What is percent error?
Percent error measures how far an experimental, observed or measured value is from an accepted reference value, relative to the size of that reference. It is widely used in science labs, measurement work and numerical comparisons because a raw error by itself does not tell you whether the miss is large or small compared with the quantity being measured.
The calculator follows the common laboratory convention: subtract the accepted value from the experimental value, take the absolute value of that difference, divide by the magnitude of the accepted value, and multiply by 100. The result is nonnegative because percent error normally describes the size of the error rather than its direction. Chemistry LibreTexts presents the same experimental-versus-accepted structure for percent error.
Alongside the main percentage, this tool keeps the underlying quantities visible. You can see the signed difference, absolute error, relative error and whether the measurement is above, below or exactly equal to the accepted reference. That makes it easier to audit your arithmetic instead of receiving only a final percentage.
How to use the Percent Error Calculator
- Enter the experimental value. This is the value you measured, observed, estimated or obtained from your calculation.
- Enter the accepted value. This is the reference, theoretical, known or agreed value you are comparing against.
- Calculate. The blue result card reports the standard absolute percent error.
- Check the supporting results. Absolute error shows the raw magnitude of the difference, relative error shows the error as a decimal ratio, and signed error preserves whether your result was high or low.
- Read the transparent formula. The breakdown substitutes your actual numbers so you can verify the calculation for homework, a lab report or your own analysis.
Percent error formula
The standard magnitude form used by this calculator is:
The vertical bars mean absolute value. They remove the sign from the difference, so a measurement that is 2 units too high and one that is 2 units too low have the same absolute error. Dividing by the accepted value scales that error relative to the reference. Multiplying by 100 converts the relative error from a decimal ratio to a percentage.
Why is the accepted value in the denominator?
Percent error asks how large the error is relative to the accepted reference. For example, an absolute error of 2 is substantial when the accepted value is 10 but tiny when the accepted value is 10,000. Using the accepted value as the baseline makes those situations distinguishable: 20% in the first case versus 0.02% in the second.
Why use the magnitude of the accepted value?
When the goal is a nonnegative error magnitude, using |accepted| keeps the denominator positive even if a legitimate reference value is negative. Some disciplines or instructors may define a signed relative-error convention differently. For a class or formal report, follow the convention specified by your instructor, laboratory manual or standard. This calculator reports the common absolute percent-error magnitude as its primary answer and separately reports direction.
Worked percent error examples
Measured 9.72 m/s², accepted 9.81 m/s². Absolute error = 0.09. Percent error = 0.09 ÷ 9.81 × 100 ≈ 0.917%.
Measured 105, accepted 100. Absolute error = 5. Percent error = 5 ÷ 100 × 100 = 5%. Signed error is +5%.
Measured 48, accepted 50. Absolute error = 2. Percent error = 2 ÷ 50 × 100 = 4%. Signed error is −4%.
Example: experimental density
Suppose an experimental density is 2.45 g/cm³ and the accepted density is 2.70 g/cm³. The absolute difference is |2.45 − 2.70| = 0.25 g/cm³. Divide by 2.70 and multiply by 100: the percent error is approximately 9.26%. This is the same style of worked measurement example used in introductory chemistry explanations of percent error.
Example: a very small raw error can still matter
Imagine an accepted value of 0.050 and an experimental value of 0.052. The absolute error is only 0.002, but relative to 0.050 that is 0.002 ÷ 0.050 = 0.04, or 4%. Raw error and percent error answer different questions: raw error gives the size of the gap in the original units, while percent error puts that gap in context.
Absolute percent error versus signed error
The standard percent-error magnitude uses an absolute value and therefore cannot be negative. However, the sign of experimental − accepted still contains useful information. A positive signed error means the experimental value is above the accepted reference; a negative signed error means it is below.
| Measure | Formula | What it tells you |
|---|---|---|
| Signed difference | experimental − accepted | Direction and raw size of the deviation |
| Absolute error | |experimental − accepted| | Raw magnitude of the deviation |
| Relative error | |error| ÷ |accepted| | Error as a fraction of the reference |
| Percent error | relative error × 100% | Error magnitude on a percentage scale |
Keeping magnitude and direction separate avoids a common reporting problem. Saying an experiment has “−4% percent error” can conflict with a course that defines percent error as an absolute magnitude. Reporting 4% percent error, measurement below accepted value communicates both pieces clearly.
What happens when the accepted value is zero?
If the accepted value is 0, the ordinary percent-error formula is undefined because it requires division by zero. The calculator will still show the signed and absolute difference, but it will not invent a percentage. This is a mathematical limitation of relative-to-reference percentages, not a calculator failure.
When the reference is zero, use a measure that fits the purpose of your analysis—for example, report the absolute error in the original units or use a domain-specific error metric with a meaningful nonzero scale. Do not replace the zero denominator with an arbitrary tiny number simply to force a percentage; that changes the question and can create a misleadingly huge result.
