DAY 15 — Combined Standard Uncertainty

What is the main topic of Day 15?

Combined Standard Uncertainty

How is combined standard uncertainty defined?

Combined standard uncertainty u_c is the standard uncertainty obtained by combining the standard uncertainties associated with the input quantities in a measurement model

Which VIM reference is given for combined standard uncertainty?

VIM3 2.31

What are examples of input quantities that may affect a measurement result?

Reference standard UUT indication repeatability resolution environmental effects corrections and other relevant inputs

What must happen to each uncertainty component before combination?

Each component is evaluated as a standard uncertainty before combination

What does the measurement model describe?

The measurement model describes the mathematical relationship between the measurand and the input quantities

What is the general measurement model shown?

Y = f(X₁ X₂ ... Xₙ)

What is the simple calibration model shown?

Y = X_ref + C

In the simple calibration model what does Y represent?

The measurement result

In the simple calibration model what does X_ref represent?

The reference value

In the simple calibration model what does C represent?

The correction

What does a sensitivity coefficient show?

It shows how much the result changes when an input quantity changes

What is the sensitivity coefficient formula?

c_i = ∂f/∂X_i

What question can be used to understand a sensitivity coefficient?

How strongly does this input affect my result?

What reference is given for the sensitivity coefficient?

GUM JCGM 100:2008

How is the contribution from each input expressed?

u_i(y) = |c_i|u(x_i)

Does a large input uncertainty always mean a large contribution?

No because the sensitivity coefficient may be small

What is the combined standard uncertainty formula for uncorrelated inputs?

u_c(y) = √[Σ from i=1 to N c_i²u²(x_i)]

What condition applies to the RSS formula?

The input quantities are independent or uncorrelated

What does RSS mean in the Day 15 infographic?

Root Sum Square

What does the simplified RSS formula become when all sensitivity coefficients are 1?

u_c = √(u₁² + u₂² + ... + u_N²)

What are the four standard uncertainty values in the temperature calibration example?

0.050 °C 0.040 °C 0.030 °C and 0.020 °C

What is the sensitivity coefficient for the temperature calibration example?

c_i = 1

What is the combined standard uncertainty for the temperature calibration example?

u_c = 0.0735 °C

How is the combined standard uncertainty calculated in the example?

u_c = √(0.050² + 0.040² + 0.030² + 0.020²)

What is the squared contribution of the reference standard?

0.00250

What is the squared contribution of repeatability?

0.00160

What is the squared contribution of resolution?

0.00090

What is the squared contribution of environment?

0.00040

What is the sum of the squared contributions in the example?

0.0054

What expanded uncertainty is shown in the Day 15 example?

U = 2u_c = 0.147 °C with k = 2

What happens when significant correlations exist?

Covariance terms must be considered

What is the correlated-input uncertainty formula shown?

u_c²(y) = Σ c_i²u²(x_i) + 2ΣΣ c_i c_j u(x_i)u(x_j)r(x_i x_j)

What does r(x_i x_j) represent?

The correlation coefficient

What range is shown for the correlation coefficient?

−1 ≤ r ≤ +1

What does u(x_i x_j) represent?

The covariance of x_i and x_j

What effect can positive correlation have on u_c?

Positive correlation can increase u_c

What effect can negative correlation have on u_c?

Negative correlation can reduce u_c

What determines the distribution to choose?

The distribution depends on the information and measurement model not just the value

What does a rectangular distribution represent?

Equal likelihood within ±a

What is the standard uncertainty formula for a rectangular distribution?

u = a/√3

What rectangular-distribution example is shown?

±0.10 °C gives u = 0.058 °C

What does the normal distribution represent?

Standard deviation or a supported coverage interval

What is the standard uncertainty formula for the normal distribution example?

u = U/k

What normal-distribution example is shown?

U = ±0.20 °C with k = 2 gives u = 0.100 °C

What coverage probability is shown for the normal example?

95% with k = 1.96

What does the triangular distribution represent?

The center is more likely than the limits

What is the standard uncertainty formula for a triangular distribution?

u = a/√6

What triangular-distribution example is shown?

±0.12 °C gives u = 0.049 °C

Can the same limit produce the same standard uncertainty for different distributions?

No

What standard uncertainty results from ±0.10 °C with a rectangular distribution?

0.058 °C

What standard uncertainty results from ±0.10 °C with a normal distribution using k = 2?

0.050 °C

What standard uncertainty results from ±0.10 °C with a triangular distribution?

0.041 °C

What warning is given about distribution assumptions?

The distribution assumption changes the standard uncertainty

What does the calibrator's mindset section say not to ask only?

What is the final uncertainty?

What should the calibrator ask about the measurement model?

What measurement model produced it what contributes to it and are any inputs correlated?

What are the calibration red flags listed?

Just add all the uncertainties RSS always works the largest uncertainty is automatically the final uncertainty all sensitivity coefficients are 1 correlation does not matter u_c and U are the same and the calculator gave a number so the model must be correct

What references are listed for Day 15?

GUM JCGM 100:2008 GUM Amendment 1:2026 VIM3 JCGM 200:2012 ISO/IEC 17025:2017 Clause 7.6 ILAC P14:09/2020 and the applicable calibration method/procedure

What technical challenge is given on Day 15?

A calibration includes u₁ = 0.05 u₂ = 0.04 u₃ = 0.03 and u₄ = 0.02 °C and asks for the combined standard uncertainty assuming c_i = 1 and no correlation

What witty challenge is given on Day 15?

If someone says Just RSS everything what should you check before pressing the calculator button?

What is the overall learning theme of Day 15?

Individual standard uncertainty components become one combined standard uncertainty through the measurement model