Type B Distribution Models

Many contributions, such as certificate uncertainty or specification bounds, are evaluated using Type B methods. The classification follows the evaluation method. Environmental variation, drift, or repeatability can instead be evaluated statistically when adequate observation data are available; a physical source is not inherently Type A or Type B.

According to GUM Clause 4.3, a Type B evaluation relies on the metrologist's scientific judgment using all available information, including:

• Previous calibration data and historical control charts
• Manufacturer technical specifications
• Calibrated values and expanded uncertainties from accredited calibration certificates
• Uncertainties assigned to reference data taken from physical handbooks
• Digital display resolution and quantization limits

To combine a Type B uncertainty component with Type A components in a combined uncertainty budget, the non-statistical tolerance or limit $\pm a$ must be converted into an equivalent standard uncertainty $u(x_i)$, which represents one standard deviation ($\sigma$). This conversion requires selecting an appropriate mathematical model: the probability distribution.

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Master Summary of Probability Distributions in Metrology

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Detailed Mathematical Derivations and Applications

The Rectangular (Uniform) Distribution

The rectangular distribution is the default model under the GUM when a quantity is known to lie within upper and lower bounds $[-a, +a]$, but there is no specific knowledge about the probability of values within those bounds.

Mathematical Derivation of Variance

The probability density function (PDF) is constant across the interval:

$$p(x) = \begin{cases} \frac{1}{2a}, & -a \le x \le +a \\ 0, & |x| > a \end{cases}$$

Because the distribution is symmetric, the mean is zero ($E[X] = 0$). The variance is the expected value of $x^2$:

$$V(x) = \sigma^2 = \int_{-a}^{+a} x^2 p(x) dx = \int_{-a}^{+a} x^2 \left(\frac{1}{2a}\right) dx = \frac{1}{2a} \left[ \frac{x^3}{3} \right]_{-a}^{+a}$$ $$V(x) = \frac{1}{2a} \left( \frac{a^3}{3} - \frac{(-a)^3}{3} \right) = \frac{1}{2a} \left( \frac{2a^3}{3} \right) = \frac{a^2}{3}$$

Taking the square root yields the standard uncertainty:

$$u(x) = \sqrt{\frac{a^2}{3}} = \frac{a}{\sqrt{3}} \approx \frac{a}{1.73205} \approx 0.5774 a$$

Primary Calibration Applications

1. Digital Display Resolution (Quantization): A digital instrument rounds its internal estimate to the nearest display increment (least significant digit, LSD, or resolution $d$). The maximum rounding error is half of the resolution: $a = d/2 = 0.5\text{ LSD}$.

$$u_{\text{resolution}} = \frac{d / 2}{\sqrt{3}} = \frac{d}{2\sqrt{3}} = \frac{d}{\sqrt{12}} \approx 0.2887 d$$

Example: A digital indicator displays readings to $0.001\text{ mm}$ ($1\ \mu\text{m}$). The semi-range is $a = 0.0005\text{ mm}$. The standard uncertainty due to resolution is:

$$u_{\text{res}} = \frac{0.0005\text{ mm}}{\sqrt{3}} = 0.000289\text{ mm} = 0.289\ \mu\text{m}$$

1. Manufacturer Specifications Without Stated Confidence: When an equipment manual quotes an accuracy of "$\pm 0.05\%$ of reading" with no mention of a normal distribution or confidence level, a rectangular model can be justified when reliable bounds are known and no better distribution information is available with half-width $a = 0.05\% \times \text{Reading}$.
2. Environmental Temperature Stability: A laboratory thermostat maintains room temperature within $\pm 1.5^\circ\text{C}$ of setpoint. If no continuous logging histogram is available, assume a rectangular distribution with $a = 1.5^\circ\text{C}$: $u = 1.5 / \sqrt{3} = 0.866^\circ\text{C}$.

The Triangular Distribution

The triangular distribution applies when values are known to cluster near the center of the interval $[-a, +a]$ and diminish linearly toward the extremes.

Mathematical Derivation of Variance

By convolution, the sum (sum or difference) of two independent rectangular distributions with identical half-widths $a/2$ forms an exact symmetric triangle with base $[-a, +a]$.

The probability density function is:

$$p(x) = \begin{cases} \frac{1}{a} \left(1 - \frac{|x|}{a}\right), & -a \le x \le +a \\ 0, & |x| > a \end{cases}$$

Integrating $x^2 p(x)$ across the interval yields the variance:

$$V(x) = \sigma^2 = \int_{-a}^{+a} x^2 \frac{1}{a}\left(1 - \frac{|x|}{a}\right) dx = 2 \int_0^a \left(\frac{x^2}{a} - \frac{x^3}{a^2}\right) dx = 2 \left[ \frac{a^3}{3a} - \frac{a^4}{4a^2} \right] = 2 \left( \frac{a^2}{3} - \frac{a^2}{4} \right) = \frac{a^2}{6}$$

Taking the square root yields the standard uncertainty:

$$u(x) = \sqrt{\frac{a^2}{6}} = \frac{a}{\sqrt{6}} \approx \frac{a}{2.44949} \approx 0.4082 a$$

1. A supported triangular model: Two independent, equal uniform components sum to an exact triangle. A backlash mechanism or controlled bath does not automatically have that shape; inspect the physics and data before selecting the divisor.

