Propagation, Covariance, and Effective Degrees of Freedom

Once all individual standard uncertainties—both Type A and Type B—have been quantified and converted into equivalent standard deviations ($u(x_i)$), the metrologist must combine them into a single comprehensive metric: the combined standard uncertainty ($u_c(y)$).

This synthesis is governed by GUM Clause 5.1, known formally as the Law of Propagation of Uncertainty.

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The Law of Propagation of Uncertainty

In metrology, the measurand $Y$ is rarely measured directly; it is typically determined from $N$ other input quantities $X_1, X_2, \dots, X_N$ through a defined functional relationship:

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

Sensitivity Coefficients ($c_i$)

Each input quantity $X_i$ affects the output $Y$ according to its partial derivative, termed the sensitivity coefficient ($c_i$):

$$c_i = \frac{\partial f}{\partial x_i} = \left. \frac{\partial f}{\partial X_i} \right|_{X_1 = x_1, X_2 = x_2, \dots, X_N = x_N}$$

The sensitivity coefficient describes how much the output estimate $y$ changes per unit change in input estimate $x_i$. The uncertainty contribution from component $i$ to the output is:

$$u_i(y) = |c_i| \cdot u(x_i)$$

The Root Sum Square (RSS) Method for Uncorrelated Inputs

When all input quantities are mutually uncorrelated in the stated first-order model, the first-order Taylor series approximation of the combined variance $u_c^2(y)$ is the sum of the squared individual contributions:

$$u_c^2(y) = \sum_{i=1}^N \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i) = \sum_{i=1}^N c_i^2 u^2(x_i) = \sum_{i=1}^N u_i^2(y)$$

Taking the square root yields the celebrated Root Sum Square (RSS) formula:

$$u_c(y) = \sqrt{\sum_{i=1}^N [c_i \cdot u(x_i)]^2} = \sqrt{u_1^2(y) + u_2^2(y) + \dots + u_N^2(y)}$$

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Correlated Inputs and Covariance Terms

When two or more input quantities are physically or metrologically dependent, the simple RSS formula is insufficient. Examples of correlation include:

• Two reference instruments calibrated against the same master standard during the same calibration run
• Two temperature probes sharing the same digital readout thermometer
• Gage blocks evaluated using the same mechanical comparator whose thermal expansion drifts with ambient room temperature

The General Propagation Equation with Covariance

Under GUM Clause 5.2, when correlation exists between input quantities $X_i$ and $X_j$, the combined variance must include cross-product covariance terms:

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

Where $u(x_i, x_j)$ is the estimated covariance, defined in terms of the correlation coefficient $r(x_i, x_j)$:

$$u(x_i, x_j) = r(x_i, x_j) \cdot u(x_i) \cdot u(x_j), \quad \text{where } -1 \le r(x_i, x_j) \le +1$$

Metrological Impact of Correlation

Consider a differential measurement where $Y = X_1 - X_2$ ($c_1 = +1, c_2 = -1$):

$$u_c^2(y) = u^2(x_1) + u^2(x_2) - 2 r(x_1, x_2) u(x_1) u(x_2)$$

• Positive Correlation ($r > 0$): If both instruments drift in the same direction due to a common temperature change, the covariance term is subtracted, reducing the combined uncertainty! In the limit where $u(x_1) = u(x_2)$ and $r = +1$, the common-mode uncertainty completely cancels ($u_c(y) = 0$).
• Negative Correlation ($r < 0$): If the errors move in opposite directions, the negative signs multiply to form a positive addition, increasing the combined uncertainty beyond the uncorrelated RSS value.

Effective Degrees of Freedom: The Welch-Satterthwaite Equation

Degrees of freedom can be added for appropriate pooled independent variance estimates, but not arbitrarily for every uncertainty model. However, when combining variances from distributions with vastly different magnitudes and degrees of freedom, simple addition fails.

To determine the reliability of the combined standard uncertainty $u_c(y)$, metrologists compute the effective degrees of freedom ($\nu_{\text{eff}}$) using the Welch-Satterthwaite equation:

$$\nu_{\text{eff}} = \frac{u_c^4(y)}{\sum_{i=1}^N \frac{u_i^4(y)}{\nu_i}} = \frac{u_c^4(y)}{\sum_{i=1}^N \frac{[c_i u(x_i)]^4}{\nu_i}}$$

Where:

• $u_c(y)$ is the combined standard uncertainty
• $u_i(y) = c_i u(x_i)$ is the individual uncertainty contribution from input $i$
• $\nu_i$ is the degrees of freedom associated with component $i$

Determining Degrees of Freedom ($\nu_i$) for Budget Components

1. Type A Components: For a series of $n$ repeated measurements, $\nu_i = n - 1$. For pooled standard deviations, $\nu_i = \sum (n_j - 1)$.
2. Type B Components (Standard Practice): When a Type B limit is derived from a limit or uncertainty estimate whose reliability justifies negligible uncertainty in that estimate (do not assume this from an NMI name alone), the relative uncertainty of the uncertainty is essentially zero, and the degrees of freedom are taken as infinite:

$$\nu_i \to \infty \implies \frac{u_i^4}{\nu_i} = \frac{u_i^4}{\infty} = 0$$

1. Type B Components with Subjective Limits: If a technician estimates a Type B limit with an estimated relative uncertainty $R = \Delta u(x_i) / u(x_i)$ (for example, 25% relative uncertainty in the estimated standard uncertainty), GUM formula G.3 provides:

$$\nu_i \approx \frac{1}{2} \left( \frac{\Delta u(x_i)}{u(x_i)} \right)^{-2} = \frac{1}{2 (0.25)^2} = \frac{1}{2 \times 0.0625} = 8$$

Worked covariance comparison

Suppose two input estimates each have the same output units, with standard uncertainties of 3 units and 4 units. If the output is their difference and they are uncorrelated, variance is nine plus sixteen, giving a standard uncertainty of five units. With a supported correlation coefficient of positive one-half, the covariance term subtracts twelve square units. Variance becomes thirteen and the standard uncertainty is about 3.606 units.

If the output were the sum, both sensitivity coefficients would be positive. The same covariance would add twelve square units, producing variance thirty-seven and standard uncertainty about 6.083 units. Correlation therefore cannot be described as always increasing or always reducing uncertainty; its effect depends on both the correlation and the signs of the sensitivities.

Perfect cancellation in an ideal difference applies only to the modeled common contribution. Independent resolution, repeatability, imperfect matching, or other residual effects can remain. Establish covariance from suitable information rather than assuming that instruments from the same supplier share identical errors. Ordinary component-wise Welch–Satterthwaite calculations also need review when correlated variance estimates are involved.