Short Answer
Measurement uncertainty is often misunderstood, even by experienced engineers and scientists. It is not the same as an error, nor is it a simple tolerance range. In this comprehensive guide, we break down what measurement uncertainty actually is, how to evaluate it according to international standards, and how to report it correctly—so you can avoid the pitfalls that lead to incorrect conclusions and non-compliant results.
Rule Statement
The formal definition of measurement uncertainty, as given in the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008, clause 2.2.3), is:
“Non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used.”
This definition emphasizes three key aspects:
- Dispersion – uncertainty is a spread, not a single point.
- Quantity values – it applies to the values of a measurand, not to the measurement process itself.
- Based on the information used – uncertainty depends on the available knowledge, including both statistical data and scientific judgment.
Uncertainty is not an error. An error is the difference between a measured value and the true value; it is a single, often unknown quantity. Uncertainty, on the other hand, is a parameter that describes the range within which the true value is expected to lie, with a given level of confidence. It is a measure of doubt, not a mistake.
Worked Examples
Example 1: Simple Length Measurement
Suppose you measure the length of a metal rod using a ruler with 1 mm divisions. You take five readings: 152 mm, 153 mm, 152 mm, 154 mm, 153 mm.
Step 1: Evaluate Type A uncertainty (statistical)
Calculate the mean: (152+153+152+154+153)/5 = 152.8 mm. The standard deviation of the mean (standard uncertainty) is the experimental standard deviation divided by √5. The sample standard deviation is approximately 0.84 mm, so the standard uncertainty is 0.84/√5 ≈ 0.38 mm.
Step 2: Evaluate Type B uncertainty (non-statistical)
The ruler’s resolution is 1 mm. Assuming a rectangular distribution, the standard uncertainty from resolution is 1/(2√3) ≈ 0.29 mm. If the ruler has a calibration certificate stating an expanded uncertainty of 0.2 mm (k=2), the standard uncertainty is 0.1 mm. Combine these Type B components: √(0.29² + 0.1²) ≈ 0.31 mm.
Step 3: Combine standard uncertainties
Combine Type A and Type B: √(0.38² + 0.31²) ≈ 0.49 mm. This is the combined standard uncertainty, uc.
Step 4: Expanded uncertainty
For a 95% confidence level, use a coverage factor k=2 (assuming normal distribution). Expanded uncertainty U = 2 × 0.49 ≈ 0.98 mm. Report as 152.8 ± 1.0 mm (k=2).
Example 2: Calibration of a Thermometer
A thermometer is calibrated against a reference standard. The reference standard has a certified uncertainty of 0.05 °C (k=2). The thermometer is read 10 times, giving a standard deviation of 0.07 °C. The resolution of the thermometer is 0.1 °C.
Type A: uA = 0.07/√10 ≈ 0.022 °C.
Type B: From reference standard: 0.05/2 = 0.025 °C. From resolution: 0.1/(2√3) ≈ 0.029 °C. Combined Type B: √(0.025² + 0.029²) ≈ 0.038 °C.
Combined standard uncertainty: √(0.022² + 0.038²) ≈ 0.044 °C. Expanded (k=2): 0.088 °C, report as 0.09 °C.
Counter-Examples
Treating Uncertainty as an Error
Wrong: “The measurement uncertainty is ±0.5 mm, so the true value could be 0.5 mm away.” This implies the uncertainty is an error bound, but uncertainty is a probabilistic dispersion, not a hard limit.
Using Tolerance Instead of Uncertainty
Wrong: “The part has a tolerance of ±0.1 mm, so the measurement uncertainty is ±0.1 mm.” Tolerance is a specification for the product, not a property of the measurement. Uncertainty must be evaluated from the measurement process itself.
Ignoring Type B Contributions
Wrong: “I only report the standard deviation of my readings as the uncertainty.” This neglects systematic effects like calibration, resolution, and environmental factors. The GUM requires a complete uncertainty budget.
