Estimating measurement uncertainties

# Type B evaluation

If a measurement laboratory was to be granted with unlimited resources and time, it could carry out an exhaustive statistical study of every imaginable cause of uncertainty, using for instance different types of devices made by different manufacturers, with different measurement methods, different operating procedures and different approximations in the different measurement theoretical models.

It would then be possible to estimate the uncertainties associated with all these causes through the statistical analysis of the series of observations, and to characterize the uncertainty due to each cause with a statistically evaluated standard deviation. Eventually, all components of the uncertainty would result from Type A evaluations.

But since such a study is financially impossible, numerous components of the uncertainty have to be evaluated through all the other feasible means. The information that will be analysed can include:

• results from previous measurements,

• general or empirical knowledge of the behavior of the instruments used,

• specifications of the manufacturer,

• calibration certificates,

• the uncertainty attributed to reference quantity values mentioned in studies, textbooks or norms.

Thus for an estimate xi of a quantity Xi which has not been obtained from replicate observations, the estimated variance u2(xi) or the standard uncertainty u(xi) est évaluée par un is evaluated through a scientific judgement based on all the available information regarding the possible variability of  Xi . The standard uncertainty evaluated by this mean is called Type B standard uncertainty.

In practice, a balance of all the errors must be carried out. These errors are divided into:

• Systematic measurement errors (cf. VIM §3.14) such as the parallax error when reading a needle dial, the zero adjustment of a device, method errors, components ageing, etc.

• Random error of measurement (cf. VIM §3.13) such as reading errors or errors due to the device itself, or to exterior conditions (temperature, thermal expansion, atmospheric pressure, humidity, etc.).

## A priori probability laws

In order to express Type B uncertainty under the form of a standard deviation, probability laws must be used. Table 2 presents the most commonly used probability laws, referring here to a distribution of values of a random variable of mean and of range .

Table 2 : Probability laws commonly used to evaluate Type B uncertainties [zoom...]

Generally, if the manufacturer provides the standard uncertainty, it is used directly.

If very little information is available on an input quantity and its supposed variation interval comes under the form:

• while standard uncertainty is : .

• while standard uncertainty is : .

considering a uniform law in the variation interval of the quantity.

## Examples of Type B uncertainties

• Resolution of a measuring instrument

The graduation of a measuring instrument or a digital display device is a source of uncertainty. If the resolution of the indicating device is δx, lthe value of the input signal producing a given indication X can lie with equal probability anywhere in the interval , the input signal is then described by a rectangular probability law of width δx and of standard deviation called resolution uncertainty.

• Class of an instrument

The maximum permissible measurement error (cf. VIM §5.21) gives the extreme variation limits of the indication obtained from a measuring instrument, the class of which is defined by the interval . The associated standard uncertainty is then .

• Hysteresis

The different indications of a measuring instrument can be a fixed quantity, depending on the successive readings being made in ascending or decreasing order of values. Most of the time the direction of the hysteresis cannot be observed. If the width of the range of the possible readings due to this cause is δx, the standard uncertainty due to this hysteresis is .

• Temperature variations

One of the main influences on the quantity of a measuring system is the temperature of the means of measurement's environment (room, air-conditioned compound, case, etc.). Insofar as the temperature would vary between two extremes in a quasi-sinusoidal way, the law of probability associated with this influence quantity is the derivative of the arcsine function. If the temperature variations are such as , then the standard uncertainty due to temperature variations is .