Percent Error Formula
Percent error tells you how far a measured value strays from a known or accepted value. A lower percent error means your measurement was more accurate.
PE = |Measured - Actual| / |Actual| x 100
Solve for Measured Value
Given the accepted value and the percent error, find the range of possible measured values.
Measured = Actual +/- (PE/100) x |Actual|
Solve for Actual Value
Given a measured value and the percent error, work backward to find the accepted value.
Actual = Measured / (1 +/- PE/100)
How It Works
Percent error quantifies how far an experimental measurement deviates from a known or accepted value. The formula divides the absolute difference between the measured and accepted values by the absolute value of the accepted value, then multiplies by 100 to express the result as a percentage. A smaller percent error indicates a more accurate measurement, while a larger value signals systematic or random experimental problems.
Example Problem
A student measures the boiling point of water at 99.1 °C. The accepted value is 100.0 °C. Calculate the percent error of the measurement.
- Identify the measured value (99.1 °C) and the accepted value (100.0 °C).
- Subtract the measured value from the accepted value: 99.1 - 100.0 = -0.9.
- Take the absolute value of the difference: |-0.9| = 0.9.
- Divide the absolute difference by the absolute value of the accepted value: 0.9 / 100.0 = 0.009.
- Multiply by 100 to convert to a percentage: 0.009 x 100 = 0.9.
- The percent error is 0.9 %, meaning the student's measurement was very close to the true boiling point.
When to Use Each Variable
- Solve for Percent Error — when you have both a measured value and an accepted value, e.g., checking the accuracy of a lab measurement against a textbook constant.
- Solve for Measured Value — when you know the accepted value and want to find the measurement range corresponding to a given percent error tolerance.
- Solve for Actual Value — when you know the measured value and percent error and need to back-calculate the accepted standard.
Key Concepts
Percent error quantifies measurement accuracy by comparing a measured or experimental value to a known, accepted value. It uses the absolute difference divided by the accepted value, then multiplied by 100. The absolute value ensures the result is always non-negative. A lower percent error indicates greater measurement accuracy. Unlike percent difference (which compares two equal-status measurements), percent error requires one value to be the established truth — making it the standard metric in science labs, calibration, and quality control.
Applications
- Chemistry labs: evaluating student measurements of boiling points, densities, or molar masses against accepted values
- Manufacturing: verifying that machined parts fall within specified dimensional tolerances
- Physics experiments: assessing the accuracy of measurements like gravitational acceleration or speed of sound
- Calibration: quantifying instrument drift by comparing readings to a certified reference standard
- Pharmaceuticals: ensuring drug concentrations fall within regulatory acceptable error margins
Common Mistakes
- Using percent error when no accepted value exists — if both values are experimental, use percent difference instead
- Dividing by the measured value instead of the accepted value — percent error always uses the accepted (theoretical) value in the denominator
- Reporting a negative percent error — the standard formula uses absolute value, so the result should always be zero or positive
- Confusing percent error with relative error — relative error is the decimal fraction (without the x100 step), while percent error is expressed as a percentage
Frequently Asked Questions
What does percent error measure in an experiment?
Percent error measures the accuracy of an experimental measurement by expressing how far the measured value deviates from the known or accepted value as a percentage. It answers the question: relative to the true value, how large was your measurement error?
How do you determine if your measurement is accurate?
Calculate the percent error between your measured value and the accepted value. A lower percent error indicates higher accuracy. Acceptable thresholds depend on the field — under 5% is typical for student labs, under 1% for precision instruments, and under 0.1% for pharmaceutical or metrology applications.
What is the difference between percent error and percent difference?
Percent error compares a measurement to a known standard. Percent difference compares two equal-status measurements and divides by their average. Use percent error when one value is the accepted truth; use percent difference when neither value is more authoritative.
Is a percent error of 5% acceptable?
It depends on the field. In a chemistry lab, 5% is often acceptable for student experiments. In precision manufacturing or pharmaceutical work, tolerances of less than 1% are common. Always check the specific guidelines for your discipline.
Can percent error be over 100%?
Yes. If the measured value is more than double or less than zero compared to the accepted value, the percent error exceeds 100%. This usually indicates a major experimental problem such as a flawed procedure or instrument malfunction.
Why does the percent error formula use absolute value?
The absolute value ensures the result is always non-negative, focusing on the magnitude of the error rather than its direction. Whether you overshoot or undershoot the accepted value, the percent error is the same. Some fields use signed error to distinguish overestimation from underestimation, but the standard formula uses absolute value.
What happens if the actual (accepted) value is zero?
If the accepted value is zero, the percent error formula produces a division by zero and is undefined. In this case, alternative metrics like absolute error or relative error with a different reference point should be used instead.
Percent Error Formula
Percent error quantifies the accuracy of a measurement by comparing it to a known or accepted value:
PE = |Measured − Actual| / |Actual| × 100
Where:
- PE — percent error, expressed as a percentage (%)
- Measured — the experimentally obtained value (also called observed or experimental value)
- Actual — the accepted, theoretical, or true value
The absolute value in the numerator ensures percent error is always non-negative. The denominator uses the absolute value of the accepted value, which means the formula works correctly even when the accepted value is negative (e.g., temperatures below zero).
Worked Examples
Chemistry Lab
How accurate was the titration measurement?
A student titrates a solution and determines the concentration to be 0.118 M. The accepted value is 0.120 M. What is the percent error?
- Measured = 0.118 M, Actual = 0.120 M
- Difference: |0.118 − 0.120| = 0.002
- PE = (0.002 / 0.120) × 100
- PE = 1.6667 %
A percent error under 2% is excellent for a student titration. Common sources of error include imprecise endpoint detection and solution preparation.
Physics Experiment
How close was the pendulum gravity measurement to the accepted value?
Using a simple pendulum, a student measures gravitational acceleration as 9.72 m/s². The accepted value is 9.81 m/s². What is the percent error?
- Measured = 9.72 m/s², Actual = 9.81 m/s²
- Difference: |9.72 − 9.81| = 0.09
- PE = (0.09 / 9.81) × 100
- PE = 0.9174 %
Sub-1% error is strong for a pendulum experiment. Air resistance, string stretch, and timing precision are the main error sources.
Manufacturing QC
Does the machined part pass the tolerance check?
A machined shaft is measured at 25.08 mm. The design specification calls for 25.00 mm with a tolerance of 0.5% percent error. Does the part pass?
- Measured = 25.08 mm, Actual = 25.00 mm
- Difference: |25.08 − 25.00| = 0.08
- PE = (0.08 / 25.00) × 100
- PE = 0.32 %
At 0.32%, the part is within the 0.5% tolerance and passes QC. Parts exceeding the tolerance are rejected or reworked.
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