Percent Error Calculator

The percent error formula, and what it is actually asking

Percent error answers one narrow question: how big is the gap between what you measured and the value everyone agrees is correct, expressed as a share of that correct value? The formula is percent error = |measured − accepted| / |accepted| × 100. The vertical bars mean absolute value, so the answer is always positive. It is a statement about accuracy — closeness to the truth — not about precision, which is how tightly your repeated trials cluster together. A rifle that puts every shot in the same spot two inches left of the bullseye is precise and inaccurate, and only percent error notices.

Worked example: measuring gravity in a first-year lab

You drop a steel ball down a 1.2 m tube, time the fall with a photogate, and calculate g = 9.5 m/s². The accepted value at sea level is 9.81 m/s². Step one: subtract, 9.5 − 9.81 = −0.31. Step two: take the absolute value, 0.31 m/s² — that is your absolute error, and it keeps the units. Step three: divide by the accepted value, 0.31 / 9.81 = 0.0316. Step four: multiply by 100 for 3.16%, which you would write up as 3.2%.

Now read it. Your result is 3.2% off, and it is low, not high. That direction is the interesting part: air resistance, a photogate that triggers slightly late at the top, and a ball released with a small downward nudge all push the measured g in a specific direction. A 3.2% error that is consistently low across ten trials points at a systematic problem in the apparatus, not at bad luck.

Percent error vs percent difference vs percent change

Three formulas, three different questions, and marks lost every term for grabbing the wrong one. The dividing line is what sits in the denominator and whether one of your two numbers is authoritative.

Notice the three answers are close but not equal, and only percent change carries a sign. If two lab partners get 9.5 and 9.7 with no textbook value in play, that is a percent difference of |9.5 − 9.7| / 9.6 × 100 = 2.08%. If a share price moves from 9.50 to 9.81, that is a percent change of +3.26%. Same raw numbers, different denominators, different meanings.

Where the error came from: a working taxonomy

Systematic error (bias)

Every reading is shifted the same way by the same cause. A balance never re-zeroed reads 2 g heavy on all fifty samples. A ruler with a worn end adds a consistent millimetre. A thermometer calibrated at the wrong pressure reads 1.5 °C high all afternoon. Averaging more trials does not help — the bias is baked into every one of them. The fingerprint is a percent error whose direction never flips.

Random error (scatter)

Reaction time on a hand-held stopwatch, bench vibration, judging the last digit between two scale marks, small draughts across a balance pan. These push readings both directions with no pattern, so the mean of many trials converges toward the true value while the spread tells you the precision. If your percent error is small but your trials scatter widely, random error dominates.

Gross error (mistakes)

A transposed digit, a unit left in grams instead of kilograms, a formula entered with the wrong bracket. A percent error above 50% is almost never subtle physics — check arithmetic and units first. The classic is a factor-of-10 slip that shows up as roughly 900% error.

Reporting it well

Give one or two significant figures: 3.2%, not 3.1600407747%. An uncertainty estimate carrying ten digits is a claim you cannot support. Quote the absolute error alongside the percentage when units carry meaning — 0.31 m/s² tells a reader something 3.2% alone does not. Phrase results in full sentences: "The measured value of 9.5 m/s² was 3.2% below the accepted 9.81 m/s², consistent with air resistance and late photogate triggering." That single sentence names the number, the direction and a plausible cause, which is what most rubrics reward.

On the accuracy bands this calculator shows: under 1% excellent, 1–5% good, 5–10% acceptable for an introductory lab, above 10% worth rechecking the method. Those are teaching conventions, not physics. An analytical chemistry titration may demand under 0.5%, while a rough estimate of a building's height by shadow length is fine at 15%. Match the standard to the experiment.

Limitations worth knowing

Percent error is undefined when the accepted value is zero, because the division has no meaning — report the absolute error instead, which is why this tool refuses to invent a number there. It also inflates dramatically for accepted values near zero: measuring a 0.01 °C reference as 0.02 °C is a 100% error but a 0.01 °C miss. And it says nothing about repeatability; a single lucky trial can land at 0.4% while the underlying method is unreliable. Pair percent error with the spread of your repeated trials before claiming a good measurement.