Percent error vs percent difference vs percent change
These three calculations look similar because they all involve differences and percentages, but they answer different questions.
| Calculation | Use it when… | Reference / denominator |
|---|---|---|
| Percent error | One value is an accepted or reference value | Accepted/reference value |
| Percent difference | You compare two peer measurements and neither is the accepted truth | Usually an average of the two values |
| Percent change | A quantity changes from an old value to a new value | Original/old value |
Choosing the denominator is not a cosmetic detail. If you have a known reference and want measurement accuracy, percent error is appropriate. If two experimental measurements have equal status, percent difference is usually the better comparison. If you are measuring growth or decline over time, percent change uses the starting value as its baseline.
Percent error, accuracy and precision
Percent error is commonly used as an indicator of accuracy: how close a result is to an accepted value. It does not by itself measure precision, which concerns how closely repeated measurements agree with one another. OpenStax laboratory material distinguishes accuracy against a known accepted value from the spread or repeatability of measurements.
A set of measurements can be tightly clustered but consistently far from the accepted value—precise but inaccurate. Another set can be spread out around the accepted value—less precise even if its average is close. To evaluate repeated data, measures such as standard deviation or relative standard deviation may be more informative than a single percent-error calculation.
Is a lower percent error always better?
For closeness to a valid accepted reference, a smaller percent error means a smaller relative deviation. But there is no universal percentage that automatically means “good” for every experiment. Acceptable error depends on the measurement method, instrument resolution, uncertainty, field, assignment requirements and practical tolerance. Avoid applying a generic 5%, 10% or other cutoff unless your course, procedure or standard actually specifies one.
Percent error with negative values
Negative experimental or reference values can occur in legitimate contexts, such as temperatures on some scales, changes in signed quantities or coordinate data. The magnitude form of percent error uses the absolute difference and the magnitude of the accepted value, producing a nonnegative result. The signed difference still tells you whether the experimental number is numerically above or below the accepted number.
Be cautious when a percentage based on a signed or near-zero reference has weak practical meaning. The arithmetic may be valid while the interpretation is not useful. Context should determine whether percent error is the right metric.
Rounding and significant figures
For hand calculations, keep extra digits through the subtraction and division steps, then round the final percentage according to the reporting rules you are expected to follow. Rounding intermediate values too aggressively can noticeably change the final answer, especially when the error is small.
This calculator retains more internal precision than it displays. The displayed result is rounded for readability, while your original input values are not altered. If a laboratory report has specific significant-figure requirements, use those requirements for the final reported value.
Common percent error mistakes
- Using the experimental value as the denominator. Standard percent error uses the accepted/reference value as the baseline.
- Forgetting the absolute value. The usual percent-error magnitude is nonnegative; direction can be reported separately.
- Confusing percent error with percent difference. Percent difference is intended for two peer values without a single accepted reference.
- Confusing percent error with percent change. Percent change uses the original value as its baseline and describes change over time or between states.
- Ignoring units before subtraction. Convert values to compatible units first.
- Dividing by zero. If the accepted value is 0, ordinary percent error is undefined.
- Rounding too early. Keep sufficient precision until the final step.
- Assuming a universal acceptable-error threshold. What counts as acceptable depends on context and requirements.
How to calculate percent error by hand
Start with your experimental and accepted values. Subtract accepted from experimental. Take the absolute value of the result. Divide that absolute error by the magnitude of the accepted value. Finally, multiply by 100 and attach the percent sign.
This sequence is short, but writing each step helps prevent denominator and sign mistakes. The calculator's breakdown is designed to mirror those steps so the result remains easy to audit.
Frequently asked questions
What is the formula for percent error?
Percent error = |experimental − accepted| ÷ |accepted| × 100%. It compares the magnitude of the error with the accepted reference value.
Can percent error be negative?
Standard absolute percent error is not negative because the numerator uses an absolute value. A signed error can be reported separately to show whether the experimental value is above or below the reference.
Can percent error be greater than 100%?
Yes. If the absolute error is larger than the magnitude of the accepted value, percent error exceeds 100%. There is no mathematical upper limit when the accepted value is nonzero.
What does 0% error mean?
It means the experimental value exactly equals the accepted value at the precision of the entered numbers, so their difference is zero.
Why is percent error undefined when the accepted value is zero?
The formula divides by the accepted value. Division by zero is undefined, so a relative percentage cannot be calculated from that baseline.
What is the difference between absolute error and percent error?
Absolute error is the raw magnitude of the difference in the original units. Percent error scales that difference relative to the accepted value and expresses it as a percentage.
Should I use percent error or percent difference?
Use percent error when one value is the accepted reference. Use percent difference when comparing two measurements of equal status and neither is treated as the true reference.
Does a small percent error prove an experiment is precise?
No. A small percent error indicates closeness to the accepted value for that result. Precision concerns repeatability or spread across repeated measurements.