Percent Error Calculator

What Is Percent Error?

Percent error — also called percentage error or approximation error — is a measure of how far an experimental (observed) value deviates from a theoretical (accepted) value. It expresses the discrepancy as a percentage of the true value, making it easy to judge the accuracy of a measurement regardless of scale.

If you measured the boiling point of water at 101.5°C in a lab experiment, and the accepted value is 100°C, your percent error is 1.5%. That single number tells a chemistry teacher, quality inspector, or research scientist exactly how accurate your measurement was — far more useful than saying you were "off by 1.5 degrees."

Percent error is one of the most fundamental percentage calculations used in science, engineering, and manufacturing in the United States. It appears in every high school and college chemistry and physics course, and it is a core metric in quality control programs regulated by the FDA, NIST, and EPA.

Our calculator above handles three modes: compute percent error from known values, solve for the experimental value given a target error, or solve for the theoretical value. The Batch Compare tab lets you enter multiple measurements at once and see which trial was most and least accurate.

The Percent Error Formula

The standard percent error formula is:

Or in compact notation:

Where:

Variable	Meaning	Other Names
E	Experimental (observed) value	Measured, observed, estimated, approximate
T	Theoretical (accepted) value	True, accepted, expected, exact, actual, known
|…|	Absolute value	Ensures the result is always non-negative

The absolute value in the numerator ensures that percent error is always reported as a positive number, regardless of whether the experimental value overshoots or undershoots the theoretical value. The order of subtraction does not matter: |E − T| = |T − E|.

Important: The formula is undefined when T = 0, because you cannot divide by zero. In such cases, use absolute error or mean absolute error instead.

How to Calculate Percent Error Step by Step

Follow these four steps to calculate percent error by hand or with our calculator:

1. Subtract the theoretical value from the experimental value: E − T
2. Take the absolute value of the result: |E − T|
3. Divide by the absolute value of the theoretical value: |E − T| / |T|
4. Multiply by 100 to convert to a percentage

Worked Example 1: Boiling Point of Water (Chemistry)

A student measures the boiling point of water at 101.5°C. The accepted value is 100.0°C.

1. Subtract: 101.5 − 100.0 = 1.5
2. Absolute value: |1.5| = 1.5
3. Divide: 1.5 ÷ 100.0 = 0.015
4. Multiply: 0.015 × 100 = 1.5%

The student's measurement has a 1.5% error — well within the <5% threshold typically expected in general chemistry labs.

Worked Example 2: Gravitational Acceleration (Physics)

A physics student measures gravitational acceleration as 9.72 m/s². The accepted value is 9.807 m/s².

1. Subtract: 9.72 − 9.807 = −0.087
2. Absolute value: |−0.087| = 0.087
3. Divide: 0.087 ÷ 9.807 = 0.008871
4. Multiply: 0.008871 × 100 = 0.887%

A 0.887% error in a gravitational acceleration experiment is excellent — well below the 2% threshold expected in introductory physics.

Worked Example 3: Density of Aluminum (Engineering)

An engineer measures the density of an aluminum sample as 2.75 g/cm³. The accepted density of aluminum is 2.70 g/cm³.

1. Subtract: 2.75 − 2.70 = 0.05
2. Absolute value: |0.05| = 0.05
3. Divide: 0.05 ÷ 2.70 = 0.01852
4. Multiply: 0.01852 × 100 = 1.852%

A 1.852% error would be acceptable for a classroom demonstration but might raise concerns in aerospace manufacturing where tolerances under 1% are required.

Percent Error vs. Percent Difference vs. Percent Change

These three percentage calculations are frequently confused. Here is how they differ:

Metric	Formula	When to Use
Percent Error	|E − T| / |T| × 100	When you know the true/accepted value and want to evaluate measurement accuracy
Percent Difference	|A − B| / ((A + B) / 2) × 100	When comparing two experimental values and neither is the "true" value
Percent Change	(New − Old) / |Old| × 100	When measuring change over time from an original value to a new value

Absolute Error vs. Relative Error vs. Percent Error

These three metrics form a hierarchy — each builds on the previous one:

Metric	Formula	Units	What It Tells You
Absolute Error	|E − T|	Same as measurement	The raw size of the error in the original units (e.g., 1.5°C, 0.087 m/s²)
Relative Error	|E − T| / |T|	Unitless (decimal)	The error as a fraction of the true value (e.g., 0.015)
Percent Error	|E − T| / |T| × 100	Percentage (%)	The relative error expressed as a percentage (e.g., 1.5%)

Absolute error is the most basic — it tells you the raw distance between your measurement and the truth. But a 1.5°C error means very different things when measuring the boiling point of water (100°C) versus the melting point of iron (1,538°C). Relative error normalizes the error to the true value, and percent error simply converts that to an easily understood percentage.

