Machinery Noise Guide: Uncertainty budgets : Sound in Air : Acoustics : Science + Technology : National Physical Laboratory

Machinery Noise Guide: Uncertainty budgets

Uncertainty budgets

General calculation of uncertainty

When reporting a result of a measurement some indication of the quality of the result should be given. Without such an indication measurement results cannot be compared amongst themselves or with values in standards. The generally accepted method is to evaluate and express its uncertainty. This variation in measured values may be described as the doubt that exists about the result of the measurement. Unlike an error, which is the difference between measured values and the true value, an uncertainty is the quantification of the doubt associated with a measurement result.

In general, uncertainties are distinguished by how the values are estimated:

Type A: evaluated by statistical means (sometimes called random uncertainties),
for example Standard deviation of measurement repeatability
Type B: evaluated by other means (sometimes called systematic uncertainties)
for example pistonphone calibration.

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:

Equation 1

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:

Equation 2

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 artifact such as a sound calibrator.

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:

Equation 4

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:

Equation 5

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.

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:

Equation 6

where c_i is the sensitivity coefficient. For the purposes of this guide and for the uncerainty budgets in the is sound power standards, values of c_i are assumed to be unity.

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.