Rule Statement

Let measured be the experimentally obtained value and true be the accepted or reference value. The three error metrics are defined as follows:

• Absolute Error: |measured − true| — the magnitude of the difference, expressed in the same units as the measurement.
• Relative Error: |measured − true| / |true| — a dimensionless ratio that normalizes the error by the true value.
• Percent Error: relative error × 100% — a scaled version of relative error for intuitive communication.

All three metrics assume the true value is known. In practice, the true value may be a reference standard, a theoretical value, or a highly accurate measurement. When the true value is zero, relative and percent error are undefined; only absolute error remains meaningful.

Worked Examples

Example 1: Small-Scale Measurement

True value = 10.0, measured = 10.5.

1. Absolute error = |10.5 − 10.0| = 0.5
2. Relative error = 0.5 / 10.0 = 0.05
3. Percent error = 0.05 × 100% = 5%

Example 2: Large-Scale Measurement

True value = 100.0, measured = 105.0.

1. Absolute error = |105.0 − 100.0| = 5.0
2. Relative error = 5.0 / 100.0 = 0.05
3. Percent error = 5%

Notice that the relative and percent errors are identical in both examples, while the absolute error differs by a factor of ten. This illustrates why relative error is preferred when comparing measurements of different magnitudes.

Example 3: Small True Value

True value = 0.1, measured = 0.2.

1. Absolute error = 0.1
2. Relative error = 0.1 / 0.1 = 1.0
3. Percent error = 100%

Here, a small absolute error (0.1) corresponds to a 100% relative error, demonstrating that relative error amplifies when the true value is small.

Counter-Examples

Common misinterpretations can lead to incorrect conclusions. Consider the following counter-examples:

• Comparing absolute errors across different units: A length measurement of 1 cm ± 0.1 cm and a mass measurement of 1 kg ± 0.1 kg both have an absolute error of 0.1, but the units differ. Comparing these absolute errors is meaningless; relative errors (10% vs 0.01%) are appropriate.
• Using percent error when true value is near zero: If true = 0.01 and measured = 0.02, the percent error is 100%, but the absolute error is only 0.01. In many applications, the absolute error is more relevant than the percent error when the true value is tiny.
• Ignoring the sign of error: Some fields use signed error (measured − true) to indicate bias. While absolute error discards sign, signed error is useful for detecting systematic over- or under-estimation. Always clarify whether you are reporting magnitude or signed deviation.

Convention Comparison Table

Standards Citation

The Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008) is the de facto standard for expressing measurement uncertainty. Although GUM focuses on uncertainty rather than error, it defines related concepts. Section 5.1.6 states: “The relative standard uncertainty is the standard uncertainty divided by the absolute value of the measured quantity.” This aligns with the definition of relative error when the error is treated as an approximation of uncertainty.

“The error of a measurement is the measured value minus the true value.” — GUM, Section 2.16

Additionally, NIST Technical Note 1297 (1994) provides guidelines for evaluating and expressing uncertainty. Section 5.1 discusses the use of relative standard uncertainty, reinforcing the importance of relative measures in metrology. For industrial applications, ISO 5725-1 (Accuracy and precision of measurement methods) defines terms like trueness and precision, which are directly related to error analysis.

Common Mistakes

• Using absolute error to compare measurements of different magnitudes: Always use relative or percent error when scales differ.
• Dividing by zero: Relative and percent error are undefined when the true value is zero. Report absolute error instead.
• Forgetting absolute value: Error is typically reported as a magnitude; omitting the absolute value can lead to negative errors that cancel out in averaging.
• Rounding intermediate results: Always compute errors with full precision and round only the final answer to the appropriate number of significant figures.
• Confusing error with uncertainty: Error is the difference from a true value; uncertainty characterizes the dispersion of measurements. They are related but not identical.

Practice Problems

Problem 1

A balance reads 12.34 g for a standard mass of 12.30 g. Calculate the absolute, relative, and percent error.

Solution: Absolute = |12.34 − 12.30| = 0.04 g; Relative = 0.04 / 12.30 ≈ 0.00325; Percent ≈ 0.325%.

Problem 2

True value = 250.0, measured = 255.0. What is the percent error?

Solution: Absolute = 5.0; Relative = 5.0 / 250.0 = 0.02; Percent = 2%.

Problem 3

Why is relative error undefined when the true value is zero? Provide an alternative.

Solution: Division by zero is undefined. Use absolute error or a different metric like standard deviation if appropriate.

Quick Reference Table

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 1994.
• ISO 5725-1:1994, Accuracy (trueness and precision) of measurement methods and results — Part 1: General principles and definitions.
• IUPAC, Compendium of Analytical Nomenclature, 3rd ed., 1998.

For further reading on related topics, see our articles on error propagation and measurement uncertainty.

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