Devin Alex

Measurement uncertainty is a non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand. It is an acknowledgment that no measurement is ever exact. Instead, every measurement is an estimate of the true value, and the uncertainty bounds the region in which the true value is believed to lie with a specified level of confidence. Understanding and quantifying this dispersion is the cornerstone of modern metrology.

The authoritative framework for evaluating and expressing measurement uncertainty is the Guide to the Expression of Uncertainty in Measurement (GUM), published by the Joint Committee for Guides in Metrology (JCGM). The GUM established a rigorous, statistically sound methodology that is universally applied across all scientific and industrial disciplines, ensuring consistency and comparability of measurement results worldwide.

The GUM Framework

The GUM methodology departs from the historical approach of classifying errors as strictly "random" or "systematic." Instead, it focuses on evaluating uncertainty components based on how they are evaluated. These components are grouped into two distinct categories: Type A and Type B.

Type A Evaluation

The evaluation of an uncertainty component by the statistical analysis of a series of observations. Learn about experimental standard deviation and degrees of freedom.

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Type B Evaluation

The evaluation of an uncertainty component by means other than statistical analysis. Learn about probability distributions and assigning limits.

Explore Type B

Combined and Expanded Uncertainty

Once all individual standard uncertainty components (both Type A and Type B) have been evaluated, they must be combined to represent the overall uncertainty of the measurement result.

The Combined Standard Uncertainty ($u_c$) is obtained by combining the individual standard uncertainties using the law of propagation of uncertainty, which is essentially the root-sum-square method for uncorrelated input quantities. This provides a standard deviation that characterizes the dispersion of the final measurement result.

However, for practical reporting, especially in calibration certificates and commercial transactions, it is necessary to define an interval around the measurement result that encompasses a large fraction of the distribution of values that could reasonably be attributed to the measurand.

This is achieved by calculating the Expanded Uncertainty ($U$). The expanded uncertainty is obtained by multiplying the combined standard uncertainty by a coverage factor ($k$).

The Coverage Factor ($k$)

The coverage factor $k$ is chosen based on the desired level of confidence. Assuming a normal (Gaussian) distribution of the measurement result:

A coverage factor of $k = 1$ provides a level of confidence of approximately 68.27%.
A coverage factor of $k = 2$ provides a level of confidence of approximately 95.45%. This is the standard practice in most industrial metrology and calibration applications.
A coverage factor of $k = 3$ provides a level of confidence of approximately 99.73%.

A complete measurement result is properly stated as: $Y = y \pm U$, accompanied by the coverage factor $k$ and the corresponding level of confidence.

Sensitivity Coefficients and the Uncertainty Budget

In most practical measurements, the final result (the measurand, $Y$) is not measured directly but is determined from a functional relationship involving several input quantities ($X_1, X_2, ..., X_N$):

$Y = f(X_1, X_2, \dots, X_N)$

Each input quantity has its own standard uncertainty ($u(x_i)$). To calculate the combined standard uncertainty of the output estimate $y$, one must understand how strongly a change in each input quantity affects the final result. This is quantified by the Sensitivity Coefficient ($c_i$).

Mathematically, the sensitivity coefficient is the partial derivative of the measurement function with respect to the specific input quantity, evaluated at the estimates of the input quantities:

$c_i = \frac{\partial f}{\partial x_i}$

The contribution of an individual input to the final combined uncertainty is the product of its standard uncertainty and its sensitivity coefficient: $u_i(y) = |c_i| \cdot u(x_i)$. The law of propagation of uncertainty (for uncorrelated inputs) then becomes a root-sum-square of these individual contributions:

$u_c(y) = \sqrt{\sum_{i=1}^{N} [c_i u(x_i)]^2}$

An Uncertainty Budget is a tabular summary of this entire process. It lists all sources of uncertainty, their standard uncertainties, probability distributions, sensitivity coefficients, degrees of freedom, and their final contribution to the combined standard uncertainty. It is an essential document for peer review and accreditation.

Correlated vs. Uncorrelated Input Quantities

The simplified root-sum-square equation above assumes that the input quantities are entirely independent (uncorrelated). However, if two input quantities are dependent upon each other (for example, if they were both measured using the same uncalibrated instrument) they are correlated.

When correlation exists, the law of propagation of uncertainty must include covariance terms. Covariance ($u(x_i, x_j)$) describes how two variables change together. If variables are positively correlated, their combined uncertainty is larger than if they were independent; if negatively correlated, the combined uncertainty can actually be smaller.

The complete equation including correlation is:

$u_c^2(y) = \sum_{i=1}^{N} c_i^2 u^2(x_i) + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} c_i c_j u(x_i, x_j)$

Identifying and correctly accounting for correlation is one of the more challenging and critical aspects of advanced uncertainty analysis.

The Welch-Satterthwaite Equation

When choosing the coverage factor $k$ to calculate expanded uncertainty, one must know the degrees of freedom associated with the combined standard uncertainty.

A Type A evaluation from $n$ repeated measurements has $n - 1$ degrees of freedom. A rigorous Type B evaluation often has infinite degrees of freedom (if the limits are considered absolutely certain). However, when combining multiple components with different degrees of freedom, we must calculate the Effective Degrees of Freedom ($\nu_{eff}$) of the combined standard uncertainty.

This is accomplished using the Welch-Satterthwaite equation:

$\nu_{eff} = \frac{u_c^4(y)}{\sum_{i=1}^{N} \frac{u_i^4(y)}{\nu_i}}$

Once $\nu_{eff}$ is calculated, the appropriate coverage factor $k$ for the desired confidence level (typically 95.45%) is found using the Student's t-distribution table. If $\nu_{eff}$ is large (e.g., > 30), $k \approx 2$ is usually a safe and standard approximation.

Monte Carlo Methods

GUM Supplement 1. Learn how the propagation of probability density functions (PDFs) overcomes the limitations of the Taylor series approximation.

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CMC Uncertainty Budgets

Learn how to identify errors, apply sensitivity coefficients, and construct a practical Measurement Uncertainty Budget for ISO/IEC 17025 accredited calibrations.

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Uncertainty Analysis