Percent Error Calculator
This percent error calculator compares a measured value against the accepted value and returns the absolute error, the relative error, the signed error, the percent error and the accuracy together. Switch between an always-positive result and a signed one that shows direction.
How to use the percent error calculator
- Enter the accepted value. This is the true, theoretical or reference figure — the one from the data table, the textbook, the certificate or the standard.
- Enter the measured value. This is what your experiment, instrument or estimate actually produced.
- Read the outputs. Absolute error in the original units, relative error as a decimal, percent error, and accuracy. Everything updates as you type.
- Choose absolute or signed. Absolute is always positive and answers “how far off?”. Signed keeps the plus or minus and answers “how far off, and in which direction?”.
The percent error formula
Each piece of the output is a stage of that calculation:
- Absolute error =
|measured − accepted|. This carries units. If you measured in g/cm³, the absolute error is in g/cm³. - Signed error =
measured − accepted, keeping the sign. Negative means you measured low, positive means you measured high. - Relative error =
absolute error ÷ |accepted|. A pure number with no units, because the units cancel. - Percent error = relative error × 100.
- Accuracy =
100 − percent error, expressed as a percentage.
The absolute value bars matter twice over. The one in the numerator makes the error positive. The one in the denominator handles a negative accepted value, such as a reference temperature of −40°C, where dividing by a bare negative would flip the sign of the whole result. If absolute values are new to you, the absolute value calculator covers them on their own.
Relative error is what makes the number worth reporting. An absolute error of 0.05 means nothing until you know what it is 0.05 of. Against 2.70 it is 1.85%; against 0.10 it would be 50%.
A worked lab example
Example: density of aluminum
You measure a block of aluminum at 2.65 g/cm³. The accepted value is 2.70 g/cm³.
Report it as 1.85%, or as −1.85% if your write-up asks for direction. The minus sign says the measurement came in below the accepted value, which points at a specific set of causes: a sample with trapped air, an under-reading balance, or a volume measured too large.
Example: boiling point of water
A thermometer reads 98.2°C where the accepted value is 100°C.
A consistent negative reading across several trials suggests the thermometer needs calibrating rather than that any one trial went wrong.
Example: acceleration due to gravity
A pendulum experiment gives 9.63 m/s² against an accepted 9.81 m/s².
Keep the working at full precision and round only the final percentage. Rounding 0.018349 to 0.02 first would report 2%, which overstates the error by nearly a tenth of its own size. The significant figures calculator helps you decide how many digits the result deserves.
| Measured | Accepted | Absolute error | Percent error | Signed |
|---|---|---|---|---|
| 2.55 | 2.70 | 0.15 | 5.56% | −5.56% |
| 2.65 | 2.70 | 0.05 | 1.85% | −1.85% |
| 2.70 | 2.70 | 0.00 | 0.00% | 0.00% |
| 2.75 | 2.70 | 0.05 | 1.85% | +1.85% |
| 2.85 | 2.70 | 0.15 | 5.56% | +5.56% |
| 3.24 | 2.70 | 0.54 | 20.00% | +20.00% |
When the signed error tells you more
The standard convention takes the absolute value, so percent error is normally reported as a positive number. That convention exists because in most write-ups the question is simply how close you got, and the sign would add noise.
Keep the sign when the direction is diagnostic:
- Repeated trials. Errors of −1.8%, −2.1% and −1.9% are a systematic error — something in the setup is biased low. Errors of −1.8%, +2.1% and −0.4% are random scatter. Take the absolute value of each and both sets look identical, and the more useful conclusion disappears.
- Calibration. To correct an instrument you have to know which way to adjust it.
- Averaging errors across trials. Signed errors average toward zero when the error is random. Absolute errors never do, so their average is always positive and always overstates the bias.
- Comparing methods. Two methods that miss by the same amount in opposite directions are not making the same mistake.
The trade-off is that a signed error cannot be averaged with an absolute one, and mixing the two in a single table makes it impossible to tell what the mean means. Pick one convention for a write-up and state which you used.
Percent error, percent difference and percent change
Three formulas that look alike and answer different questions. Choosing the wrong one is the most common way a correct measurement gets reported as a wrong number.
percent difference = |A − B| ÷ ((A + B) ÷ 2) × 100
percent change = (new − old) ÷ |old| × 100
| Percent error | Percent difference | Percent change | |
|---|---|---|---|
| Denominator | The accepted value | The mean of both values | The earlier value |
| Use it when | One value is known to be correct | Neither value is authoritative | One value came before the other |
| Signed? | Usually absolute | Always absolute | Always signed |
| Typical case | Lab result against a data book | Two students’ measurements | Price this year against last year |
| On 2.65 and 2.70 | 1.85% | 1.87% | −1.85% |
Run the bottom row out to see why the denominators matter. Percent error divides 0.05 by 2.70 and gets 1.85%. Percent difference divides 0.05 by the mean of 2.675 and gets 1.87%. Percent change divides −0.05 by the earlier value 2.70 and gets −1.85%.
