Percent Error Calculator

The accepted value is the denominator, and that is the whole definition

Percent error exists because an absolute error on its own says very little. Being 1 cm out is nothing when measuring a room and a disaster when measuring a bearing, and the only way to tell those apart is to divide by the size of the thing being measured. Which size, though, is not a free choice: it is the accepted value, the one that is not in doubt. Your measurement is the quantity under suspicion, and dividing by a suspect number would make the score depend on the error it is supposed to be reporting.

The difference is small on a good measurement and grows quickly on a bad one. A reading of 105 against an accepted 100 is 5% out; divide the same 5 by the reading instead and you get 4.762%, which flatters the measurement for no reason other than that it was too high. On a reading of 200 against 100 the two calculations give 100% and 50% — the same experiment scored twice as well by the wrong divisor.

No authority is cited for the arithmetic because none exists to cite: this is elementary algebra, written out in full above rather than asserted. What is not arithmetic — how many figures to report, and what tolerance is acceptable — is covered further down, and the answer there is that it belongs to the standard you are working to, not to the calculator.

Four steps, and one habit worth keeping

The calculator shows this working for whatever numbers you enter; here it is once in full for a free-fall experiment reading 9.80 m/s² against the accepted 9.81.

The habit is this: note the sign before you drop it. Percent error is defined as a magnitude, so the absolute value in step two is correct and standard. But “0.102% low” and “0.102% high” are different findings about an apparatus, and a column of readings that are all low points at something systematic — a mis-zeroed balance, a slow timer, a thermometer reading below ambient — that a column of magnitudes hides completely. The calculator keeps the signed figure beside the magnitude for exactly this reason.

Every row in the reference table below is computed by the same engine as the calculator above:

Report the figures your measurement earned, not the ones the screen offers

The unrounded percent error for the free-fall example is 0.101937%. Reporting it that way claims the experiment resolved the acceleration to eight figures, which a stopwatch and a metre rule did not. The reading carried three significant figures, so the error is reported to three: 0.102%. That is why the significant-figure control on the calculator is an input beside the measurements rather than a display setting hidden away from them.

Two rules cover almost every case. Leading zeros are placeholders and never significant, so 0.00450 carries three figures, not five. Trailing zeros after a decimal point are significant, which is the entire difference between writing 9.8 and 9.80 — the second claims to have resolved the hundredths place and found nothing there. A trailing zero with no decimal point, as in “100”, is ambiguous by convention and is treated here as not significant; if you mean three figures, write 1.00 × 10².

Rounding itself is done half-away-from-zero on the digit that matters, which is the rule taught alongside significant figures: 0.101937% to three figures is 0.102%, and 0.125 to two is 0.13 rather than the 0.12 that a naive round-half-up on a binary float can produce. The calculator keeps the unrounded value underneath the rounded one so you can see what was discarded.

Percent error, percentage difference, and percentage change are three different questions

All three subtract two numbers and divide. They differ only in what they divide by, and that difference decides which one your situation calls for:

• Percent error divides by the accepted value. Use it when one of the two numbers is known to be right and the other is being judged against it — a measurement against a published constant, an estimate against an outcome, a forecast against what happened.
• Percentage difference divides by the mean of the two. Use it when neither value is a reference and the comparison should read the same whichever one you write first — two instruments, two suppliers’ quotes, two lab duplicates. That is the Percentage Difference Calculator.
• Percentage change divides by the earlier value. Use it when the two numbers are the same thing at two points in time. That is the Percentage Change Calculator.

The same pair of numbers gives three different answers, which is why picking the wrong one is not a rounding issue. Take 20 and 30: as a percent error of a measured 30 against an accepted 20 it is 50%; as a percentage difference between two peers it is 40%; as a change from 20 to 30 it is 50%, and as a change from 30 to 20 it is −33.33%. Only the second is symmetric, and only the first assumes one of the numbers is beyond question.

A quick test: if swapping the two numbers over ought to change the answer, you want percent error or percentage change. If it ought not to, you want percentage difference.

A small percent error is not the same as a good experiment

Percent error measures accuracy — how close you landed to the truth. It says nothing at all about precision, which is how close your repeated attempts landed to each other, and the two come apart in both directions.

Take three readings of 11 against an accepted value of 10. The spread is exactly zero, so the method is perfectly repeatable, and yet every reading is 10% out. That is the signature of a systematic error — something in the apparatus is consistently wrong, and repeating the measurement will never reveal it because it repeats too. Now take 8, 12 and 10 against the same accepted 10: the mean is exactly right and the percent error of the mean is 0%, but the readings scatter by a standard deviation of 2 and no single one of them should be trusted.