Convention Comparison Table
| Aspect | GUM (JCGM 100:2008) | NIST TN 1297 | ISO 5725 | ASTM E29 |
|---|---|---|---|---|
| Definition of uncertainty | Parameter characterizing dispersion | Same as GUM | Parameter associated with a result that characterizes the dispersion | Not explicitly defined; focuses on rounding and significant figures |
| Coverage factor | k=2 for 95% (default) | k=2 for 95% (recommended) | k=2 for 95% (common) | Not applicable |
| Reporting significant digits | Uncertainty to 2 significant digits; result rounded to match | Same as GUM | Uncertainty to 1 or 2 significant digits | Rules for rounding test results to a specified number of digits |
| Type A vs Type B | Explicitly defined | Explicitly defined | Uses repeatability and reproducibility | Not applicable |
| Combining components | Root-sum-square (RSS) of standard uncertainties | Same as GUM | ANOVA-based | Not applicable |
Standards Citation
The following standards and guides provide the authoritative basis for evaluating and reporting measurement uncertainty:
- GUM (JCGM 100:2008) – Evaluation of measurement data — Guide to the expression of uncertainty in measurement. Clauses 2.2.3 (definition), 4.2 (Type A), 4.3 (Type B), 5.1 (combined standard uncertainty), 6.2 (expanded uncertainty).
- NIST TN 1297 – Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. Provides practical implementation of GUM, with specific guidance on significant figures (Section 7).
- ISO 5725-1 – Accuracy (trueness and precision) of measurement methods and results. Defines repeatability and reproducibility, which are components of uncertainty.
- ASTM E29 – Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. Governs rounding of test results, which is directly related to how uncertainty is reported.
Common Mistakes
- Omitting the coverage factor – Reporting “± 0.5” without specifying k or confidence level is meaningless.
- Rounding the result before the uncertainty – Always round the uncertainty first, then round the measured value to the same decimal place.
- Using the standard deviation of the sample instead of the standard deviation of the mean – The uncertainty of the mean is smaller than the spread of individual readings.
- Forgetting to include resolution effects – Every digital or analog instrument contributes a Type B uncertainty from its resolution.
- Double-counting components – For example, including both a calibration correction and its uncertainty when the correction is already applied.
Practice Problems
Problem 1: A balance has a calibration certificate stating an expanded uncertainty of 0.02 g (k=2). The balance’s resolution is 0.01 g. You weigh a sample 4 times: 10.23, 10.25, 10.22, 10.24 g. Compute the combined standard uncertainty and report the result with expanded uncertainty (k=2).
Solution: Type A: mean = 10.235 g, sample std = 0.0129 g, uA = 0.0129/√4 = 0.00645 g. Type B: calibration: 0.02/2 = 0.01 g; resolution: 0.01/(2√3) ≈ 0.00289 g. Combined Type B: √(0.01² + 0.00289²) ≈ 0.0104 g. Combined standard: √(0.00645² + 0.0104²) ≈ 0.0122 g. Expanded: 0.0244 g, round to 0.024 g. Result: 10.235 ± 0.024 g (k=2).
Problem 2: A pH meter has a stated accuracy of ±0.02 pH units. You take 3 readings: 7.01, 7.02, 7.01. Estimate the standard uncertainty (assume the accuracy is a rectangular distribution).
Solution: Type A: mean = 7.0133, sample std = 0.00577, uA = 0.00577/√3 ≈ 0.00333. Type B: accuracy as rectangular: 0.02/√3 ≈ 0.01155. Combined: √(0.00333² + 0.01155²) ≈ 0.0120 pH.
Quick Reference Table
| Symbol | Term | Definition / Formula |
|---|---|---|
| u | Standard uncertainty | Uncertainty expressed as a standard deviation |
| uA | Type A uncertainty | Evaluated by statistical analysis of repeated observations |
| uB | Type B uncertainty | Evaluated by other means (calibration, resolution, etc.) |
| uc | Combined standard uncertainty | √(Σ ui²) for independent components |
| U | Expanded uncertainty | U = k × uc |
| k | Coverage factor | Typically 2 for 95% confidence |
Sources & Further Reading
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
- NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
- ISO 5725-1:1994, Accuracy (trueness and precision) of measurement methods and results.
- ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
For more on rounding and significant figures, see our Significant Figures Rules and Rounding Methods articles.
FAQ
What is the difference between uncertainty and error?
Error is the difference between a measured value and the true value; it is a single, often unknown quantity. Uncertainty is a parameter that characterizes the dispersion of values that could reasonably be attributed to the measurand. Error is a concept; uncertainty is a quantitative expression of doubt.
How many significant figures should I use for uncertainty?
According to NIST TN 1297 and GUM, report uncertainty to two significant figures, then round the measured value to the same decimal place. For example, if uncertainty is 0.0234, report 0.023, and if the measured value is 12.3456, report 12.346.
Can uncertainty be zero?
No. Every measurement has some uncertainty, even if it is extremely small. The resolution of the instrument alone introduces a non-zero uncertainty. A zero uncertainty would imply perfect knowledge, which is impossible in practice.
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