Sensitivity Coefficients in Uncertainty Budgets

From the blog of https://www.engineering.com/sensitivity-coefficients-in-uncertainty-budgets/

Introduction to Sensitivity Coefficients

Key Concept

When analyzing measurement uncertainty, sensitivity coefficients tell us how much an error in an input affects the final result.

Simple Case: Direct Measurements

For a measurement with three error sources:

\[Y = y + x_1 + x_2 + x_3\]

where:

• Y is the measurement result

• y is the true value (unknown)

• x₁, x₂, x₃ are errors from different sources

In this case, sensitivity coefficients = 1 because each error directly maps to the result.

The law of propagation of uncertainty states:

\[U_C = \sqrt{\sum_{i=1}^{n} c_i^2u_i^2}\]

where:

• U_C is combined uncertainty

• c_i is sensitivity coefficient for each source

• u_i is standard uncertainty for each source

Practical Example: Building Height Measurement

Fig. 1 Measuring building height using clinometer and tape measure

The height H is calculated as:

\[H = h_1 + L \times \tan(\theta)\]

where:

• h₁ = height of clinometer

• L = horizontal distance

• θ = measured angle

Sensitivity Coefficients

1. For clinometer height (h₁):

   \[\frac{\partial H}{\partial h_1} = 1\]

2. For horizontal distance (L):

   \[\frac{\partial H}{\partial L} = \tan(\theta)\]

3. For angle (θ):

   \[\frac{\partial H}{\partial \theta} = L \times \sec^2(\theta)\]

Example Calculations

For measurements:

• h₁ = 1.65 m

• L = 10 m

• θ = 58°

Results:

• ΔL = 10mm → ΔH = 16mm (sensitivity = 1.6)

• Δθ = 0.5° → ΔH = 316mm (sensitivity = 632 mm/deg)

Special Case: Temperature Effects

For thermal expansion:

\[\frac{\partial L}{\partial T} = \alpha L\]

Where:

• α = coefficient of thermal expansion (e.g., 12×10⁻⁶/°C for steel)

• L = measured length

Fig. 2 Non-linear sensitivity in cosine error

Key Points to Remember

1. Sensitivity coefficients = 1 when:

   • Measuring directly

   • Using Type A evaluations (repeatability studies)

2. Must calculate sensitivity coefficients when:

   • Combining multiple measurements mathematically

   • Dealing with temperature effects

   • Working with angular measurements

   • Considering environmental influences

3. Units must be consistent:

   • Dimensionless when input/output units match

   • Include proper conversion when units differ

Best Practice

When sensitivity is not constant, evaluate at the uncertainty value to get worst-case scenario, but be careful to:

1. Check if sensitivity keeps increasing with error

2. Consider using expanded uncertainty value

Uncertainty Budget Example

Source	Value	Distribution	Divisor	Sensitivity	Standard Uncertainty
L	±50.5mm	Normal (95%)	2	1.6	40.4mm
h₁	±8.75mm	Normal (95%)	2	1.0	4.4mm
θ	±1°	Normal	1	632mm/deg	632mm

The combined standard uncertainty:

\[u_c = \sqrt{(40.4)^2 + (4.4)^2 + (632)^2} = 633.4\text{ mm}\]