Instantly calculate percent error by entering your experimental value and theoretical value into the calculator above. The tool applies the standard percent error formula, returns the result as a percentage, and shows the step-by-step solution so you can verify every number before dropping it into a lab report or homework assignment.
Percent error tells you how far a measured result is from the accepted or true value, expressed as a percentage. A smaller percent error means your measurement is closer to the expected value. A larger one signals a meaningful discrepancy worth investigating.
Calculate Percent Error with This Percentage Error Calculator
Enter two numbers to calculate percent error:
- Theoretical value (also called the accepted value or true value). This is the reference number from a textbook, data sheet, or established measurement.
- Experimental value (also called the observed value or measured value). This is the number you recorded during your experiment or measurement.
The free percent error calculator handles the subtraction, absolute value, and division automatically. It then multiplies by 100 to give you a clean percentage. Results are estimates for learning and verification purposes, not a substitute for professional analysis.
Percent Error Formula
The standard percent error formula is:
Percent Error = (|Experimental Value − Theoretical Value| / |Theoretical Value|) × 100
That single line does three things: finds the absolute difference between two values, divides by the reference value, and converts the ratio to a percentage.
Theoretical Value and Experimental Value in the Formula
The theoretical value is what you expect based on known science or an established standard. The experimental value is what you actually measured. The gap between them is the deviation the formula quantifies.
For example, if a chemistry textbook lists the boiling point of water at 100 °C and your thermometer reads 101.3 °C, the theoretical value is 100 and the experimental value is 101.3.
Absolute Value, Subtract, and Multiply by 100
The formula starts by asking you to subtract the theoretical value from the experimental value. You then take the absolute value of that difference, which strips away the negative sign and keeps the result always positive.
Why absolute value? The direction of the error (over or under) matters in some contexts, but percent error focuses on magnitude. The absolute value of the difference ensures you get a clean, comparable number.
Finally, you multiply by 100 to convert the decimal ratio into a percentage. Without this step, you would get 0.013 instead of 1.3%.
The Denominator in the Percent Error Formula
The denominator is the absolute value of the theoretical value. It acts as the reference point that scales the error. Dividing by a larger theoretical value produces a smaller percent error for the same absolute difference, which makes sense: being off by 1 gram matters more when the expected mass is 5 grams than when it is 500 grams.
If the theoretical value is zero, the denominator becomes zero and the formula is undefined. The FAQ section below covers that edge case.
Step-by-Step Solution to Calculate Percentage Error
Follow these steps to calculate percent error by hand:
- Identify your values. Write down the theoretical value and the experimental value.
- Subtract. Compute Experimental Value − Theoretical Value.
- Take the absolute value. Drop any negative sign from the result.
- Divide. Divide the absolute difference by the absolute value of the theoretical value.
- Multiply by 100. Convert the decimal to a percentage.
That five-step process is exactly what the calculator automates.
Step-by-Step Solution Example Using Lab Results
Suppose your lab partner measured gravitational acceleration as 10.2 m/s² and the accepted value is 9.8 m/s².
- Theoretical value = 9.8 m/s². Experimental value = 10.2 m/s².
- 10.2 − 9.8 = 0.4
- |0.4| = 0.4
- 0.4 / |9.8| = 0.04082
- 0.04082 × 100 = 4.08%
The percent error for this measurement is approximately 4.08%. You can confirm this by entering both numbers into the calculator above.
Percentage Error vs. Percent Difference vs. Margin of Error
These three terms sound similar but answer different questions.
| Metric | What it answers | Reference point |
|---|---|---|
| Percent error | How far is my measurement from the true value? | One known (theoretical) value |
| Percent difference | How far apart are two measured values? | The average of the two values |
| Margin of error | How much could a survey or sample result vary? | Statistical confidence interval |
Percent error requires a true or accepted value. Percent difference does not; it compares two values of equal standing. Margin of error comes from statistics and sampling, not from lab measurement.
Relative Error, Absolute Error, and Approximation Error
- Absolute error is the simple absolute difference between the measured value and the true value. It keeps the original units (grams, meters, seconds).
- Relative error is the absolute error divided by the true value. It is a unitless ratio.
- Percent error is relative error multiplied by 100. It is the most common way to report relative error because percentages are easier to interpret at a glance.
Approximation error is a broader term that covers any deviation introduced by rounding, simplifying, or estimating. Percent error is one way to quantify approximation error.
