The difference between percent error vs percent difference comes down to whether you have a true value. Percent error compares a measurement with an accepted value and divides by that accepted value. Percent difference compares two measurements of equal standing and divides by their average. Use error when one number is “right”; use difference when neither is.
Both formulas look almost identical, which is exactly why lab reports, engineering memos and homework sets mix them up. The denominator is the whole story. Pick the wrong one and your answer can be off by several percentage points, and a grader or reviewer will notice. This guide sets out both formulas, adds percent change (the third member of the family), works through real lab numbers and explains how absolute error, relative error and significant figures fit around them.
For a quick answer, plug your numbers into the percent error calculator or the percentage difference calculator. The rest of this page explains which one you should be using and why.
Percent error: measuring against a known value
Percent error tells you how far a measured or estimated value is from an accepted, theoretical or reference value, as a share of that reference.
Formula: percent error = |measured − accepted| ÷ |accepted| × 100
The accepted value goes in the denominator because it is the yardstick. It might be a physical constant (the speed of light, the density of pure aluminum), a value printed on a certified reference material, or the theoretical yield calculated from a balanced equation. NIST maintains the CODATA fundamental physical constants that most “accepted values” in physics and chemistry courses ultimately trace back to.
Worked example: measuring gravity
You time a pendulum and calculate g = 9.62 m/s². The accepted local value is 9.81 m/s².
- Absolute error: |9.62 − 9.81| = 0.19 m/s²
- Divide by the accepted value: 0.19 ÷ 9.81 = 0.01937
- Multiply by 100: 1.94%
Your percent error is about 1.9%. Many instructors want the signed version as well: (9.62 − 9.81) ÷ 9.81 × 100 = −1.94%. The minus sign tells you the measurement came in low, which is a hint about systematic error. A pendulum that consistently reads low might point to timing too few swings, so your reaction time inflates the period.
Worked example: density of aluminum
A student measures a block of aluminum and gets 2.58 g/cm³. The accepted value is 2.70 g/cm³. Absolute error is 0.12 g/cm³, and 0.12 ÷ 2.70 × 100 = 4.44%. Four and a half percent is respectable for a ruler-and-balance measurement, but it is large enough that you would look for air bubbles, a chipped edge or a balance that was not zeroed.
Percent difference: comparing two equal measurements
Sometimes there is no “true” value. Two lab partners measure the same spring constant, two labs test the same sample, or you compare readings from two thermometers. Neither number deserves to be the reference, so percent difference divides by their average.
Formula: percent difference = |A − B| ÷ ((A + B) ÷ 2) × 100
Because the average sits in the denominator, swapping A and B gives the same answer. That symmetry is the point.
Worked example: two lab groups
Group 1 measures g = 9.62 m/s². Group 2 measures 9.75 m/s².
- Difference: |9.62 − 9.75| = 0.13
- Average: (9.62 + 9.75) ÷ 2 = 9.685
- 0.13 ÷ 9.685 × 100 = 1.34%
The two groups agree to within about 1.3%. Note that this says nothing about whether either group is accurate. Two measurements can agree closely and both be wrong in the same direction. If you also have the accepted value, report percent error for each group separately: 1.94% and 0.61%.
Percent change: when one value comes first
Percent change is the third formula, and it belongs in the comparison because people often reach for it by accident. It measures how much a value moved from a starting point.
Formula: percent change = (new − old) ÷ |old| × 100
Use it when there is a clear before and after: last year’s sales and this year’s, a reaction rate before and after heating, a patient’s blood pressure before and after treatment. Unlike percent difference, it is not symmetric. A rise from 50 to 75 is a 50% increase, but a fall from 75 to 50 is a 33.3% decrease. The percent difference between 50 and 75 is 40% either way. Our percentage change calculator handles the before-and-after case, and the guide on how to calculate percentage covers it in more depth.
Side-by-side comparison
| Percent error | Percent difference | Percent change | |
|---|---|---|---|
| Formula | |measured − accepted| ÷ |accepted| × 100 | |A − B| ÷ average(A, B) × 100 | (new − old) ÷ |old| × 100 |
| Denominator | Accepted (true) value | Mean of the two values | Original value |
| Use when | A reference value exists | Two values of equal standing | Clear before and after |
| Symmetric? | No | Yes | No |
| Signed? | Usually reported as absolute; signed version shows direction | Always positive | Positive for increase, negative for decrease |
| Upper limit | None | 200% | None for increases; −100% for decreases |
| Example | 9.62 vs 9.81: 1.94% | 9.62 vs 9.75: 1.34% | 50 to 75: +50% |
The 200% cap on percent difference surprises people. If one value is 0 and the other is anything positive, the difference equals the value and the average is half of it, so the result is always 200%. Values of 10 and 30 give a 100% difference (20 ÷ 20), while percent change from 10 to 30 is 200%. When values are that far apart, percent difference stops being very informative and you are better off reporting the two numbers directly.
