3.2 The proper use of the pool of available information for a Type B evaluation of standard uncertainty calls for insight based on experience and general knowledge, and is a skill that can be learned with practice. It should be recognized that a Type B evaluation of standard uncertainty can be as reliable as a Type A evaluation, especially in a measurement situation where a Type A evaluation is based on a comparatively small number of statistically independent observations.NOTE
A calibration certificate states that the mass of a stainless steel mass standard ms of nominal value one kilogram is 1 000,000 325 g and that “the uncertainty of this value is 240 μg at the three standard deviation level”. The standard uncertainty of the mass standard is then simply . This corresponds to a relative standard uncertainty of . The estimated variance is
3.4 The quoted uncertainty of Xi is not necessarily given as a multiple of a standard deviation as in 3.3. Instead, one may find it stated that the quoted uncertainty defines an interval having a 90, 95, or 99 percent level of confidence . Unless otherwise indicated, one may assume that a normal distribution (A.8) was used to calculate the quoted uncertainty, and recover the standard uncertainty of Xi by dividing the quoted uncertainty by the appropriate factor for the normal distribution. The factors corresponding to the above three levels of confidence are 1.64; 1.96; and 2.58.
There would be no need for such an assumption if the uncertainty had been given in accordance with the recommendations of this Guide regarding the reporting of uncertainty, which stress that the coverage factor used is always to be given.
A calibration certificate states that the resistance of a standard resistor Rs of nominal value ten ohms is 10,000 742 Ω±129 μΩ at 23 °C and that “the quoted uncertainty of 129 μΩ defines an interval having a level of confidence of 99 percent”. The standard uncertainty of the resistor may be taken as , which corresponds to a relative standard uncertainty of . The estimated variance is
3.5 Consider the case where, based on the available information, one can state that“there is a fifty-fifty chance that the value of the input quantity Xi lies in the interval a_ to a+ ” (in other words, the probability that Xi lies within this interval is 0.5 or 50 percent). If it can be assumed that the distribution of possible values of Xi is approximately normal, then the best estimate Xi of can be taken to be the midpoint of the interval. Further, if the half-width of the interval is denoted by , one can take , because for a normal distribution with expectation μ and standard deviation σ the interval encompasses approximately 50 percent of the distribution.
A machinist determining the dimensions of a part estimates that its length lies, with probability 0.5, in the interval 10.07 mm to 10.15 mm, and reports that l=(10.11±0.04) mm, meaning that 0.04 mm defines an interval having a level of confidence of 50 percent. Then a = 0.04 mm, and if one assumes a normal distribution for the possible values of l, the standard uncertainty of the length is u(l)=1.48 × 0.04 mm ≈ 0.06 mm and the estimated variance is
It would give the value of u(xi) considerably more significance than is obviously warranted if one were to use the actual normal deviate 0.96742 corresponding to probability p=2/3,that is, if one were to write
3.7 In other cases, it may be possible to estimate only bounds (upper and lower limits) for Xi in particular, to state that “the probability that the value of Xi lies within the interval a_ to a+ for all practical purposes is equal to one and the probability that Xi lies outside this interval is essentially zero”. If there is no specific knowledge about the possible values of Xi within the interval, one can only assume that it is equally probable for Xi to lie anywhere within it (a uniform or rectangular distribution of possible values. Then , the expectation or expected value of Xi, is the midpoint of the interval, , with associated variance
NOTE When a component of uncertainty determined in this manner contributes significantly to the uncertainty of α measurement result, it is prudent to obtain additional data for its further evaluation.
A handbook gives the value of the coefficient of linear thermal expansion of pure copper at 20°c , α20(Cu) , as 16.52x10-6°c-1 and simply states that “the error in this value should not exceed 0.40x10-6°c-1”. Based on this limited information, it is not unreasonable to assume that the value of α20(Cu) lies with equal probability in the interval 16.12x10-6°c-1 to 16.92x10-6°c-1, and that it is very unlikely that α20(Cu) lies outside this interval. The variance of this symmetric rectangular distribution of possible values of of half-width a= 0.40x10-6°c-1 is then, from Equation (7),u2(α20) = , and the standard uncertainty is u(α20)=(0.40x10-6°c-1)/√3=0.23x10-6°c-1.
