Fraction denominators must be nonzero.
Each numeric part may contain up to 50 digits. Scientific exponents must stay from −100 through 100. The calculator converts entries to reduced rational numbers backed by integers, then performs subtraction, absolute value, division and multiplication without binary floating-point drift.
Calculations stay in the current browser page. Recent history clears on reload. The result panel appears below the action buttons on desktop and mobile screens.
Practical Uses of Percentage Error
Laboratory measurements
Compare an experimental result with an accepted constant or reference sample. Keep units and test conditions consistent, and report uncertainty when the method requires it.
Calibration checks
Compare an instrument reading with a certified standard. Percent error summarizes relative deviation, but pass or fail still depends on the stated tolerance and calibration procedure.
Forecast and estimate review
Compare a forecast with the final recorded outcome. For repeated forecasts, one percent-error result is not enough. Methods such as mean absolute percentage error need rules for zero actual values and multiple observations.
Manufacturing and quality control
Compare a measured dimension or output with its nominal target. Use the drawing, specification or control plan to decide whether the deviation is acceptable.
Percent Error Is Not the Same as Tolerance or Uncertainty
Percent error measures distance from a selected reference. Tolerance states an allowed range. Measurement uncertainty describes doubt around a measured result. Statistical significance evaluates evidence under a statistical model. These ideas are related in some workflows, but one does not replace another.
A low percent error does not prove that a method is unbiased, repeatable or correctly calibrated. A high result does not identify the cause. Review sampling, instrument resolution, environmental conditions, transcription, unit conversion and the quality of the accepted value.
Common Percent Error Mistakes
- Swapping the observed and accepted values.
- Dividing by the observation instead of the accepted-value magnitude.
- Forgetting absolute value and reporting a negative main percent error.
- Using zero as the accepted value without a defined alternative method.
- Mixing units, scales, time periods or test conditions.
- Using percentage difference when one value is a clear benchmark.
- Rounding the inputs or intermediate error before the final step.
- Treating percent error as an automatic pass-or-fail decision.
- Ignoring reference uncertainty or significant-figure rules.
- Using an absolute percentage without also noting a meaningful sign reversal.
Related Calculators
Frequently Asked Questions
What is the formula for percent error?
Subtract the accepted value from the observed value, take the absolute result, divide by the magnitude of the accepted value, then multiply by 100.
What is the percent error for observed 98 and accepted 100?
The absolute error is 2. Divide 2 by 100 and multiply by 100, which gives a percent error of 2%.
Why does the accepted value go in the denominator?
The accepted value is the benchmark used to scale the error. Dividing by its magnitude expresses the gap relative to the reference.
Does input order matter in a percent error calculation?
Yes. Enter the measured or observed result first and the accepted reference second. Swapping them changes the denominator and usually changes the percentage.
Can percent error be negative?
The standard absolute percent error is nonnegative. The signed error may be negative when the observation is below the accepted value.
Can percent error be greater than 100%?
Yes. It exceeds 100% when the absolute error is larger than the magnitude of the accepted value.
What happens when the accepted value is zero?
The standard percent-error formula is undefined because it would divide by zero. Use absolute error or a domain-approved alternative.
What happens when the observed value is zero?
If the accepted value is nonzero, the absolute error equals the accepted-value magnitude and the percent error is 100%.
Can I calculate percent error with negative values?
Yes, when the comparison is meaningful. This calculator divides by the magnitude of the accepted value so the main percentage remains nonnegative.
Is percent error the same as percentage difference?
No. Percent error uses an accepted value as the denominator. Percentage difference treats both values as peers and uses their average magnitude.
How many decimal places should I use?
Follow the reporting rule for your course, method or organization. Keep full precision during the calculation and round only the final result.
Does a low percent error prove a measurement is accurate?
No. It shows closeness to one selected reference. It does not prove repeatability, remove uncertainty or confirm that the reference and method are suitable.
Disclaimer: This calculator provides a mathematical comparison for education and general planning. Verify the accepted value, units, method, uncertainty, tolerance and rounding rules before using the result for scientific, engineering, medical, financial or other important work.