Absolute error vs relative error
Every percent error starts as an absolute error: the raw gap between measured and accepted values, in the original units. Absolute error alone does not tell you whether a result is good. Being off by 2 mm when you measure a 50 mm bolt is a 4% error. Being off by 2 mm on a 2 m doorframe is 0.1%. Same absolute error, completely different quality.
Relative error divides the absolute error by the accepted value, giving a unitless fraction (0.0194 in the gravity example). Percent error is simply relative error times 100. The relative error calculator gives all three versions at once, and the absolute uncertainty calculator converts between absolute and relative uncertainty when you are working with instrument tolerances rather than a known true value.
Error and uncertainty are not the same thing, by the way. Error is the difference from a true value, which you can only calculate if you know it. Uncertainty is your estimate of how wrong you could be, which you can state even when the true value is unknown. NIST’s Technical Note 1297 on evaluating and expressing measurement uncertainty is the standard reference if your course or job goes further down this road.
Accuracy, precision and repeated measurements
Percent error measures accuracy: closeness to the true value. Precision is about how tightly repeated measurements cluster, and it is a different property. Suppose you measure g five times and get 9.62, 9.75, 9.70, 9.68 and 9.71 m/s².
- Mean: 9.692 m/s²
- Percent error of the mean: |9.692 − 9.81| ÷ 9.81 × 100 = 1.20%
- Sample standard deviation: 0.048 m/s², which is 0.49% of the mean
The spread (about 0.5%) is smaller than the error (about 1.2%). That pattern means your measurements are precise but slightly biased, so the problem is systematic rather than random. Averaging more trials will not fix it; changing the method might. You can get the spread with the standard deviation calculator, and the standard deviation guide explains what the number means.
Significant figures and rounding
A percent error should not claim more precision than the measurements behind it. If your measured and accepted values have three significant figures, report the percent error to two or at most three: 1.9% rather than 1.9368%. The absolute error 0.19 has only two significant figures after the subtraction, and that limits everything downstream.
Two practical rules:
- Do the arithmetic with full precision and round only the final answer. Rounding intermediate steps can shift the last digit.
- When subtraction cancels leading digits (9.81 − 9.62), count significant figures in the result, not the inputs. The significant figures calculator handles the counting.
Common mistakes
- Dividing by the measured value. Percent error always divides by the accepted value. Dividing by 9.62 instead of 9.81 gives 1.98% instead of 1.94%.
- Using percent difference when a true value exists. If the textbook gives you an accepted value, the assignment almost certainly wants percent error.
- Using percent error to compare two students’ results. Neither is the reference; use percent difference.
- Dropping the sign when it matters. The absolute value is standard, but the signed error tells you whether you are systematically high or low, which is the useful diagnostic.
- Reporting six decimal places. Match the precision of your data.
- Calling percent yield a percent error. Percent yield (actual ÷ theoretical × 100) is related but answers a different question. The percent yield calculator covers that one; the molarity guide shows where these numbers come from in a real lab.
Frequently asked questions
Is percent error the same as percent difference?
No. Percent error divides by an accepted true value, while percent difference divides by the average of two measured values. They give similar results only when the values are close together.
Can percent error be negative?
The standard formula uses an absolute value, so it is always positive. The signed version, (measured − accepted) ÷ accepted × 100, is negative when your measurement is too low.
What is a good percent error?
It depends on the instrument and the experiment. In introductory labs, under 5% is usually considered good and under 1% excellent; in analytical chemistry or calibration work, tolerances can be far tighter.
Why does percent difference use the average?
When neither value is a reference, dividing by either one would favour it. The average treats both equally, so the result is the same no matter which value you call A.
When should I use percent change instead?
Use percent change whenever there is an obvious starting value and a later value, such as prices over time or a measurement before and after a treatment.
The quick test is simple: if one number is known to be correct, divide by it and call it percent error. If both numbers are measurements, divide by their average and call it percent difference. If one came before the other, divide by the first and call it percent change. Get the denominator right and the rest is one subtraction and one division.