The percent error calculator above answers a specific question: how far is a measurement from the value it should have been? It takes your measured or experimental result, compares it against the accepted, theoretical or published value, and returns the error as a percentage, alongside the absolute error, the relative error and the direction of the discrepancy. The substituted formula is printed with your own numbers so you can copy the working straight into a lab report.
It is worth being precise about what this page is not, because three similar-sounding tools get confused constantly. Percent error compares a measurement to a known correct value. If you want a fraction of a number — "what is 15% of 240" — that is our percentage calculator. If you want the change between two values over time, such as revenue rising from one month to the next, that is our percentage change calculator. Percent error is the one where one of the two numbers is authoritative and the other is being judged against it. Arb Digital keeps them as separate tools because mixing them up produces answers that look reasonable and mean nothing.
What This Percent Error Calculator Does
Enter the value you measured and the value you should have got. The headline figure is percent error, reported by default as a positive number because the standard convention in science education is to take the absolute value — the size of the discrepancy matters more than its direction for judging experimental quality. Switch to signed mode when your teacher, standard or protocol asks for direction to be preserved, in which case a negative result means your measurement fell below the accepted value.
The grid breaks the calculation into its parts. Absolute error is the raw difference in the original units, which is what you would quote alongside an instrument's precision. Relative error is the same difference expressed as a decimal fraction of the accepted value; percent error is simply relative error multiplied by 100. Accuracy is reported as 100% minus the percent error, which is a common way of summarising how close a result landed. The direction cell states plainly whether you measured high or low.
The calculator also handles the case that breaks the formula: an accepted value of zero. Percent error is undefined when the accepted value is zero, because the division has no meaning, and the tool says so rather than returning an infinity or a misleading large number.
How to Use It
- Enter your measured value. The reading from your experiment, instrument, model or estimate. Use the same units for both fields.
- Enter the accepted value. The reference figure from a textbook, a standards body, a theoretical calculation or a calibrated instrument.
- Choose absolute or signed. Absolute is standard for school and undergraduate lab work; signed is useful when you are diagnosing a systematic bias.
- Set the decimal places to match the precision of your source data. Reporting six decimals from a two-figure measurement overstates what you know.
- Copy the working line into your write-up. It shows the substituted formula rather than just the result.
The Formula and How It's Calculated
The standard formula is percent error = |measured − accepted| ÷ |accepted| × 100. Work the numerator first to get the absolute error, divide by the accepted value to get the relative error, then multiply by 100 to express it as a percentage.
Using the default values, a pendulum experiment measures the acceleration due to gravity as 9.72 m/s² against an accepted value of 9.81 m/s² — the standard acceleration of gravity is defined by NIST as exactly 9.80665 m/s², usually rounded to 9.81 in school work. The absolute error is |9.72 − 9.81| = 0.09. The relative error is 0.09 ÷ 9.81 = 0.009174. The percent error is 0.009174 × 100 = 0.92%. That is a good result for a school pendulum experiment, and the direction tells you the measurement came in low. In signed form the answer would be −0.92%.
The denominator is always the accepted value, never the measured one. This matters more than it appears: if you divide 0.09 by 9.72 instead you get 0.93%, which is close here only because the two values are close. When a measurement is badly off — say 5 against an accepted 10 — dividing by the accepted value gives 50% while dividing by the measurement gives 100%. Only the first is percent error. The reasoning behind using the reference value as the denominator, and the wider vocabulary of measurement uncertainty, is set out in the NIST guidance on evaluating measurement uncertainty.
Percent Error Is Not Percentage Difference
There is a fourth calculation people confuse with this one, and it deserves its own paragraph. Percentage difference compares two values where neither is authoritative — two independent measurements of the same thing, or two instruments checked against each other. It divides by the average of the two values rather than by one of them, precisely because there is no reason to prefer either as the denominator.
Use percent error when one number is right by definition: a published constant, a certified reference material, a theoretical prediction, a legally defined standard. Use percentage difference when you have two peers. Reporting percent error against a value that is itself uncertain quietly claims an authority that the reference does not have, which is the kind of thing that gets flagged in a methods review.
What Counts as a Good Percent Error
There is no universal threshold, and any page that gives you one without qualification is guessing. Acceptability depends entirely on what is being measured and with what. A school experiment measuring gravity with a stopwatch and a piece of string might reasonably land within 5%. A calibrated analytical balance in a pharmaceutical lab would be a serious problem at 5% and is expected to be far better than 0.1%. An opinion poll with a stated margin of error of three points cannot meaningfully claim sub-1% accuracy no matter what arithmetic you perform on it.
