Percent Error and Percent Difference, Explained
After reading this you can compute the percent error of a measurement against an accepted value, decide when percent difference is the honest choice instead, and read both numbers without fooling yourself.
What these numbers actually measure
Suppose you measure the acceleration of gravity in a lab and get 9.7 m/s². The accepted value is 9.81 m/s². You are off, but by how much relative to the size of the quantity? The raw gap is 0.11 m/s². That number alone does not tell you whether your work was good. A gap of 0.11 would be terrible if the true value were 0.5, and excellent if it were 9810.
Percent error rescales the gap by the size of the accepted value, so you can compare across quantities of different magnitudes. It answers one question: how large is my discrepancy compared to the thing I am trying to measure?
There is a second, related question. Sometimes there is no accepted value. You measure the same rod twice with two instruments and get 10.1 cm and 9.9 cm. Neither is "correct". You still want to know how far apart they are relative to their size. That is percent difference. Same arithmetic spirit, different denominator, and a different meaning.
When to use each one
The choice hinges on a single fact: do you have a value you trust as the reference?
- Percent error
- Use when one value is an accepted, theoretical, or certified reference and the other is your measurement. The reference goes in the denominator because it is the standard you judge against.
- Percent difference
- Use when two values are on equal footing: two trials, two instruments, two students, two methods. No value is privileged, so you divide by their average.
If you use percent difference when a true value exists, you understate your error by burying the reference in an average with your own (possibly wrong) number. If you use percent error when no reference exists, you are pretending one of two arbitrary measurements is truth. Both mistakes are common in student reports.
A quick test: could a physicist look up your reference value in a table or a datasheet? If yes, use percent error against that table value. If the "reference" is just your other reading, use percent difference.
The formulas and the intuition
Percent error compares your measured value to the accepted value:
Here x_{\text{meas}} is what you measured and x_{\text{true}} is the accepted value. The numerator is the absolute size of the discrepancy. The denominator is the absolute value of the true quantity, which sets the scale. The absolute value bars mean the result is never negative: percent error reports magnitude, not direction.
Percent difference compares two peers by dividing the gap by their average:
The numerator is again the gap. The denominator is the mean of the two values, which is the natural scale when neither one is special. Because the formula is symmetric, swapping x_1 and x_2 gives the same answer. That symmetry is the whole point: neither value is the reference.
Some fields drop the absolute value in percent error to keep the sign, calling +2% an overestimate and -2% an underestimate. That is fine as long as you say so. The default here reports magnitude.
Worked example with the demo values
Gravity in the lab: 9.7 versus 9.81
These are the tool's default values. Measured x_{\text{meas}} = 9.7 m/s², accepted x_{\text{true}} = 9.81 m/s².
- Find the discrepancy:
9.7 - 9.81 = -0.11. Take the absolute value:0.11. - Divide by the absolute accepted value:
0.11 / 9.81 = 0.011213. - Multiply by 100:
0.011213 × 100 = 1.121%.
So your percent error is about 1.12%. For a hand-timed pendulum or a dropped ball measured with a stopwatch, that is a good result.
Now treat the same two numbers as peers and compute percent difference, using the tool's other defaults 10.1 and 9.9 for a clean contrast:
- Gap:
|10.1 - 9.9| = 0.2. - Average:
(10.1 + 9.9) / 2 = 10.0. - Divide and scale:
0.2 / 10.0 × 100 = 2.000%.
Two readings of 10.1 and 9.9 differ by 2.00%. Notice the honesty: no value was called correct.
Watch the denominator change the answer
The denominator is where most confusion lives. For the same gap of 0.2, percent error and percent difference disagree whenever the two values are far apart, because one divides by the reference and the other by the average. When the values are close, the two formulas nearly agree. Try shifting the second value below and watch both outputs move.
