Percent Error Calculator

About Percent Error Calculator

How Does Percent Error Work?

Percent error measures how far an experimental (measured) value is from a known correct (theoretical or accepted) value, expressed as a percentage. The formula has three components, each building on the last:

Type	Formula	What It Measures
Absolute error	|Experimental - Theoretical|	The raw difference in the same units as the measurement
Relative error	|Experimental - Theoretical| / |Theoretical|	The error as a unitless fraction of the true value
Percent error	(|Experimental - Theoretical| / |Theoretical|) x 100%	The error as a percentage of the true value

The absolute value bars are important. Percent error is always positive (or zero) because the direction of the error does not matter for this calculation - only the magnitude. Some textbooks omit the absolute value bars and allow signed percent error, which can indicate whether the measurement was above or below the accepted value, but the unsigned version is far more common in lab reports.

Worked example: You measured the boiling point of water as 99.1 degrees C at sea level. The accepted value is 100.0 degrees C (at standard atmospheric pressure, 1 atm or 101.325 kPa).

• Absolute error: |99.1 - 100.0| = 0.9 degrees C
• Relative error: 0.9 / 100.0 = 0.009
• Percent error: 0.009 x 100% = 0.9%

A 0.9% error on a boiling point measurement is excellent for a student lab - the small deviation could come from altitude (boiling point drops roughly 0.5 degrees C per 150 m of elevation), impurities in the water, or thermometer calibration.

What Is a "Good" Percent Error?

The acceptable threshold depends entirely on what is being measured and what equipment is available. A 2% error on a student gravity experiment is solid work, but that same 2% on a pharmaceutical dosage measurement would be a serious problem. Here is a general guide:

Percent Error	Rating	Context
Under 1%	Excellent	Precision instruments, professional research labs, NIST-calibrated equipment
1 - 5%	Good	Student chemistry and physics labs with decent equipment, titrations, calorimetry
5 - 10%	Acceptable	Student labs with basic equipment, field measurements, biological experiments
10 - 20%	Poor	Rough estimates, low-precision instruments, first attempts at a new technique
Over 20%	High	Likely a systematic error, calculation mistake, or equipment malfunction

NIST Technical Note 1297 provides the formal framework for evaluating measurement uncertainty in professional settings. The guide distinguishes between Type A uncertainty (evaluated by statistical analysis of repeated observations) and Type B uncertainty (evaluated by other means, such as manufacturer specifications or calibration certificates). In student labs, percent error is the more practical metric since it gives a single number showing how close the result landed to the accepted value.

Percent Error vs Percent Difference vs Percent Change

These three formulas are often confused in lab reports and homework. Each answers a different question:

	Percent Error	Percent Difference	Percent Change
Compares	Measured value to a known true value	Two measured values to each other	An old value to a new value
Reference	The theoretical (accepted) value	The average of the two values	The old (original) value
Formula	|exp - theo| / |theo| x 100%	|a - b| / ((a+b)/2) x 100%	(new - old) / |old| x 100%
Use when	There is a known correct answer	Both values are measurements with no "correct" one	Tracking change over time
Can be negative?	No (uses absolute value)	No (uses absolute value)	Yes (decrease = negative)

The key distinction: percent error requires a known accepted value. If two students each measure the density of a metal sample and get 2.68 and 2.73 g/cm cubed, but neither knows the true density, they should use percent difference (with the average as the reference). If they later look up the accepted density of aluminium (2.70 g/cm cubed), they can each calculate their percent error.

Sources of Error in Lab Measurements

Understanding why percent error exists helps improve future measurements. Errors fall into three categories:

• Systematic errors: Consistent bias in one direction. Examples include a miscalibrated instrument that always reads 0.2 degrees high, parallax when reading a meniscus, or impure reagents that affect reaction yield. These cause percent error to be consistently positive or negative and cannot be reduced by repeating the measurement - the instrument or method itself needs correcting.
• Random errors: Unpredictable variation in both directions. Examples include temperature fluctuations during a reaction, reading precision limited by scale divisions, or timing a reaction by hand with a stopwatch. These cause scatter around the true value and can be reduced by averaging multiple trials. The standard deviation calculator helps quantify the spread of repeated measurements.
• Human errors (blunders): Mistakes like misreading a scale, spilling a solution, transposing digits when recording data, or using the wrong units. These are not true "errors" in the scientific sense - they are mistakes that should be identified and the affected data point discarded rather than averaged in.

A common exam question asks students to explain why their percent error is not zero. Good lab reports identify specific error sources and explain which direction they would push the result. Saying "human error" without specifics is generally marked down because it does not demonstrate understanding of the measurement process.

