What percent error actually tells you
Percent error answers one narrow question: how big is the gap between what you measured and what the value is supposed to be, expressed as a share of the value it is supposed to be. It exists because raw differences are meaningless on their own. Missing a length by 2 mm is excellent work on a 3 m beam and terrible work on a 5 mm gauge block. Dividing by the accepted value strips out the scale, so a chemistry titration and an astronomical distance can be judged on the same axis.
The word accepted is doing real work in that sentence. Percent error assumes you already know the right answer from an independent authority: a defined constant such as standard gravity, a certified reference material, a value from a CODATA table, or a prediction your theory makes before you take the reading. If you have two measurements and no reason to trust one over the other, percent error is the wrong tool and you want the symmetric percentage difference instead.
Percent error is also not uncertainty. The Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008) reserves error for the difference between a result and a reference value, and uncertainty for the spread of values you could reasonably attribute to the measurand. Error is a single number you can only compute when the reference is known; uncertainty is a property of your procedure that you can quote even when no reference exists. A lab report usually wants both, and they are not interchangeable.
The formula, and why the denominator is the accepted value
Start with the absolute error, which is just the size of the gap:
absolute error = |measured − accepted|
Divide it by the magnitude of the accepted value to get the relative error, a pure dimensionless number, and multiply by 100 to express that as a percentage:
percent error = 100 × |measured − accepted| ÷ |accepted|
Three details in that expression are worth defending.
The absolute value bars in the numerator make percent error non-negative, which is the convention almost every lab manual uses. That discards information, so this calculator also reports the signed percent error, which is negative when you read low and positive when you read high. A systematic instrument bias shows up as a consistent sign across repeated trials, and you lose that diagnostic entirely if you only ever look at magnitudes.
The denominator is the accepted value. Dividing by the measured value instead gives a different answer, and the difference is not small when the error is large. Measuring 100 where 95 is accepted gives 5 ÷ 95 = 5.2632%; measuring 95 where 100 is accepted gives 5 ÷ 100 = 5.0000%. Same 5-unit gap, different percentages, because the reference changed. Percent error is deliberately asymmetric, and that asymmetry is what makes it a statement about your measurement rather than about the pair.
The denominator is a magnitude. When the accepted value is negative — a temperature of −20 °C, an enthalpy change of −285.8 kJ/mol — dividing by the signed value would flip the sign of the result and make a high reading look low. Taking |accepted| keeps percent error non-negative and lets the signed output carry the direction, which is why this calculator uses it.
When the accepted value is zero, none of this works. There is no scale to divide by, and the calculator returns a dash rather than an infinity. Quote the absolute error on its own, or shift to a scale whose zero is physically meaningful — kelvin rather than celsius, for instance.
Worked example: measuring g with a pendulum
You time a simple pendulum in a first-year physics lab and derive a local acceleration due to gravity of 9.80 m/s². The accepted value you were told to compare against is 9.80665 m/s², the standard acceleration of free fall fixed by the 3rd General Conference on Weights and Measures in 1901 and reproduced in the SI Brochure.
- Signed difference. 9.80 − 9.80665 = −0.00665 m/s². The minus sign says you read low.
- Absolute error. |−0.00665| = 0.00665 m/s².
- Relative error. 0.00665 ÷ 9.80665 = 0.000678111. Check it the other way: 9.80665 × 0.000678111 = 0.0066500, which is the numerator, so the division is right.
- Percent error. 100 × 0.000678111 = 0.0678%.
- Signed percent error. 100 × (−0.00665 ÷ 9.80665) = −0.0678%.
Seven hundredths of one percent from a stopwatch and a piece of string is a good result, and the sign tells you something a magnitude would hide. Timing errors in a pendulum experiment are usually biased: a human reaction delay that lengthens the measured period pushes the derived g down, because g = 4π²L/T² falls as T rises. A negative signed error is exactly the direction that bias predicts, so this result is consistent with the dominant systematic effect rather than with random scatter.
Now check the same result against a tolerance. If your lab requires agreement within 1%, the passing window around 9.80665 is 9.80665 ± 0.0980665, or 9.7086 to 9.9047. Your 9.80 sits comfortably inside. At a 0.1% tolerance the window narrows to 9.7968 to 9.8165, and 9.80 still passes — but only by 0.003 m/s².
How to read the number you get
There is no universal threshold for a good percent error, because the honest benchmark is the precision the method can deliver, not a round number. Judge your result against three things instead.
The instrument's own specification. A digital multimeter rated ±0.5% of reading cannot be expected to do better than 0.5%, so a 0.3% percent error is at the limit of what the hardware can resolve and a 0.05% result is probably luck rather than skill. Read the tolerance band table this calculator produces as a pass/fail check against whatever specification actually applies.