3.8 In 3.7, the upper and lower bounds a- and a- for the input quantity Xi may not be symmetric with respect to its best estimate xi; more specifically, if the lower bound is written as a- = xi - b- and the upper bound as a+ = xi - b+ , then b- ≠ b+. Since in this case xi (assumed to be the expectation of xi ) is not at the centre of the interval a- to a+ , the probability distribution of xi cannot be uniform throughout the interval. However, there may not be enough information available to choose an appropriate distribution; different models will lead to different expressions for the variance. In the absence of such information, the simplest approximation is
which is the variance of a rectangular distribution with full width .
If in Example 1 of 3.7 the value of the coefficient is given in the handbook as α20(Cu)16.52x10-6°c-1, and it is stated that “the smallest possible value is and the largest possible value is ”, then , and, from Equation (8), .
In many practical measurement situations where the bounds are asymmetric, it may be appropriate to apply a correction to the estimate xi of magnitude (b+ - b-)/2so that the new estimate xi of xi is at the midpoint of the bounds: . This reduces the situation to the case of 3.7, with new values .
Based on the principle of maximum entropy, the probability density function in the asymmetric case may be shown to be , with and . This leads to the variance ; and for .
3.9 In 3.7, because there was no specific knowledge about the possible values of xi within its estimated bounds a- to a+ , one could only assume that it was equally probable for xi to take any value within those bounds, with zero probability of being outside them. Such step function discontinuities in a probability distribution are often unphysical. In many cases, it is more realistic to expect that values near the bounds are less likely than those near the midpoint. It is then reasonable to replace the symmetric rectangular distribution with a symmetric trapezoidal distribution having equal sloping sides (an isosceles trapezoid), a base of width a+ - a-=2a , and a top of width 2αβ where ,this trapezoidal distribution approaches the rectangular distribution of 3.7, while for β= 0, it is a triangular distribution.Assuming such a trapezoidal distribution for xi , one finds that the expectation of and its associated variance is
which becomes for the triangular distribution, β = 0,
For a normal distribution with expectation μ and standard deviation σ , the interval μ±3σ encompasses approximately 99.73 percent of the distribution. Thus, if the upper and lower bounds a+ and a- define 99.73 percent limits rather than 100 percent limits, and can be assumed to be approximately normally distributed rather than there being no specific knowledge about xi between the bounds as in 3.7, then . By comparison, the variance of a symmetric rectangular distribution of half-width[Equation (7)] and that of a symmetric triangular distribution of half-width [Equation (9b)]. The magnitudes of the variances of the three distributions are surprisingly similar in view of the large differences in the amount of information required to justify them.
The trapezoidal distribution is equivalent to the convolution of two rectangular distributions, one with a half-width a1 equal to the mean half-width of the trapezoid, , the other with a half-width a2 equal to the mean width of one of the triangular portions of the trapezoid, . The variance of the distribution is . The convolved distribution can be interpreted as a rectangular distribution whose width 2a2 has itself an uncertainty represented by a rectangular distribution of width and models the fact that the bounds on an input quantity are not exactly known. But even if a2 is as large as 30 percent of a1, u exceeds a1√3 by less than 5 percent.
3.10 It is important not to “double-count” uncertainty components. If a component of uncertainty arising from a particular effect is obtained from a Type B evaluation, it should be included as an independent component of uncertainty in the calculation of the combined standard uncertainty of the measurement result only to the extent that the effect does not contribute to the observed variability of the observations. This is because the uncertainty due to that portion of the effect that contributes to the observed variability is already included in the component of uncertainty obtained from the statistical analysis of the observations.
3.11 The discussion of Type B evaluation of standard uncertainty in 3.3 to 3.9 is meant only to be indicative. Further, evaluations of uncertainty should be based on quantitative data to the maximum extent possible.