Reading and interpreting the result
A percent error is a diagnostic, not a grade. What counts as good depends entirely on the experiment. A stopwatch-timed free fall giving 1.1% is fine. A gravimeter that should read to five digits and gives 1.1% is broken.
| Percent error | Typical reading | Example setting |
|---|---|---|
| < 0.1% | Precision instrument working well | Analytical balance, calibrated meter |
| 0.1% to 2% | Careful hand measurement | Stopwatch timing, ruler and protractor |
| 2% to 10% | Rough lab or crude method | Eyeballed pendulum, quick estimate |
| > 10% | Something likely wrong | Unit slip, misread scale, bad setup |
Percent error tells you the size of the discrepancy. It does not tell you the source. A 50% error usually means a mistake in method or arithmetic, not bad luck. Always ask whether the number is plausible before trusting it.
Common mistakes
Most errors in computing "error" come from four places.
- Dividing by the measured value instead of the true value. For
9.7and9.81, dividing by9.7gives1.134%instead of the correct1.121%. Small here, large when the values differ more. The accepted value belongs in the denominator. - Confusing the two formulas. Using the average denominator when a reference exists understates your error. For
9.7vs9.81, the average is9.755, giving1.128%instead of1.121%. The bias grows as your measurement drifts from truth. - Forgetting the absolute value. A raw signed result can be negative, which is fine if you meant to keep the sign, but a "percent error" of
-1.12%reported as a magnitude confuses readers. - Reporting too many digits. If your measurement has three significant figures, a percent error of
1.12%is honest and1.12130%is not. The precision of the answer cannot exceed the precision of the inputs.
Percent error is not an uncertainty. It is the gap between one measurement and one accepted value. If you need to combine the uncertainties of several measured quantities through a formula, that is a different calculation. See the Error Propagation Calculator.
Related tools
Percent error shows up wherever you compare a measurement to a predicted value, so these calculators pair naturally with it.
When you predict an angle or focal length and then measure it, check the two against each other. The Snell's Law and Refraction Calculator and the Lens and Mirror Equation Calculator both give clean theoretical values to compare a bench measurement against. In modern physics labs, compare a measured photon energy to the value from the Photon Energy Calculator, a measured frequency shift to the Doppler Effect Calculator, or a decay count to the Radioactive Decay and Half-Life Calculator. For a particle beam, the de Broglie Wavelength Calculator gives the predicted wavelength. High-speed predictions come from the Special Relativity Calculator, and electrostatic force from the Coulomb's Law Calculator. In each case, percent error turns "close" into a number.
Frequently asked questions
Can percent error be greater than 100%?
Yes. If your measurement is more than double the true value, the gap exceeds the true value and the percent error passes 100%. Measuring 25 against a true 10 gives |25-10|/10 × 100 = 150%. A result that large almost always signals a mistake in setup or units.
Why divide by the average for percent difference?
Because neither value is the reference, so neither one should sit alone in the denominator. The average is the symmetric choice: it treats both values equally and gives the same answer no matter which you list first.
What if the true value is zero?
Percent error is undefined when the accepted value is zero, because you would divide by zero. When the reference is genuinely zero, report the absolute error (the raw gap in real units) instead of a percentage.
Is a smaller percent error always better?
For accuracy, yes, a smaller gap from the accepted value is better. But a suspiciously tiny percent error can also mean you copied the expected answer or fudged the data. A believable lab result has some error, and you should be able to explain where it came from.
How does percent error relate to significant figures?
Round the percent error to match the precision of your inputs. If the measured value has three significant figures, quoting the error to two or three figures is honest. Extra digits imply a precision you do not have.
Related tools
Radioactive Decay & Half-Life Calculator
Amount remaining, activity and radiometric dating.
Coulomb's Law Calculator
Electrostatic force between two charges.
Doppler Effect Calculator
Observed frequency for a moving source or observer.
Error Propagation Calculator
Combine measurement uncertainties through a formula.
Lens & Mirror Equation Calculator
Image distance and magnification from 1/f = 1/dₒ + 1/dᵢ.
Photon Energy Calculator (E = hf)
Energy, frequency and wavelength of a photon.