The right way to judge a percent error is against the precision of your instrument and the tolerance of your task. If your ruler resolves to a millimetre and your object is 90 mm long, your measurement cannot be better than roughly 1% no matter how careful you are, so a 0.92% error is at the limit of what the equipment can even detect. Reporting a smaller error than your instrument can resolve is a sign of false precision, which our significant figures calculator can help you avoid by keeping your reported digits honest.
Random Error, Systematic Error, and Why the Sign Matters
A single percent error tells you the size of one discrepancy. It does not tell you where the discrepancy came from, and that distinction is the useful part of an error analysis. Random error scatters results either side of the true value: repeat the experiment and you get 9.72, then 9.88, then 9.79. Averaging repeated trials reduces random error, which is why lab protocols insist on multiple runs.
Systematic error pushes every result the same way. A stopwatch started late, a balance that was never tared, a ruler with a worn end — these produce measurements that are consistently low or consistently high, and averaging more trials does not help at all. The way to detect one is to run the calculation in signed mode across several trials. If the sign is random, you are seeing noise. If every trial comes back negative, you have a bias to hunt down. Our standard deviation calculator quantifies the scatter, and our mean, median and mode calculator gives you the central value to compare against the accepted figure.
When the Accepted Value Is Zero or Very Small
Percent error breaks down completely when the accepted value is zero, because dividing by zero is undefined. It also becomes unstable when the accepted value is merely small. If the true value is 0.01 and you measure 0.02, the absolute error is a tiny 0.01 but the percent error is a dramatic 100%. Neither number is wrong; they answer different questions, and quoting only the percentage makes a trivial discrepancy sound catastrophic.
In these situations the honest approach is to report the absolute error in its original units, optionally alongside the instrument's stated resolution, and to say why the percentage was not used. Fields that regularly measure near zero — trace contaminant analysis, temperature differences around a freezing point, net figures that can cross from positive to negative — generally standardise on absolute error or on error relative to a full-scale reading rather than to the measured quantity itself.
Arb Digital maintains a growing library of free maths, statistics, finance and conversion tools. Browse the collection, or tell us which calculator you keep wishing existed.
Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Dividing by the measured value. The denominator is always the accepted value. Swapping them produces a different number that is not percent error.
- Confusing it with percentage change. Change compares a value to its own earlier state; error compares a measurement to a known truth. Different questions, different tools.
- Using percent error when neither value is authoritative. Two independent measurements call for percentage difference, which divides by their average instead.
- Reporting more decimal places than the instrument supports. A percent error quoted to four decimals from a two-figure measurement is false precision.
- Treating a small percent error as proof of a good experiment. A close result from a badly designed method can be luck, or two errors cancelling out.
Related Free Tools From Arb Digital
For a plain fraction of a number use the percentage calculator, and for growth or decline between two figures use the percentage change calculator — both answer different questions from this page. To analyse a set of repeated trials, use the standard deviation calculator and the mean, median and mode calculator. Keep your reported precision honest with the significant figures calculator, and see the full free online tools hub for everything else.
Frequently Asked Questions
Percent error equals the absolute difference between the measured and accepted values, divided by the accepted value, multiplied by 100. Measuring 9.72 against an accepted 9.81 gives 0.09 divided by 9.81, which is 0.009174, or 0.92 percent.
No. Percent error compares a measurement against a value known to be correct, so the accepted value is always the denominator. Percentage change compares a value to its own earlier state and uses the starting value as the denominator.
Only if you deliberately keep the sign. The standard convention takes the absolute value so the result is always positive, because the size of the discrepancy is what matters. Keeping the sign is useful when you are looking for a systematic bias across repeated trials.
It depends entirely on the measurement and the instrument. A school experiment using a stopwatch might reasonably land within five percent, while an analytical balance in a laboratory would be expected to perform far better than a tenth of a percent. Judge the result against your equipment's precision.
Because the formula divides by the accepted value, and division by zero has no defined result. When the true value is zero or very close to it, report the absolute error in its original units instead of a percentage.
Absolute error is the raw difference between the two values in the original units. Relative error divides that difference by the accepted value, giving a unitless fraction. Multiplying relative error by 100 produces percent error.
Accuracy is usually quoted as 100 percent minus the percent error, so a 0.92 percent error corresponds to about 99.08 percent accuracy. Accuracy describes closeness to the true value, while precision describes how tightly repeated measurements cluster together.
