1. Establish what the information actually means
The notation ±a may mean bounds, expanded uncertainty, tolerance or maximum error. Check the definition, unit, applicable conditions and whether a is the half-width before applying a divisor. A plus-minus sign alone does not establish a distribution.
Type A and Type B identify evaluation methods, not distribution names. A Type B evaluation may use a normal, rectangular or other model. A product tolerance does not automatically become measurement uncertainty. Any proposed range must concern an input to the measurement model and be supported by relevant information.
2. Rectangular: credible bounds and equal plausibility
A rectangular distribution from −a to +a assumes credible bounds and equal probability throughout. Its mean is zero and standard uncertainty is a/√3. Here a is the half-width, not the full width or expanded uncertainty.
This is a useful limited-information model, for example for a small unknown departure from reference conditions. It is not automatically justified by every specification. For asymmetric bounds, determine the interval's centre and half-width; a non-zero expected value should be reflected in the input estimate or an appropriate correction.
3. Triangular: evidence favours the centre
A symmetric triangular distribution shares the bounds ±a, but its density increases linearly towards the centre. Standard uncertainty is a/√6. This requires evidence favouring central values over those near the limits; bounds alone are insufficient.
Do not replace a rectangular model with a triangular one just to reduce the contribution. Document the evidence for the distribution's shape. An asymmetric triangle or a displaced most probable value requires the appropriate model parameters; the simple a/√6 expression no longer applies, and the expected value must also be considered.
4. Normal distributions and calibration certificates
A normal distribution has a mean and standard deviation, but no finite hard bounds. Thus ±a cannot describe its complete range. When a certificate specifies U and k, u = U/k recovers the standard uncertainty used for that U.
Check the stated coverage probability and the basis for k. Dividing by a known k does not establish normality: the factor may reflect limited degrees of freedom, for example. If the document supplies only an interval or an ambiguous ± statement, request clarification rather than filling the gap with an assumed factor of 2.
5. Resolution: where does q/√12 come from?
When a display rounds to the nearest increment q, the difference between its indication and the pre-rounded value lies within ±q/2. A uniform distribution for that difference gives u = (q/2)/√3 = q/√12. This is a particular quantization model, not a rule for every measurement involving a digital display.
Truncation rather than rounding changes the interval and expected value, potentially requiring a correction. Identical successive indications do not justify independent rounding errors or dividing this contribution by √n. When readings fluctuate, check whether their observed spread already contains part of the resolution effect before introducing a separate contribution.
6. The input distribution is not the output distribution
Propagate standard uncertainties through the model using sensitivities and any covariances. A dominant rectangular contribution can produce non-normal output. Several independent contributions may support normal approximation, but their number alone does not establish its adequacy.
Expanded uncertainty U = k·u꜀ requires a justified coverage factor; k = 2 does not always mean 95% coverage. For strong nonlinearity, asymmetry or a dominant unusual distribution, consider propagating distributions, for example by Monte Carlo simulation. The resulting coverage interval may be asymmetric and poorly represented by y ± U.
Remember this
The divisor follows from evidence and the chosen model. Bounds ±a do not automatically imply k = 2, and a display increment q is not itself a standard uncertainty.
Sources and further reading
This guide explains fundamental GUM principles. Always adapt your model to the method and the information available about the measurement.
Practise with your own measurement.
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