Reverse Mode: Finding the Acceptable Range

If you know the accepted value and the maximum acceptable percent error, you can work backwards to find the range of experimental values that would pass:

Range = Theoretical +/- (Percent Error / 100) x Theoretical

Example: The accepted density of aluminium is 2.70 g/cm cubed. A lab instructor allows a maximum 5% error.

• Tolerance: 5% x 2.70 = 0.135 g/cm cubed
• Acceptable range: 2.70 +/- 0.135 = 2.565 to 2.835 g/cm cubed
• Any measurement between 2.57 and 2.84 g/cm cubed (rounded to 3 significant figures) would be within 5%

This reverse calculation is useful when setting tolerances for manufacturing, quality control, or lab grading rubrics. A teacher grading 30 lab reports can quickly determine the acceptable range rather than calculating each student's percent error individually.

Common Lab Values and Expected Errors

These are standard measurements performed in introductory chemistry and physics courses. The accepted values come from NIST and other standards bodies:

Measurement	Accepted Value	Typical Student Result	Typical % Error
Boiling point of water	100.0 degrees C (at 1 atm)	99-101 degrees C	0-1%
Density of water (20 degrees C)	0.998 g/cm cubed	0.97-1.03 g/cm cubed	0-3%
Acceleration due to gravity	9.81 m/s squared	9.5-10.1 m/s squared	0-3%
Speed of sound in air (20 degrees C)	343 m/s	330-355 m/s	0-4%
Density of aluminium	2.70 g/cm cubed	2.6-2.8 g/cm cubed	0-4%
Specific heat of water	4.184 J/(g x degrees C)	3.9-4.4 J/(g x degrees C)	0-5%
Speed of light	2.998 x 10 to the 8 m/s	N/A (not measured in student labs)	N/A

The accepted value for gravitational acceleration (9.80665 m/s squared, often rounded to 9.81) is the standard defined by the 3rd General Conference on Weights and Measures. Local values vary slightly - from about 9.78 m/s squared at the equator to 9.83 m/s squared at the poles - so a measured value of 9.79 m/s squared might actually be more accurate than 9.81 for a lab near the equator.

When Is Percent Error Undefined?

Percent error cannot be calculated when the theoretical value is exactly zero. The formula requires dividing by the theoretical value, so a zero denominator makes the result undefined. This comes up in real experiments - for example, measuring the expected temperature change in a perfectly insulated system (theoretical change = 0) or testing whether a substance has zero conductivity.

In these cases, alternatives include reporting the absolute error directly, using percent difference (if comparing two experimental values), or reframing the question so the theoretical value is non-zero. Some fields use a modified formula with the average of both values in the denominator, which avoids the zero-division problem but is technically percent difference rather than percent error.

Error Propagation: Combining Multiple Measurements

Most real experiments involve several measured quantities combined in a formula. When calculating density, for example, both the mass and volume measurements carry uncertainty, and that uncertainty propagates into the final result. The standard approach uses the root-sum-of-squares (RSS) method:

For a result R that depends on measurements x and y, the relative uncertainty in R is approximately the square root of (relative uncertainty in x) squared plus (relative uncertainty in y) squared. This applies when x and y are multiplied or divided. For addition and subtraction, absolute uncertainties are combined instead.

Example: Measuring density where mass = 15.2 +/- 0.1 g and volume = 5.6 +/- 0.2 cm cubed:

• Relative uncertainty in mass: 0.1 / 15.2 = 0.66%
• Relative uncertainty in volume: 0.2 / 5.6 = 3.57%
• Combined relative uncertainty: square root of (0.66 squared + 3.57 squared) = 3.63%
• Density = 15.2 / 5.6 = 2.71 g/cm cubed, with about 3.6% uncertainty

Notice that the volume measurement dominates the uncertainty because its relative error is much larger. Improving the volume measurement (using a more precise graduated cylinder or a burette) would reduce the overall uncertainty more than improving the mass measurement. The variance calculator can help analyse the spread of repeated measurements when evaluating Type A uncertainty.

For handling the precision of individual measurements correctly, the significant figures calculator counts and rounds to the right number of sig figs. For more complex calculations, the equation solver handles algebraic equations step by step.

Sources

• Chemistry LibreTexts - Percent Error
• NIST Technical Note 1297 - Guidelines for Evaluating and Expressing Uncertainty
• NIST - Uncertainty of Measurement Results
• Math Is Fun - Percentage Difference vs Error
• UNC Physics - Measurement and Uncertainty Analysis Guide

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