Type B Evaluation
A Type B evaluation of standard uncertainty is a method of evaluation by means other than the statistical analysis of a series of observations. Instead, it is based on scientific judgment using all relevant information available.
In a typical calibration laboratory, the majority of uncertainty components are evaluated using Type B methods. This is because we cannot statistically sample every possible source of error (like long-term drift, reference standard uncertainty, or environmental effects) during a single calibration event. Instead, we rely on a priori knowledge and assign appropriate probability distributions to these sources of error to estimate their standard deviation (standard uncertainty).
Sources of Information
Information used for Type B evaluations typically includes:
- Data provided in calibration certificates and other reports.
- Manufacturer's specifications and accuracy statements.
- Previous measurement data and historical control charts.
- Experience with or general knowledge of the behavior and properties of relevant materials and instruments.
- Uncertainties assigned to reference data taken from handbooks or publications.
Assigning Probability Distributions
The core mechanism of a Type B evaluation is determining the bounds (limits) of the uncertainty source, usually denoted as $\pm a$, and then assigning a probability distribution that best describes the likelihood of the true value falling within those bounds.
Once the distribution and limits are established, the standard uncertainty ($u_B$) is calculated by taking the square root of the variance of that specific mathematical distribution.
Common Probability Distributions
Rectangular (Uniform) Distribution
When to use: Use this when you only know the upper and lower limits ($\pm a$), and you have no reason to believe any value within those limits is more likely than another. This is the most conservative and commonly used Type B distribution.
Examples: Manufacturer specifications (when no confidence level is provided), the resolution of a digital display, environmental temperature limits of a laboratory.
Normal (Gaussian) Distribution
When to use: Use this when the uncertainty is provided with a stated level of confidence (e.g., 95%) and a coverage factor $k$. The limits do not represent hard boundaries, but rather a statistical interval.
Examples: The expanded uncertainty reported on an accredited calibration certificate.
Where $U$ is the expanded uncertainty and $k$ is the coverage factor provided on the certificate.
Triangular Distribution
When to use: Use this when you know the limits ($\pm a$), but you have strong reason to believe that values near the center (zero error) are much more likely than values near the extremes.
Examples: Analog scale reading interpolation, or an uncertainty source that is the sum of two independent rectangular distributions of similar magnitude.
U-Shaped (Arcsine) Distribution
When to use: Use this when the quantity cycles between its extremes, and values near the limits are much more likely than values in the center.
Examples: Temperature fluctuations controlled by a simple on/off thermostat, or the effect of radio-frequency (RF) mismatch (standing wave ratio) phase angles.
Degrees of Freedom for Type B
Assigning degrees of freedom ($\nu$) to a Type B evaluation can be subjective. It requires assessing the reliability of the limits $\pm a$.
If the limits are considered to be absolutely certain (a 100% probability that the value lies within them, such as the hard limits of a digital resolution), the degrees of freedom are considered infinite ($\nu = \infty$).
If there is some doubt about the reliability of the limits, a finite number of degrees of freedom can be calculated based on the relative uncertainty of the uncertainty estimate itself:
For example, if you believe your estimate of the uncertainty limit $a$ is reliable to within about 25% (i.e., $\Delta u/u = 0.25$), the degrees of freedom would be calculated as approximately $1 / (2 \cdot 0.25^2) = 8$. This ensures that less reliable Type B estimates correctly influence the final expanded uncertainty by increasing the necessary coverage factor.