Percent Error Calculator
How far a measurement sits from the value it should have been, as a percentage of that value — with the absolute error, the relative error, and the direction the reading ran. When neither number is the true one, switch to percent difference and the tool divides by the mean instead.
Measure against a known value
The two formulas differ in what they divide by, and that is the whole difference: the reference value, or the mean of the pair.
The accepted, theoretical, or certified figure — the one you are treating as correct.
The measurement you actually took.
96.5 against 100 is a gap of 3.5, taken as a fraction of 100. The measurement ran low by 3.5%.
- Absolute error
- 3.5
- Signed difference
- -3.5
- Signed percent
- -3.5%
- Relative error
- 0.035
- Reference value
- 100
- Formula applied
- |observed − expected| ÷ |expected| × 100
- The headline figure follows the textbook convention and is never negative. The signed percent beside it keeps the direction, because a gauge reading 3.5% low and one reading 3.5% high are the same error and opposite repairs.
- Percent error divides by the expected value, so the answer changes if you swap the two numbers. That asymmetry is the point: one of them is the reference.
- Percentages are reported to four decimal places; the arithmetic underneath runs at full precision and rounds once, at the end.
Three formulas that look alike and answer different questions
Nearly every mistake made with these is a mistake about which one applies, not about the arithmetic. All three divide a gap by a baseline; what separates them is which baseline is entitled to sit on the bottom.
error = |o − e| ÷ |e| × 100 • difference = |a − b| ÷ ((a + b) ÷ 2) × 100 • change = (new − old) ÷ old × 100- o, e
- The observed measurement, and the expected value you are treating as correct
- a, b
- Two measurements of the same quantity, neither entitled to be the reference
- new, old
- One quantity at two points in time — a different question entirely
Percent error: one value is correct
Use it when there is an accepted answer — a published constant, a certified reference material, a theoretical result. The denominator is that accepted value, because the question is “how far off was I, as a fraction of the right answer?” Swapping the two numbers changes the result, and it should: only one of them is the reference.
Percent difference: neither value is correct
Use it when two measurements disagree and nothing entitles either to be the baseline. Dividing by either one would make the answer depend on which you wrote down first, so the denominator is the mean of the pair. That makes the result symmetric by construction — enter the readings in the other order and the number is identical.
Percent change: one value, two times
Growth, discounts, before-and-after. It is not an error at all, it keeps its sign, and it is the formula people reach for when they mean percent error and the other way round. It lives with the percentage calculator, where the rest of the everyday percentage arithmetic is.
A reading of 96.5 against an expected 100
The absolute error is the gap itself: |96.5 − 100| = 3.5, in whatever unit the measurement is in. That number is meaningful on its own and is the only thing you can report if the reference is zero.
The relative error turns it into a fraction of the reference: 3.5 ÷ 100 = 0.035. Multiply by 100 and the percent error is 3.5%. The signed version keeps the direction — 96.5 − 100 = −3.5, so −3.5%, and the instrument is reading low. Both figures describe the same gap; only one of them tells you which way to turn the adjustment screw.
Now suppose those two numbers came from two thermometers and neither is a reference. The mean is (96.5 + 100) ÷ 2 = 98.25, and the percent difference is 3.5 ÷ 98.25 × 100 = 3.5623%. Slightly larger than the percent error, because the denominator is slightly smaller — and, unlike the percent error, identical if you swap the two readings. Getting 3.5% where 3.5623% belongs is the everyday version of this mistake: small enough to survive a marking scheme, wrong enough to be worth fixing.
What this calculator assumes
- In percent error mode, the expected value is genuinely the reference. The tool cannot tell whether it deserves that status — that is a judgement about your experiment, not arithmetic.
- Both figures are single measurements. If you averaged several readings, average them first; there is no weighting here, and no propagation of the uncertainty in each one.
- Negative measurements are compared by magnitude. A reading of −38 against a reference of −40 is 5% off, not −5% — the sign of the measurement is not the sign of the error, and only the signed percentage carries direction.
- A reference of zero is refused rather than reported as infinity. Relative error is undefined there, and the absolute error is the whole of what can honestly be said.
- Percentages are reported to four decimal places; the two figures carrying the measurement’s unit are kept to twelve significant figures, so an error far smaller than the values themselves does not round away to zero.
- Nothing here judges whether an error is acceptable. That depends on the tolerance of the instrument and the claim being made with it, neither of which is an input.
Percent error FAQ
What is the difference between percent error and percent difference?
What you divide by, and that follows from whether one of the numbers is entitled to be called correct. Percent error compares a measurement against an accepted or theoretical value and divides by that reference: |observed − expected| ÷ |expected| × 100. Percent difference compares two measurements when neither is the reference — two thermometers, two labs, two runs — and divides by the mean of the pair: |a − b| ÷ ((a + b) ÷ 2) × 100. Using the mean is what makes the answer the same whichever measurement you happen to write first.
How is percent error different from percent change?
Percent error is about agreement between two things measured now; percent change is about one thing measured twice, at different times. The formulas look nearly identical — percent change is (new − old) ÷ old × 100 — but percent change keeps its sign, because a fall of 20% and a rise of 20% are different events, while percent error is conventionally reported unsigned. If what you want is growth, discounts or a before-and-after, use the percentage calculator instead.
Can percent error be negative?
By convention, no: the definition wraps the difference in absolute-value bars, so the answer is a magnitude. That convention throws away something useful, though — whether the instrument reads high or low — so this calculator reports the signed percentage beside the conventional one. A gauge running 3.5% low and one running 3.5% high have identical percent error and opposite fixes.
What counts as a good percent error?
There is no universal threshold, and any page that gives you one is guessing. It depends entirely on the tolerance of the instrument and the claim you are making with it: 5% is unremarkable in a school titration, disqualifying in an analytical balance calibration, and meaningless without knowing which. The useful comparison is against the stated accuracy of the equipment — if the manufacturer says ±1% and you are 4% out, something is wrong beyond the instrument.
Why can't I calculate percent error when the expected value is zero?
Because a relative error is a fraction of something, and there is no fraction of zero. Dividing by zero gives infinity, and infinity is not a measurement — a 0.01 error and a 10,000 error against a reference of zero would both be reported as infinitely wrong, which tells you nothing about either. When the reference is genuinely zero, the absolute error is the only honest figure and is what you should report.
Should I use the observed or the expected value on the bottom?
The expected value, always, when you are computing percent error. It is the reference the result is a fraction of, and putting the measurement underneath instead makes the answer depend on the error you are trying to measure. If you genuinely cannot say which of the two is the reference, that is the signal to switch to percent difference and divide by the mean.
Sources and review notes
- NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — the distinction between an absolute and a relative figure used here
- BIPM, Guides in Metrology — including JCGM 100, the Guide to the Expression of Uncertainty in Measurement, on why a relative quantity needs a non-zero reference
The formulas here are definitional and have no data behind them to go stale. What goes wrong in practice is choosing between them, so the mode selector above names the situation each one belongs to rather than the formula, and the result panel says which one it applied.