Percentage Error Calculator
Percentage error measures how far an experimental or estimated value falls from a value you accept as correct. Unlike percentage difference, it has a privileged denominator: the true value.
Formula
- Measured
- = the experimental, observed or estimated value
- True
- = the accepted, theoretical or reference value
- | |
- = absolute value, used when only the magnitude of the error matters
The true value is always the denominator. That is the property that distinguishes percentage error from percentage difference, which has no privileged value.
How to use this calculator
- Enter the value your measurement or estimate produced.
- Enter the accepted, published or theoretical value you are comparing it against.
- Tick "show the sign" if you need to know whether you over- or under-estimated; leave it clear for the conventional unsigned figure.
- Press Calculate to see the percentage error, the absolute error and the implied accuracy.
Worked example
A pendulum experiment measuring g as 9.65 m/s²
A student measures gravitational acceleration as 9.65 m/s². The accepted standard value is 9.81 m/s². The true value goes in the denominator.
- 9.65 − 9.81 = −0.16
- −0.16 ÷ 9.81 = −0.016310
- × 100 = −1.63%
Answer: 1.63% error (an underestimate). Under 2% is a good result for a school pendulum experiment.
Error, difference and uncertainty are three different things
Percentage error compares your value against one you accept as correct. It requires a reference standard and it measures accuracy — how close you are to the truth.
Percentage difference compares two values where neither is authoritative. It measures agreement, not accuracy, and is the right tool for duplicate measurements.
Uncertainty is a property of your measurement process, estimated from the instrument and the method before you know the answer. A result can have small error and large uncertainty (you got lucky) or large error and small uncertainty (something systematic is wrong).
What counts as acceptable
There is no universal threshold — it depends entirely on the field and the measurement. School physics experiments generally accept 5% and treat under 2% as good. Analytical chemistry often demands better than 1%, and gravimetric methods better than 0.1%. Engineering estimates in early design may tolerate 20% or more, because the purpose is to size a problem rather than to specify a part.
The number to compare against is the tolerance stated in your protocol or specification, not a general rule of thumb. If no tolerance is stated, the percentage error on its own tells you little.
| Context | Typical acceptable error |
|---|---|
| School physics practical | Under 5% |
| Undergraduate chemistry titration | Under 1% |
| Analytical balance work | Under 0.1% |
| Early-stage engineering estimate | 10–25% |
| Construction quantity take-off | 5–10% |
Signed error and systematic bias
Convention reports percentage error unsigned, because for a single measurement the magnitude is what matters. But when you have several measurements, the sign is diagnostic. Random error scatters both ways; systematic error pushes consistently in one direction.
If five repeats all come out low, the cause is unlikely to be random — a mis-calibrated instrument, an unaccounted friction, a consistently early stopwatch. That is why this calculator offers the signed figure: a run of same-sign errors is the clearest evidence of bias you will get without a full statistical analysis.
The zero-true-value problem
When the accepted value is zero, percentage error is undefined — you would be dividing by zero, and any deviation at all is infinitely large relative to nothing. This calculator refuses rather than returning infinity.
In those cases report the absolute error in its original units, or normalise against something else meaningful: the full-scale range of the instrument, the mean of the data set, or a specified tolerance band. "Zero ± 0.02 V against a 10 V range" communicates far more than an undefined percentage.
Important considerations
- The true value must go in the denominator; reversing it changes the answer.
- Percentage error is undefined when the accepted value is zero.
- A low percentage error does not prove a good method — it can happen by chance with a large uncertainty.
- Report the number of significant figures your instrument justifies rather than every digit the calculator prints.
- Consistent same-sign errors across repeats indicate systematic bias rather than random scatter.
Common mistakes to avoid
- Dividing by the measured value. The accepted value is the reference and belongs in the denominator.
- Using percentage error when there is no true value. Use percentage difference for two equally uncertain measurements.
- Reporting more precision than the measurement supports. An error of 1.6% should not be quoted as 1.6310%.
- Treating percentage error as uncertainty. They are different concepts and can disagree completely.
- Ignoring the sign across repeated trials. That is where evidence of systematic bias hides.
Frequently asked questions
What is the percentage error formula?
Percentage error = (|measured − true| ÷ |true|) × 100. The accepted or theoretical value is always the denominator, which is what makes it a measure of accuracy rather than of agreement.
Should percentage error be positive or negative?
Convention reports it as a positive magnitude. The signed version is useful when you have several measurements, because errors that all share a sign point to systematic bias rather than random scatter.
What is a good percentage error?
It depends on the field. School experiments usually accept under 5%; analytical chemistry often requires under 1%; early engineering estimates may tolerate 20%. Compare against the tolerance in your protocol, not a general rule.
What is the difference between percentage error and percentage difference?
Percentage error divides by a known true value and measures accuracy. Percentage difference divides by the average of two values and measures agreement between two measurements where neither is authoritative.
Can I calculate percentage error if the true value is zero?
No — division by zero makes it undefined. Report the absolute error in its units instead, or normalise against the instrument's full-scale range or a stated tolerance.
Is percentage error the same as accuracy?
They are complementary. Accuracy is often quoted as 100% minus the percentage error, so a 1.63% error corresponds to 98.37% accuracy. Both describe the same single comparison against the true value.
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