Skip to main content
CalcMax

Percent Error Calculator

Result

0.6116%

Percent error (%)

Relative error (as a decimal)
0.006116
Absolute error
0.060000

A percent error calculator compares a measured value with an accepted one and reports how far off the measurement is. Three outputs come back from the same pair of numbers: the absolute error, which is the plain difference in the units you typed; the relative error, which is that difference divided by the accepted value and is the form arithmetic wants; and the percent error, which is the same ratio multiplied by a hundred. Enter 9.75 as the measured value and 9.81 as the accepted value and the panel gives 0.06, 0.006116 and 0.6116 percent. The percent error formula is the whole of the page: the difference between the two numbers, over the accepted value, times a hundred. Two details separate a correct percent error from a plausible wrong one. The first is which number goes in the denominator. The accepted value is the reference, so it is the one that divides — the question being asked is how large the error is relative to the truth, not relative to the reading. Swapping them for a measurement of 50 against an accepted value of 100 gives 100 percent instead of 50, a factor-of-two error that arrives with no warning because both numbers look reasonable. The second detail is that all three outputs are reported without a sign. An error is a distance from the reference, so overshooting by five percent and falling short by five percent produce the same reading: a measured value of 1.05 against an accepted value of 1 and a measured value of 0.95 against an accepted value of 1 both give an absolute error of 0.05 and a percent error of 5. Which direction the measurement went is in the numbers you typed, and the page leaves it there rather than inventing a sign convention that half its readers would disagree with. The page accepts negative values on both sides, because temperatures below zero, depths relative to a datum and signed balances are all real measurements, and it reports the error between a measured value of minus 50 and an accepted value of minus 100 as 50 percent — the same relative distance as 50 against 100. What it will not accept is an accepted value of exactly zero. That is not a matter of taste: dividing by the reference is what turns a difference into a relative error, and a reference of zero removes the scale entirely, so the page reports an error rather than returning infinity. A measured value of zero is fine and common — nothing detected when something was expected is a hundred percent error — and the failure is one-sided: because the denominator is the accepted value, the case that breaks is the accepted value being zero and not the measurement being zero.

One reference value, six measurements, and the symmetry of an unsigned error

Measured valueAbsolute errorPercent error (%)
1.050.055
1.020.022
1.010.011
0.990.011
0.980.022
0.950.055

The accepted value is 1 in every row, so the absolute error and the percent error are the same number and the table reads as a set of distances from a single reference. The pair of rows to look at is the second and the fifth: a measurement of 1.02 and a measurement of 0.98 are equally far off in opposite directions, and both report an absolute error of 0.02 and a percent error of 2. That symmetry is what it means for these outputs to be unsigned — a page that carried the sign would print 2 and minus 2 here, and a page that divided by the measured value instead of the reference would print 1.96 and 2.04 for the same two rows, neither of which is 2. A reference of 1 was chosen so the arithmetic is visible at a glance; the relation it shows holds for any nonzero reference.

Formula

Absolute error = |measured − accepted| Relative error = absolute error ÷ |accepted| Percent error = relative error × 100

Measured value
The reading, the result of the experiment, the number you got — whatever is being compared. It may be negative, and it may be zero, which is the reading that says nothing was detected where something was expected
Accepted value
The reference the measurement is judged against: a published constant, a standard, a true value, or a second measurement treated as the truth. It is the denominator of the percent error formula, which is why it may not be zero
Absolute error
Measured minus accepted, in the same units as the inputs, reported without a sign. It says how far off the reading was and nothing about how large that is — 0.06 is small for a mass in kilograms and enormous for a mass in tonnes
Relative error
The absolute error divided by the accepted value, which is the scale-free version of the same fact: 0.06 against 9.81 is 0.006116, about six parts in a thousand, whatever the units were
Percent error
The relative error times a hundred, which is the number most lab reports and most search queries ask for. It is the same quantity as the relative error and not a second measurement of anything

The three forms answer three different questions about one comparison. Percent error is the headline and what most people want: a result of 4.9 against an accepted 5 is 2 percent off, which is a sentence anyone can act on. Put it on a lab report, a calibration sheet, a tolerance note. Relative error is the same number before the conversion, and it is the form that goes into further arithmetic — error propagation, uncertainty budgets and any formula that combines two measurements multiply the relative errors rather than the percentages, so converting to percent and back is a detour. Absolute error is the one that carries the units and the one that gets dropped: 0.06 off a target of 9.81 tells you the measurement missed, but only the absolute error tells you that the miss was 0.06 grams and not 0.06 kilograms. Three cases cover most uses. Checking a single measurement against a standard, where the accepted value is published and the question is whether the reading is close enough. Comparing two instruments, where the same accepted value is used for both and the smaller percent error wins. And comparing a model against reality, where the accepted value is an observation and the error says how far the model drifted — the sign of that drift is in the two inputs even though the outputs do not show it, so read the numbers rather than the panel when the direction matters.

Worked examples

  1. A measurement of 9.75 against an accepted 9.81

    1. Absolute error: |9.75 − 9.81| = 0.06
    2. Relative error: 0.06 ÷ 9.81 = 0.006116
    3. Percent error: 0.006116 × 100 = 0.6116

    The default case, and the one that shows why three outputs are worth printing. The absolute error is 0.06 — the same figure a reading of 9.81 against 9.75 would give — while the percent error is 0.6116, comfortably inside a one percent tolerance. An error that sounds small in absolute terms can be large relative to the quantity, and the reverse is just as common: 0.06 is nothing against a mass of 9.81 and everything against a mass of 0.07.