When to Use a Percentage Calculator vs. a Percent Error Calculator
A general percentage calculator helps you find what percent one number is of another, compute percentage change, or work out tips and discounts. Use it for everyday math.
A percent error calculator is purpose-built for comparing a measurement to a known standard. If you have a theoretical value and an experimental value, use the percent error calculator. If you just need to find "what is 15% of 200," a standard percentage calculator is the right tool.
Accepted Value, Observed Value, and Measured Value
Different textbooks and fields use different names for the same two inputs. Here is a quick translation guide:
- Accepted value = theoretical value = true value = expected value. The established reference number.
- Observed value = measured value = experimental value. The number you recorded.
The percent error formula works identically regardless of which terminology your course uses. Plug the reference number into the "theoretical" slot and your result into the "experimental" slot.
True Value vs. Actual Value in Experiments
"True value" is the idealized, exact quantity that would be measured under perfect conditions. In practice, no measurement is perfect, so scientists rely on an "accepted value" determined by repeated, high-quality experiments.
"Actual value" is sometimes used casually to mean the accepted value, but it can also refer to the number you actually measured. Context matters. When calculating percent error, always clarify which number is your reference and which is your measurement.
Common Sources of Error in Measurement
Understanding why error occurs helps you reduce it. Errors generally fall into two categories: systematic (consistent bias in one direction) and random (unpredictable fluctuations).
Human Error, Accuracy, and Precision
Human error includes misreading a scale, starting a stopwatch late, or recording the wrong number. These mistakes are avoidable with careful technique.
Accuracy describes how close a measurement is to the true value. Precision describes how close repeated measurements are to each other. A measurement can be precise (consistent) but not accurate (biased), or accurate on average but not precise (scattered). Percent error specifically measures accuracy, not precision.
Significant Figures and Their Effect on Percentage Error
Rounding to fewer significant figures introduces rounding error into both your measurements and your percent error calculation. If your instrument reads to three significant figures, reporting your percent error to six decimal places implies false precision.
Match the significant figures in your final percent error to the least precise value in the calculation. This keeps your lab report honest and accurate.
Percent Error in Chemistry, Physics, and Lab Results
Percent error is one of the most commonly reported statistics in science lab reports. Instructors use it to assess whether experimental results are within an acceptable range of the known value.
In a chemistry or physics course, a percent error under 5% is often considered good for student-level equipment. However, acceptable thresholds depend on the experiment and the tools available.
Boiling Point, Gravitational Acceleration, and Speed of Sound Examples
Boiling point of water. The accepted value at standard pressure is 100 °C. If your thermometer reads 99.1 °C, the percent error is:
(|99.1 − 100| / 100) × 100 = 0.9%
Gravitational acceleration. The accepted value is 9.8 m/s². A free fall experiment that yields 9.6 m/s² gives:
(|9.6 − 9.8| / 9.8) × 100 ≈ 2.04%
Speed of sound in air at 20 °C. The accepted value is approximately 343 m/s. If your measurement is 338 m/s:
(|338 − 343| / 343) × 100 ≈ 1.46%
These examples show how percent error lets you compare accuracy across completely different experiments and units.
Frequently Asked Questions
Can Percentage Error Be Negative?
By the standard formula, no. The absolute value step makes percent error always positive. Some instructors and textbooks skip the absolute value to preserve the sign, which indicates whether the measurement was above or below the expected value. Unless your assignment specifically asks for a signed result, report percent error as a positive number.
What If the Theoretical Value Is Zero?
When the theoretical value is zero, the denominator becomes zero and the formula is undefined. You cannot divide by zero. In this case, percent error is not applicable.
Alternatives include reporting the absolute error directly or using percent difference if you are comparing two nonzero measured values. Ask your instructor which metric to use when the accepted value is zero.
How Is Percent Error Different from Standard Error?
Percent error compares one measurement (or one average) to a known true value. It uses the percent error formula described above.
Standard error is a statistical measure of how much a sample mean is expected to vary from the true population mean. It depends on standard deviation and sample size. The two metrics answer different questions and are not interchangeable.
How Do I Calculate Percent Error for Multiple Measurements?
First, find the mean (average) of your measurements. Then use that mean as the experimental value in the percent error formula, with the accepted value as the theoretical value.
- Add all measurements together.
- Divide by the number of measurements to get the mean.
- Apply the percent error formula: (|Mean − Theoretical Value| / |Theoretical Value|) × 100.
This approach gives a single percent error that represents the overall accuracy of your measurement set. For a sense of precision, also report the standard deviation of the individual measurements.