Acceptable Percent Error by Field

What counts as "acceptable" percent error depends entirely on the discipline and context. Here are general guidelines used in the United States:

Field	Acceptable % Error	Context
General Chemistry Lab	< 5%	High school and introductory college chemistry experiments
Analytical Chemistry	< 1%	Quantitative analysis, titrations, gravimetric analysis
Introductory Physics	< 5–10%	Mechanics, optics, and thermodynamics experiments
Advanced Physics	< 1–2%	Precision measurements (speed of sound, g, refractive index)
Mechanical Engineering	< 1%	Dimensional tolerances, material testing (per ASTM standards)
Pharmaceutical (FDA)	< 0.5–2%	Drug potency, dissolution testing (USP standards)
Environmental (EPA)	< 5–10%	Air and water quality monitoring, emissions calibration
NIST Calibration	< 0.01–0.1%	National measurement standards and reference materials
Survey/Polling	± 3–5%	Margin of error in public opinion polls
Manufacturing QC	< 0.5–1%	Production line sampling, Six Sigma processes

Common Mistakes and Misconceptions

1. Dividing by the experimental value: The formula divides by the theoretical (true) value, not the experimental one. Using the wrong denominator gives a different (incorrect) result.
2. Forgetting the absolute value: Without the absolute value, you might get a negative number. While a negative result can tell you the direction of error (underestimate vs. overestimate), the standard percent error formula always reports a positive value.
3. Confusing percent error with percent difference: Percent error requires a known true value. If you're comparing two measurements without a reference, use percent difference instead.
4. "Zero percent error" confusion: A 0% error means your measurement exactly matches the theoretical value. This is rare in practice; some error is always expected.
5. Theoretical value of zero: The formula is undefined when T = 0 because you cannot divide by zero. In these cases, report the absolute error instead.
6. Reporting too many decimal places: Match your percent error precision to the significant figures in your measurements. If you measured to 3 significant figures, report percent error to 3 significant figures.
7. Negative percent error: Some textbooks use a signed formula (without absolute value) where negative values indicate the measurement was below the true value. Always check which convention your instructor uses.

Real-World Applications in the United States

NIST Measurement Standards

The National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland, maintains the United States' national measurement standards. When NIST certifies a Standard Reference Material (SRM), the certificate states the percent error tolerance. Laboratories across the country calibrate their instruments against NIST standards, and percent error is the primary metric for evaluating whether a lab's equipment meets certification requirements.

FDA Pharmaceutical Testing

The U.S. Food and Drug Administration (FDA) requires pharmaceutical manufacturers to perform assay testing that measures the percent error of active ingredient concentration. Under USP (United States Pharmacopeia) guidelines, most drugs must have a potency within ±2% of the labeled amount. Tablets, capsules, and injectable solutions are routinely tested using percent error to ensure they contain the correct dosage.

EPA Environmental Monitoring

The Environmental Protection Agency (EPA) uses percent error as a quality control metric for air and water monitoring data. Under the Clean Air Act and Clean Water Act, continuous emissions monitoring systems (CEMS) must demonstrate calibration with percent errors typically below 5%. State environmental agencies report percent error for every calibration check submitted to the EPA.

Manufacturing and Engineering Quality Control

In automotive, aerospace, and precision manufacturing, percent error is a core metric in quality control. Engineers calculate percent error to evaluate whether machined parts meet dimensional tolerances specified by ASTM International and ISO standards. In Six Sigma methodology, percent error feeds into process capability analysis (Cpk) to determine whether production lines meet statistical quality targets.

Science Education in American Schools

Percent error is one of the most commonly taught concepts in AP Chemistry, AP Physics, and college-level science courses across the United States. Students calculate percent error in virtually every lab report, from measuring the mean value of repeated trials to comparing experimental results against textbook values. The College Board includes percent error in its AP Science curriculum frameworks.