This is why the trials mode reports two error figures rather than one. For four readings of 9.78, 9.82, 9.79 and 9.83 against 9.81, the percent error of the mean is 0.051% — random errors have partly cancelled — while the average of the four individual errors is 0.204%, four times larger, because taking each magnitude first stops them cancelling. Both are correct and they answer different questions: the first is how good the experiment was, the second is how good a single reading from it tends to be.

The spread itself belongs to the Standard Deviation Calculator, which describes a dataset in full with no accepted value involved. This page reports the sample standard deviation beside the errors so the two properties can be read together, and stops there.

What counts as a good percent error, honestly

There is no universal threshold, and any page that gives you one has invented it. Whether a percent error is acceptable is a property of the method you are following, not of the number: an analytical balance weighing a standard, a school pendulum experiment timing ten swings by hand, and a field estimate of a crowd have tolerances that differ by orders of magnitude, and all three can be done correctly.

What can be said generally is narrower and more useful. The tolerance comes from the least precise step in the measurement, not from the final arithmetic — if the timer resolves to a tenth of a second over a two-second swing, no amount of care elsewhere gets the result below a few percent. A percent error far smaller than the instrument can resolve is a warning sign rather than a triumph, and usually means the accepted value leaked into the working. And a set of errors that are all in the same direction is a different problem from a set that scatter around zero: the first is systematic and can often be found and removed, the second is random and only shrinks by repeating.

Where a number carries a grade, a certification, or a safety decision, the tolerance is written in the standard or the practical brief you were given. Read it there.

In a spreadsheet, it is ABS and a division

With the measurement in A1 and the accepted value in B1, each figure on this page is one short formula. The trap in all of them is multiplying by 100 in a cell already formatted as a percentage, which inflates the answer a hundredfold:

• Percent error: =ABS(A1-B1)/ABS(B1), with the cell formatted as Percentage. For the raw number instead, append *100 and leave the cell as a number.
• Signed percent error, keeping the high/low direction: =(A1-B1)/ABS(B1). Dropping the outer ABS is the whole change.
• To significant figures: =ROUND(x,3-1-INT(LOG10(ABS(x)))) rounds x to three significant figures. Excel’s ROUND counts decimal places, not figures, which is why the exponent has to be worked out first.
• MAPE across measured values in A1:A20 and accepted values in B1:B20: =AVERAGE(ABS(A1:A20-B1:B20)/ABS(B1:B20)), entered normally in current Excel and Google Sheets, or with Ctrl+Shift+Enter in older versions where it needs to be an array formula.

There is nothing to download for any of this and no account to create: the calculator above is a free web page that runs entirely in your browser, on desktop or mobile, and the numbers you type are never sent anywhere.

Read the guide

Everything on this page turns on which of the two numbers goes underneath the line, and that is a habit worth building somewhere with consequences attached. Why a Percentage Loss and an Equal Percentage Gain Don’t Cancel Out takes the same mechanism into money: a 25% fall followed by a 25% rise does not return you to where you started, for exactly the reason a percent error divided by the measurement is not a percent error — the base moved underneath the second calculation. It is the clearest worked demonstration of the denominator problem on this site.

Related calculators

The other three ways to compare two numbers:

More in Math, or browse all calculators.

Sources and methodology

The arithmetic cites no authority because none exists to cite — percent error is elementary algebra, written out in full above rather than asserted. What is cited below is the part that is not arithmetic: how measurement uncertainty is treated formally, which is a larger subject than this page covers, and what a spreadsheet does with the formulas in the section above.

• NIST — uncertainty of measurement results
• NIST/SEMATECH e-Handbook of Statistical Methods
• Microsoft Support — the ABS function in Excel
• How we calculate — methodology

Method. Every figure here — the calculator, the step-by-step working, the reference table, and the worked comparisons in the prose — is produced in your browser by one engine, src/lib/percent-error.ts, so nothing on the page can disagree with the tool above it. That engine is checked on every change against 89 hand-written assertions, including the cases most likely to be got wrong: that the accepted value is the denominator and not the measurement, that a negative accepted value still yields a positive magnitude, that an accepted value of zero has no percentage answer at all rather than an infinite one, that significant-figure rounding goes half-away-from-zero, that the error of the mean and the mean of the errors stay separate numbers, and that a pair with a zero accepted value is dropped from a MAPE rather than silently scored as zero. The count and the per-case breakdown are published on the formula verification page.