Percent difference is symmetric: swap A and B and the answer is unchanged, which is exactly what you want when neither measurement outranks the other. Percent error is not symmetric, and that is deliberate — the accepted value is the fixed reference, so it always goes on the bottom.
Why a zero accepted value breaks the formula
If the accepted value is 0, percent error is undefined. The formula divides by it, and division by zero has no answer. This is not a limitation of the calculator; there is nothing to compute.
The reason is what a percentage means. A percent error asks what fraction of the accepted value the error represents. When the accepted value is zero there is no quantity to take a fraction of, and any error at all — 0.001 or 1,000 — is infinitely large relative to zero.
Values very close to zero are almost as bad. An accepted value of 0.01 with an absolute error of 0.005 gives a 50% error, which sounds catastrophic and may describe a perfectly good measurement. Percent error becomes unstable as the reference approaches zero.
What to report instead when the accepted value is zero or near it:
- The absolute error on its own, with units. “The measured offset was 0.004 V from an expected 0 V” is clear and complete.
- The error relative to the instrument’s full scale, which is how meter accuracy is usually specified.
- The error relative to the measurement range, if your experiment defines one.
The same caution applies to any accepted value small enough that a rounding step in your working could shift the percentage noticeably. The rounding calculator shows how much a given rounding rule moves a figure.
Common mistakes
- Dividing by the measured value. The accepted value goes in the denominator. Swapping them changes 1.85% into 1.89% here, and much more when the error is large.
- Forgetting to multiply by 100. A relative error of 0.0185 is 1.85%, not 0.0185%.
- Reporting a negative percent error when the convention is absolute. State which convention you are using.
- Rounding before the final step. Carry full precision through the division and round once at the end.
- Using percent error when there is no accepted value. If neither figure is authoritative, use percent difference.
- Comparing values in different units. Convert first — 2.70 g/cm³ and 2,700 kg/m³ are the same density, but the raw numbers look wildly different.
- Treating a small percent error as proof the method was sound. Two errors in opposite directions can cancel and leave a flattering number.
- Assuming accuracy cannot be negative. If the percent error exceeds 100%,
100 − errorgoes below zero, which simply means the measurement missed by more than the accepted value itself.
Frequently asked questions
What is the percent error if I measured 2.65 and the accepted value is 2.70?
1.85%. The difference is 0.05, and 0.05 ÷ 2.70 = 0.018518, which is 1.85% once multiplied by 100. The accuracy is 98.15%. As a signed error it is −1.85%, since the measurement came in below the accepted value. This is the standard aluminum density case from a first-year lab.
What is the percent error formula?
percent error = |measured − accepted| ÷ |accepted| × 100. Subtract the accepted value from the measured one, drop the sign, divide by the accepted value, and multiply by 100. The accepted value goes in the denominator in every version of the formula. The bars around it handle negative reference values such as a temperature below zero.
Can percent error be negative?
Only if you are deliberately reporting the signed form. The standard formula uses absolute values, so it never is. The signed version, (measured − accepted) ÷ |accepted| × 100, is negative when you measured low and positive when you measured high. That direction is worth keeping when you are calibrating or looking for systematic bias.
What is a good percent error?
There is no universal threshold. What counts as acceptable depends on the instrument’s precision, the method, and the tolerance your lab or course sets. A 5% error can be excellent with a ruler and a stopwatch and unacceptable with an analytical balance. Compare your result against the tolerance you were given rather than against a general rule.
What is the difference between percent error and percent difference?
The denominator. Percent error divides by the accepted value, because one figure is taken as correct. Percent difference divides by the mean of the two values, because neither outranks the other. On 2.65 and 2.70 that is 1.85% against 1.87%. Percent difference is symmetric; percent error is not, by design.
Why is percent error undefined when the accepted value is zero?
Because the formula divides by the accepted value, and dividing by zero has no answer. A percentage expresses the error as a fraction of the reference, and there is no fraction of zero to take. Report the absolute error with its units instead, or express the error against the instrument’s full scale.
What is the percent error for a boiling point measured at 98.2°C?
1.8%, against an accepted 100°C. The absolute error is 1.8°C, and 1.8 ÷ 100 × 100 = 1.8%, giving 98.2% accuracy. The signed error is −1.8%. If several trials all read low by a similar amount, the thermometer is likely miscalibrated rather than the trials being individually unlucky.
How do I convert percent error to accuracy?
Subtract it from 100. A 1.85% error is 98.15% accuracy; a 5% error is 95% accuracy. If the percent error is above 100% the accuracy comes out negative, which is a real result rather than an error — it means the measurement missed by more than the accepted value itself, as 6.5 against an accepted 2.7 would.
Does the order of measured and accepted matter?
Yes, in the denominator. The numerator is inside absolute value bars so the order there makes no difference, but the accepted value must be the one you divide by. With 2.65 and 2.70 the correct answer is 1.85%; dividing by 2.65 instead gives 1.89%. The gap widens as the error grows.
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