The repeatability of your own trials. If five repeats scatter over a range of 3% and your percent error is 1%, the disagreement is buried inside your own noise and you cannot claim a systematic problem. If five repeats agree to 0.1% and your percent error is 1%, something is biased, and the sign of the signed percent error is the first clue to what.
What the discipline treats as acceptable. Rough teaching-lab rules of thumb are that under 1% is good, 1–5% is ordinary, and over 10% usually means a mistake in method or arithmetic rather than in technique. Treat these as rules of thumb; they are not standards, and a percent error of 15% on a hand-timed experiment can be more defensible than 2% on an instrument rated to 0.01%.
One diagnostic is nearly free. Recompute the percent error after correcting for the single largest systematic effect you can name. If a 4% error drops to 0.4%, you have found the cause; if it barely moves, you have not.
Percent error against percentage difference for the same pairs
| Measured | Accepted | Absolute error | Percent error | Percentage difference |
|---|---|---|---|---|
| 105 | 100 | 5 | 5.0000% | 4.8780% |
| 95 | 100 | 5 | 5.0000% | 5.1282% |
| 100 | 95 | 5 | 5.2632% | 5.1282% |
| 2.9 | 3.0 | 0.1 | 3.3333% | 3.3898% |
| 1.62 | 1.60 | 0.02 | 1.2500% | 1.2422% |
| 0.5 | 0.4 | 0.1 | 25.0000% | 22.2222% |
Percent error = 100·|m − a|/|a|. Percentage difference = 100·|m − a|/((m + a)/2). Row 3 and row 2 share a percentage difference of 5.1282% because that measure does not care which value you call the reference.
Mistakes that make a reported percent error wrong
- Dividing by the measured value. The single most common slip. It gives a smaller number when you read high and a larger one when you read low, which quietly flatters half of all results.
- Mixed units. Comparing 980 cm/s² against 9.80665 m/s² produces a percent error near 9,893%. If your answer is in the thousands of percent, convert before you do anything else.
- Reporting more digits than the measurement supports. If you measured to three significant figures, a percent error quoted to six is fiction. Round it the way you would round any derived quantity — the significant figures calculator and the rounding calculator both apply.
- Averaging percent errors across trials instead of averaging the trials. Compute the mean measurement first, then take one percent error against the accepted value. Averaging the individual percent errors discards the cancellation of random scatter.
- Treating percent error as an uncertainty. It is a comparison to a known answer. Outside a teaching lab you usually do not have one, which is why professional work quotes a combined standard uncertainty instead.
- Using it when neither value is authoritative. Two instruments disagreeing is a percentage difference problem, not a percent error problem.
Where percent error sits among the other percentage measures
Four related quantities get confused with each other constantly, and the distinction is entirely in the denominator.
Percent error divides by the accepted value. Use it when one of the two numbers is authoritative.
Percent change divides by the earlier value and keeps its sign. Use it for the same quantity measured at two times — a price, a population, a concentration during a reaction. It is the same arithmetic as signed percent error, but the reference is chronological rather than authoritative.
Percentage difference divides by the mean of the two values and is therefore symmetric: swap the inputs and the answer is unchanged. Use it for two independent measurements of the same thing, such as duplicate assay results.
Percent recovery, common in analytical chemistry, is 100 × measured ÷ accepted rather than a difference at all. A 98% recovery corresponds to a 2% percent error, and the two are often quoted in the same table, which is why they get mixed up.
Two further tools sit alongside these once you move from a single comparison to a set of them. Relative error expressed in parts per million is the same quantity scaled by 10⁶ instead of 100, and metrology uses it for the same reason chemistry uses percent — to keep the digits readable. And when your errors are dominated by scatter rather than bias, a standard deviation or a confidence interval describes the data far better than any single percent error can, because it says how much the answer would move if you did the experiment again.
If your calculation feeds a further computation, propagate the relative error rather than the percent error. For a product or a quotient, relative errors add in quadrature; for a sum or a difference, absolute errors do. Working in the decimal relative error keeps you from multiplying by 100 twice, which is a surprisingly frequent source of a factor-of-100 mistake in scientific notation work.
Key terms
- Absolute error
- The magnitude of the difference between the measured and accepted values, in the units of the measurement. It answers “by how much” but not “how badly”.
- Relative error
- Absolute error divided by the magnitude of the accepted value. Dimensionless. Multiply by 100 for percent, by 10⁶ for parts per million.
- Accepted value
- A reference value you are treating as correct: a defined constant, a certified reference material, a CODATA recommendation, or a theoretical prediction.
- Systematic error
- A bias that pushes every trial the same way, so it shows as a consistent sign in the signed percent error. Repeating the measurement does not reduce it.
- Random error
- Scatter that varies in sign and size between trials. Averaging many trials reduces it in proportion to the square root of the number of trials.