  2. 4.9 against an accepted 5

    1. Absolute error: |4.9 − 5| = 0.1
    2. Relative error: 0.1 ÷ 5 = 0.02
    3. Percent error: 0.02 × 100 = 2

    A round accepted value and a two percent miss, the shape most tolerance checks take. Note that the arithmetic never needed to know whether the measured value was high or low: 5.1 against 5 gives the same three outputs, because the error is a distance from the reference and the page reports it as one.

  3. 50 against an accepted 100 — the denominator test

    1. Absolute error: |50 − 100| = 50
    2. Relative error: 50 ÷ 100 = 0.5
    3. Percent error: 0.5 × 100 = 50

    This one pins the convention. Dividing by the accepted value gives 50 percent; dividing by the measured value instead would give 100 percent, and both figures appear in the wild because some sources define percentage error against the experimental value. Here the reference divides. If your course or your lab defines it the other way, the absolute error is unaffected and the other two outputs are the reciprocal of what you need.

  4. Two negative numbers: −50 against an accepted −100

    1. Absolute error: |−50 − (−100)| = 50
    2. Relative error: 50 ÷ 100 = 0.5
    3. Percent error: 0.5 × 100 = 50

    Both inputs negative, and the same 50 percent as the positive case — the distance between the two numbers is what the page measures. Subtracting a negative is where a hand calculation goes wrong most often: −50 − (−100) is 50 and not −150.

Limitations

It compares two numbers and nothing else. It does not know which of them is authoritative, so a pair typed in the wrong order produces a wrong denominator and a wrong answer with no complaint — the accepted value is the field labelled as such and the arithmetic trusts the label. It does not know the units, so it cannot tell whether an absolute error of 0.06 is excellent or useless; that judgement needs the quantity, and it is why the percent error is usually the figure worth quoting. Its outputs carry no sign, by design: the page reports the size of the error, not its direction, so it cannot be used to say whether a measurement ran high or low — read the two inputs for that. It uses a single reference value and assumes that reference is exact, which is the assumption most lab-report uses are comfortable with and the one a real uncertainty budget is not: when the accepted value has its own uncertainty, the honest comparison divides by a combined uncertainty rather than by the value itself. It has one hard failure, an accepted value of zero, and one soft edge, a measured value of zero, which is a legitimate hundred percent error and not a bug. Following the convention that the reference divides, the page reports 50 percent for 50 against 100; if your textbook divides by the measured value instead, the two figures differ by a factor of two and the absolute error is the only output that is the same either way. And the outputs are rounded — six decimals for the relative and absolute error, four for the percentage — so a comparison that is off by less than 0.0001 of a percent prints as zero.

Frequently asked questions

What is the percent error formula?
The measured value minus the accepted value, with the absolute value taken, divided by the accepted value, times a hundred. For a measurement of 4.9 against an accepted 5 the difference is 0.1, dividing by 5 gives 0.02, and multiplying by a hundred gives 2 percent. The absolute value is taken so that the error is a distance: a result that overshoots and one that falls short by the same amount give the same percent error.
Do I divide by the accepted value or the measured value?
By the accepted value — the reference is what the error is measured against, and that is the convention this page follows and the one the standard definitions of relative error use. Some courses and some lab manuals divide by the experimental value instead, which for a measurement of 50 against an accepted 100 gives 100 percent rather than 50. The two conventions differ by a factor of two in that case, so it is worth checking which one your reader expects; the absolute error is the same either way.
Can percent error be negative?
Not here. The page reports the size of the error, not its direction, because an error is a distance from the reference. A measured value of 1.05 against an accepted 1 and a measured value of 0.95 against an accepted 1 both give 5 percent. Whether the measurement ran high or low is visible in the two numbers you entered, and the page leaves that reading to you rather than applying a sign convention that varies between fields.
What is the difference between absolute, relative and percent error?
They are one quantity written three ways. Absolute error is the plain difference in the units of the measurement — 0.06 grams, say — and it is the only one of the three that carries units. Relative error is that difference divided by the accepted value, which makes it scale-free: 0.006116 whether the measurement was in grams, kilograms or tonnes. Percent error is the relative error times a hundred, so 0.6116 percent. Error propagation and uncertainty budgets use the relative form; reports and tolerance checks usually quote the percent form.
What does a percent error of zero mean?
That the measured value and the accepted value are the same number, so every one of the three outputs is zero. It is a correct result and not an error condition. In practice a true zero is rare with real instruments, and a result of zero usually means the same figure was used for both fields, or that the measurement was rounded to the accepted value's precision before it was recorded.
What if the accepted value is zero?
The page reports an error, because the accepted value is the denominator and dividing by zero has no answer. That is not an arbitrary restriction: a reference of zero removes the scale the error is being measured against, so there is no relative error to report — an absolute error of 0.06 against a reference of zero is 0.06, and what fraction of nothing that represents is not a question with an answer. A measured value of zero is a different case and is perfectly valid: it is a hundred percent error against a nonzero reference.

References

Related calculators

Percentile CalculatorValue at a percentile you choose, with its two neighbours.Z-Score CalculatorZ-score from a value, mean and standard deviation.Average CalculatorAverage of a list of numbers, with the running total.Percent Difference CalculatorPercent difference between two values, either direction.