Percent Error & Uncertainty Calculator
Calculate percent error, absolute error, and measurement uncertainty.
Percent Error & Uncertainty Calculator
Quantify the accuracy of a measurement
Percent Error
Percent error is a measure of how close an experimental value is to a true or accepted value. It is calculated as the absolute difference between the two values, divided by the true value, and multiplied by 100%.
Understanding Percent Error & Uncertainty
Quantifying the Accuracy and Precision of Measurements.
What is Percent Error?
Percent Error is a measure of how close an experimental or measured value is to a true or accepted theoretical value. It is a common way to evaluate the accuracy of a measurement.
A low percent error indicates that the experimental result is very close to the accepted value, suggesting high accuracy.
A high percent error indicates a significant discrepancy between the measured value and the true value, suggesting a potential problem with the experimental method or measurements.
Example: If a scientist knows the true melting point of a substance and their experiment yields a slightly different value, percent error quantifies how far off their measurement was.
The Formula for Percent Error
The percent error is calculated using a simple formula:
Percent Error = | (Experimental Value - Theoretical Value) / Theoretical Value | * 100%
Where:
Experimental Value: The value you measured in your experiment.
Theoretical Value: The known or accepted true value.
The absolute value `|...|` is used to ensure the percent error is always a positive number.
Example:If the accepted density of aluminum is 2.70 g/cm³ and you measure it to be 2.65 g/cm³, you can use this formula to find the percentage by which you were off.
What is Percent Uncertainty?
Percent Uncertainty is a measure of the precision of a measurement. It expresses the magnitude of the uncertainty or doubt in a measurement as a percentage of the measurement itself.
It tells you how significant the uncertainty is relative to the size of the measurement.
A low percent uncertainty means a measurement is very precise.
The formula is: Percent Uncertainty = (Uncertainty / Measured Value) * 100%
Example:A measurement of (100 ± 1) cm has a 1% uncertainty. A measurement of (10 ± 1) cm has a 10% uncertainty. Even though the absolute uncertainty is the same (1 cm), it is much more significant for the smaller measurement.
Real-World Application: Engineering and Manufacturing
Both percent error and percent uncertainty are critical concepts in science and engineering.
Scientific Experiments: Scientists report percent error to validate their results against known values. They report percent uncertainty to indicate the precision of their measurement techniques.
Manufacturing: In manufacturing a product like a piston for a car engine, there is an accepted target dimension (theoretical value). The actual manufactured pistons will have slight variations. The percent error must be kept extremely low to ensure the parts fit and function correctly.
Polling and Statistics: A political poll might report a result of '45% ± 3%'. The '± 3%' is the margin of error or uncertainty, and it could be expressed as a percent uncertainty of the measurement itself to show its relative significance.
Example:An engineer designing a bridge must work with materials that have known properties and manufacturing tolerances (uncertainties) to ensure the final structure is safe.
Key Summary
- **Percent Error** measures **accuracy**—how close you are to the true value.
- **Percent Uncertainty** measures **precision**—the relative doubt in your measurement.
- Low percent error indicates high accuracy. Low percent uncertainty indicates high precision.
- Both are essential for validating experimental results and in quality control for engineering and manufacturing.
Practice Problems
Problem: A student measures the acceleration due to gravity to be 9.4 m/s². The accepted theoretical value is 9.8 m/s². What is the student's percent error?
Use the percent error formula: |(Experimental - Theoretical) / Theoretical| * 100%.
Solution: Percent Error = | (9.4 - 9.8) / 9.8 | * 100% = | -0.4 / 9.8 | * 100% ≈ 0.0408 * 100% ≈ 4.1%.
Problem: A chemist measures the mass of a sample to be 25.0 grams with an uncertainty of ±0.2 grams. What is the percent uncertainty of this measurement?
Use the percent uncertainty formula: (Uncertainty / Measured Value) * 100%.
Solution: Percent Uncertainty = (0.2 g / 25.0 g) * 100% = 0.008 * 100% = 0.8%.
Frequently Asked Questions
What is the difference between accuracy and precision?
Accuracy is how close a measurement is to the true or accepted value (measured by percent error). Precision is how close multiple measurements of the same thing are to each other (related to uncertainty). You can be very precise but inaccurate (all your shots hit the same spot, but it's far from the bullseye).
Can percent error be negative?
Because we use the absolute value in the standard formula, percent error is always reported as a positive number. Sometimes, the sign is kept to indicate the direction of the error (i.e., whether the experimental value was too high or too low), but the magnitude is what's most important.
How do I find the 'uncertainty' of a measurement?
The uncertainty can come from several sources. For a digital instrument, it's often listed in the manual or is assumed to be ± the last digit. For an analog scale (like a ruler), it's typically half of the smallest measurement increment. If you take multiple measurements, the uncertainty can be calculated statistically (e.g., using standard deviation).
Related Science Calculators
Error Propagation Calculator
Error Propagation - Perform scientific calculations with precision and accuracy.
Annealing Temperature
Annealing Temperature - Perform scientific calculations with precision and accuracy.
Antibiotic Stock
Antibiotic Stock - Perform scientific calculations with precision and accuracy.
Arrhenius Equation Calculator
Calculate reaction rate constants and activation energy.
Battery Energy Density Calculator
Calculate battery energy density, capacity, and performance metrics.
Beer–Lambert Law Calculator
Calculate absorbance, concentration, molar absorptivity, and path length.