The U-Shaped (Arc-Sine) Distribution

The U-shaped distribution applies when a physical parameter varies cyclically or sinusoidally over time, dwelling primarily near the turnaround peaks and passing rapidly through the center.

Mathematical Derivation of Variance

Consider a variable undergoing simple harmonic oscillation with amplitude $a$:

$$x(t) = a \sin(\omega t)$$

The probability of observing a value $x$ in an infinitesimal window $dx$ is inversely proportional to the speed of the wave: $|dx/dt| = a\omega \cos(\omega t) = \omega \sqrt{a^2 - x^2}$. Normalizing over the domain $(-a, +a)$ gives the arc-sine probability density function:

$$p(x) = \frac{1}{\pi \sqrt{a^2 - x^2}}, \quad -a < x < +a$$

The variance is obtained by integrating:

$$V(x) = \sigma^2 = \int_{-a}^{+a} \frac{x^2}{\pi \sqrt{a^2 - x^2}} dx$$

Substituting $x = a \sin\theta$ (where $dx = a \cos\theta d\theta$ and $\sqrt{a^2 - x^2} = a \cos\theta$):

$$V(x) = \frac{1}{\pi} \int_{-\pi/2}^{+\pi/2} (a^2 \sin^2\theta) d\theta = \frac{a^2}{\pi} \int_{-\pi/2}^{+\pi/2} \left(\frac{1 - \cos 2\theta}{2}\right) d\theta = \frac{a^2}{\pi} \left[ \frac{\theta}{2} \right]_{-\pi/2}^{+\pi/2} = \frac{a^2}{2}$$

Taking the square root yields the standard uncertainty:

$$u(x) = \sqrt{\frac{a^2}{2}} = \frac{a}{\sqrt{2}} \approx \frac{a}{1.41421} \approx 0.7071 a$$

1. Environmental cycling: A sinusoidal temperature variation sampled at a uniformly random phase has an arcsine distribution. On/off thermostat control alone does not prove this model; an exponential heating/cooling cycle can have another distribution.

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Normal Distribution and Calibration Certificate Conversions

When a specification or calibration certificate quotes an uncertainty with a stated confidence interval, use the stated distribution and coverage information; a confidence statement alone does not prove a normal model.

Converting Expanded Uncertainty from a Calibration Certificate

Use the certificate’s stated expanded uncertainty, coverage factor, and coverage information. For a justified normal model with adequate degrees of freedom, $k=2$ corresponds to about 95.45% central coverage; it is not the interpretation of every certificate or distribution.

To enter this component into your working calibration uncertainty budget as a standard uncertainty $u(x_i)$, simply divide by the stated coverage factor:

$$u(x_i) = \frac{U}{k}$$

Hypothetical exercise: A reference resistor certificate gives the following value, uncertainty, and coverage factor. These are invented teaching inputs rather than quotations from an actual NIST report:

$$R_{\text{std}} = 10,000.042\ \Omega, \quad U = 0.024\ \Omega \quad (k = 2.00, \text{ approx. } 95\% \text{ confidence})$$

The standard uncertainty to enter into the laboratory's uncertainty budget is:

$$u(R_{\text{std}}) = \frac{0.024\ \Omega}{2.00} = 0.012\ \Omega$$

Converting Manufacturer Specifications with Stated Confidence Levels

If the stated interval is central coverage under a justified normal model, use the corresponding normal quantile. Confidence alone does not establish normality, infinite degrees of freedom, or an exact divisor. If a certificate supplies its coverage factor, use that factor and its stated interpretation. The following quantiles are normal-model examples:

• $90.00\%$ Confidence: Divisor $k = 1.645 \implies u = a / 1.645$
• $95.00\%$ Confidence: Divisor $k = 1.960 \implies u = a / 1.960$
• $95.45\%$ Confidence ($2\sigma$): Divisor $k = 2.000 \implies u = a / 2.000$
• $99.00\%$ Confidence: Divisor $k = 2.576 \implies u = a / 2.576$
• $99.73\%$ Confidence ($3\sigma$): Divisor $k = 3.000 \implies u = a / 3.000$

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Comparative Worked Calculation: Impact of Divisor Selection

To illustrate how the choice of probability distribution affects calibration budgets, consider an instrument tolerance limit of $\pm a = \pm 12.0\ \mu\text{m}$. Let us calculate the resulting standard uncertainty under all four models:

Technical reference checked October 10, 2026: JCGM uncertainty and vocabulary publications.