What is Measurement Uncertainty?
Measurement uncertainty is a parameter that characterizes the dispersion of values that could reasonably be attributed to a measured quantity. It quantifies the doubt about the result of any measurement and provides a range within which the true value is believed to lie with a specified probability.
Key Concept
No measurement is perfect. Every measurement result is only an estimate of the true value, and measurement uncertainty tells us how good that estimate is.
Why is Measurement Uncertainty Important?
For ISO 17025 Laboratories
ISO/IEC 17025:2017 Clause 7.6 requires laboratories to evaluate measurement uncertainty for all calibrations and specified types of tests.
For Decision Making
Uncertainty helps determine if measurement results are fit for purpose and supports critical decisions in quality control, research, and compliance.
Step-by-Step Guide to Calculating Measurement Uncertainty
Define the Measurand
Clearly identify what quantity you are measuring. The measurand is the specific quantity subject to measurement.
Example:
Example: Concentration of lead in drinking water, temperature of a calibration bath, mass of a reference weight.
Identify Uncertainty Sources
List all factors that could affect the measurement result. This includes equipment, environment, method, and operator.
Example:
Sources: Calibration uncertainty, resolution of instrument, temperature variations, operator technique, reference material uncertainty.
Quantify Type A Uncertainty
Calculate from repeated measurements under repeatability conditions using statistical methods.
Example:
Take 10 repeated measurements, calculate mean and standard deviation. u_A = s/√n
Quantify Type B Uncertainty
Evaluate uncertainties from calibration certificates, specifications, manufacturer data, or other information.
Example:
From calibration certificate: U = 0.5°C, k = 2 → u_B = 0.5/2 = 0.25°C
Combine Uncertainties
Calculate the combined standard uncertainty using the root sum of squares method.
Example:
u_c = √(u₁² + u₂² + u₃² + ...) where all uncertainties are in the same units
Calculate Expanded Uncertainty
Multiply the combined standard uncertainty by a coverage factor to obtain the expanded uncertainty.
Example:
For 95% confidence with sufficient degrees of freedom: U = 2 × u_c
Report Results
Present the measurement result with its uncertainty and the confidence level.
Example:
Result: 25.4°C ± 0.5°C where the reported uncertainty is an expanded uncertainty with a coverage factor k=2 corresponding to a 95% confidence level.
Type A vs Type B Uncertainty
Statistical Evaluation
Evaluation Method
Based on statistical analysis of series of observations
Common Sources
- Repeatability of measurements
- Reproducibility between operators
- Random effects in the measurement process
Calculation
u_A = s/√nwhere s = standard deviation, n = number of measurements
Non-Statistical Evaluation
Evaluation Method
Based on scientific judgment using all available information
Common Sources
- Calibration certificates
- Manufacturer specifications
- Reference data from handbooks
- Resolution of instruments
Calculation (Rectangular Distribution)
u_B = a/√3where a = half-range of possible values
Important Note
Both Type A and Type B uncertainties are expressed as standard uncertainties (with the same units as the measurand) and are combined in the same way. The distinction is only in how they are evaluated, not in how they are used or combined.
Key Measurement Uncertainty Formulas
Combined Standard Uncertainty
uc(y) = √[∑ (∂f/∂xᵢ)² · u²(xᵢ)]Where:
- uc(y) = combined standard uncertainty of measurement result y
- ∂f/∂xᵢ = sensitivity coefficient (partial derivative)
- u(xᵢ) = standard uncertainty of input quantity xᵢ
Expanded Uncertainty
U = k · uc(y)Where:
- U = expanded uncertainty
- k = coverage factor (typically 2 for 95% confidence)
- uc(y) = combined standard uncertainty
The coverage factor k is determined from the t-distribution based on the effective degrees of freedom.
Standard Deviation
s = √[∑(xᵢ - x̄)²/(n-1)]Measure of dispersion in repeated measurements
Type A Uncertainty
u_A = s/√nStandard uncertainty from repeated observations
Rectangular Distribution
u_B = a/√3When only maximum bounds are known
Real-World Measurement Uncertainty Examples
Example 1: Temperature Measurement
Scenario: Measuring temperature with a calibrated thermometer
Combined uncertainty: u_c = √(0.25² + 0.058² + 0.095²) = 0.273°C
Expanded uncertainty (k=2): U = 2 × 0.273 = 0.55°C
Final result: T = 25.4°C ± 0.6°C (k=2, 95% confidence)
Example 2: Mass Measurement
Scenario: Weighing a sample on an analytical balance
Combined uncertainty: u_c = √(0.1² + 0.058² + 0.061²) = 0.132 mg
Expanded uncertainty (k=2): U = 2 × 0.132 = 0.26 mg
Final result: m = 250.0 mg ± 0.3 mg (k=2, 95% confidence)
Calculate Measurement Uncertainty Easily
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Frequently Asked Questions
What is measurement uncertainty?
Measurement uncertainty is a quantitative indicator of the quality of measurement results. It represents the doubt about the validity of a measurement result and is expressed as a range within which the true value is believed to lie with a specified level of confidence.
What is the GUM method?
The GUM (Guide to the Expression of Uncertainty in Measurement) is an international standard (JCGM 100:2008) that provides a framework for evaluating and expressing uncertainty in measurement. It's the most widely accepted methodology used by accredited laboratories worldwide.
What is Type A uncertainty?
Type A uncertainty evaluation is based on statistical analysis of repeated observations. It's calculated from experimental data using standard deviation and is used when you have multiple measurements of the same quantity under repeatability conditions.
What is Type B uncertainty?
Type B uncertainty evaluation is based on scientific judgment using all available information, such as calibration certificates, manufacturer specifications, reference data, or previous measurement data. It doesn't rely on statistical analysis of current observations.
How is expanded uncertainty calculated?
Expanded uncertainty (U) is calculated by multiplying the combined standard uncertainty (u_c) by a coverage factor (k). For a 95% confidence level with sufficient degrees of freedom, k is typically 2. U = k × u_c.
Why is uncertainty calculation important for ISO 17025?
ISO/IEC 17025:2017 Clause 7.6 requires testing and calibration laboratories to evaluate measurement uncertainty for all calibrations and some types of tests. Proper uncertainty evaluation is essential for demonstrating technical competence and ensuring reliable measurement results.