More on Type A uncertainty

A Type A evaluation is used to obtain a value for the repeatability or random variation in a measurement. For some measurements, the random component of uncertainty may be relatively insignificant compared to Type B components. However, the Type A uncertainty should still be assessed by making repeated measurements and performing a statistical analysis of the results. If there are n independent repeated values for a quantity q then the mean value q_M is given by:

From the results of a sample of n measurements, an estimate, s(q), can be made of the standard deviation of the whole population of possible values of the measurand from the relation:

The estimated standard deviation of the uncorrected mean value of the measurand, s(q_M), is given by:

This value of s(q_M) is used as the Type A standard uncertainty, u_A.

More on Type B uncertainty

Type B components of uncertainty are those not assessed by a statistical means. Examples of Type B uncertainties include sources of systematic bias in a measurement, in other words those errors that remain constant when the measurement is repeated. Another source of Type B uncertainty originates from the previous calibration of any reference artefact such as a measuring instrument or reference transducer.

The sources of these components must be identified by assessing all of the influences that may introduce errors into the measurement. This assessment may be made in a number of ways: by consideration of the formulae or equations used in the calibration method, by theoretical analysis of the method, and by assessment of the accuracy of the instrumentation used. If necessary, further experimental work may be required to determine the magnitude of a particular component.

Having identified all Type B components of uncertainty, these must be characterised in terms of their standard deviations based on the assessed probability distributions. The Type B standard uncertainty of the component may be regarded as equal to this standard deviation. The actual probability distribution for a component may take a variety of forms, but it is generally acceptable to assign one of a few well-defined geometric distributions for which the standard deviation can be easily calculated.

Type B uncertainties: uniform distributions

Where it is possible only to assess the upper and lower bounds of an error, a uniform or rectangular distribution may be assumed. If a is the semi-range of the distribution, such that the true value is assumed to lie within ±a of the measured value with equal probability, then the standard deviation (and therefore the standard uncertainty, u_B,) is given by:

Where an instrument has been certified as conforming to a specification, then the uncertainty or tolerance limit must be taken into account. Unless the manufacturer declares a confidence level for the tolerances on the specification, it is acceptable to assume a uniform distribution, with the value of semi-range in the above equation set equal to the tolerance limit.

Type B uncertainties: normal distributions

Where a Type B component derives from previous experimental data, or where the uncertainty is obtained from the calibration certificate of an instrument, a normal distribution may be assumed (unless otherwise indicated on the certificate). In this case, the uncertainty will have been expressed either as a standard uncertainty (in which case no further calculation is required), or as an expanded uncertainty for a level of confidence with a specified coverage factor. In the latter case, the standard uncertainty may be obtained from:

where U_E is the expanded uncertainty reported on the instrument certificate and k is the coverage factor. If the certificate merely states that the level of confidence of "not less than 95%" was used, then a normal distribution with a coverage factor of k=2 may be assumed.

For more details regarding the derivation of the above relations, and for the treatment of other forms of uncertainty distribution, see [2,3].

Combining uncertainties to obtain overall uncertainty

Once the standard uncertainties, u_i, for each contribution have been derived from the analysis described above, the combined standard uncertainty, u_c, can be calculated from:

where c_i is the sensitivity coefficient. This may be derived from the partial derivative of the output quantity with respect to the input quantity, which may typically be derived from the formula or equation used to calculate the calibration result (including any correction factors used). Alternatively, it may come from a known coefficient, such as a known sensitivity variation with temperature. The calculations required to obtain sensitivity coefficients by partial differentiation may be straightforward where the functional relationship between the input and output quantities is known. However, where the relationship is not known for a particular input quantity, the coefficient may be estimated by varying the input quantity by a known amount while keeping all other inputs constant, and observing the effect on the output.

If the functional relationship between input quantities and output is a product or a quotient, it is often more convenient to express the component uncertainties as relative uncertainties, often expressed in percent (%).

The expression above for the combined uncertainty is valid only if the components of uncertainty are uncorrelated, ie the input quantities are independent. Where there are correlations between the inputs (for example where several input quantities are affected by the same influence), a different approach is required. For more detail regarding this, the reader is referred to [2,3].

In some measurement processes, there may be one component of Type B uncertainty which is dominant in magnitude such that the expanded uncertainty calculated by the above method is greater than the simple arithmetic sum of all the components. In such cases, the Type B components are combined in a different manner to that described above. For more detail, the reader is referred to [2,3].

Expanded uncertainty

It is usual to express the overall uncertainty as an expanded uncertainty for a stated level of confidence. In this case, the expanded uncertainty, U_E, may be obtained from:

where k is the coverage factor. Assuming a normal distribution, a coverage factor of k=2 will give a level of confidence of approximately 95%.

However, if the Type A uncertainty is found to be relatively large compared to other contributions and the number of repeat measurements is small, it is possible that the overall distribution will not be normal and a larger value of coverage factor will be required to obtain the desired confidence level. This is relevant only if the combined uncertainty is less then two times the Type A uncertainty. For more details, the reader is referred to [2,3].

Further guidance

More detailed information can be found in the following references:

1. A Beginners Guide to Uncertainty in Measurement, Stephanie Bell, Measurement Good Practice Guide No.11, Issue 2, 1999, NPL, UK
2. M3003. The expression of uncertainty and confidence in measurement. UKAS, Feltham, Middlesex, TW14 4UN, UK
3. Guide to the Expression of Uncertainty in Measurement (GUM), 1993, International Organization for Standardization (ISO), Geneva