G.6.1 The coverage factor kp that provides an interval having a level of confidence p close to a specified level can only be found if there is extensive knowledge of the probability distribution of each input quantity and if these distributions are combined to obtain the distribution of the output quantity. The input estimates xi and their standard uncertainties u(xi) by themselves are inadequate for this purpose.
G.6.2 Because the extensive computations required to combine probability distributions are seldom justified by the extent and reliability of the available information, an approximation to the distribution of the output quantity is acceptable. Because of the Central Limit Theorem, it is usually sufficient to assume that the probability distribution of (y−Y)/uc(y) is the t-distribution and take kp=tp(veff), with the t-factor based on an effective degrees of freedom veff of uc(y) obtained from the Welch-Satterthwaite formula, Equation (G.2b). G.6.3 To obtain veff from Equation (G.2b) requires the degrees of freedom vi for each standard uncertainty component. For a component obtained from a Type A evaluation, vi is obtained from the number of independent repeated observations upon which the corresponding input estimate is based and the number of independent quantities determined from those observations (see G.3.3). For a component obtained from a Type B evaluation, vi is obtained from the judged reliability of the value of that component [see G.4.2 and Equation (G.3)].
G.6.4 Thus the following is a summary of the preferred method of calculating an expanded uncertainty Up = kpuc(y) intended to provide an interval Y=y±Up that has an approximate level of confidence p:
1) Obtain y and uc(y) as described in Clauses 4 and 5.
2) Compute veff from the Welch-Satterthwaite formula, Equation (G.2b) (repeated here for easy reference)
( )
( )
4 c eff 4 1 N i i i u y v u y v = =∑
(G.2b)If u(xi) is obtained from a Type A evaluation, determine vi as outlined in G.3.3. If u(xi) is obtained from a Type B evaluation and it can be treated as exactly known, which is often the case in practice, vi→
∞
; otherwise, estimate vi from Equation (G.3).3) Obtain the t-factor tp(veff) for the desired level of confidence p from Table G.2. If veff is not an integer, either interpolate or truncate veff to the next lower integer.
4) Take kp=tp(veff) and calculate Up=kpuc(y).
G.6.5 In certain situations, which should not occur too frequently in practice, the conditions required by the Central Limit Theorem may not be well met and the approach of G.6.4 may lead to an unacceptable result. For example, if uc(y) is dominated by a component of uncertainty evaluated from a rectangular distribution whose bounds are assumed to be exactly known, it is possible [if t vp( eff)> 3] that y + Up and y−Up, the upper and lower limits of the interval defined by Up, could lie outside the bounds of the probability distribution of the output quantity Y. Such cases must be dealt with on an individual basis but are often amenable to an approximate analytic treatment (involving, for example, the convolution of a normal distribution with a rectangular distribution [10]).
G.6.6 For many practical measurements in a broad range of fields, the following conditions prevail:
⎯ the estimate y of the measurand Y is obtained from estimates xi of a significant number of input quantities Xi that are describable by well-behaved probability distributions, such as the normal and rectangular distributions;
⎯ the standard uncertainties u(xi) of these estimates, which may be obtained from either Type A or Type B evaluations, contribute comparable amounts to the combined standard uncertainty uc(y) of the measurement result y;
⎯ the linear approximation implied by the law of propagation of uncertainty is adequate (see 5.1.2 and
E.3.1);
⎯ the uncertainty of uc(y) is reasonably small because its effective degrees of freedom veff has a significant magnitude, say greater than 10.
Under these circumstances, the probability distribution characterized by the measurement result and its combined standard uncertainty can be assumed to be normal because of the Central Limit Theorem; and uc(y) can be taken as a reasonably reliable estimate of the standard deviation of that normal distribution because of the significant size of veff. Then, based on the discussion given in this annex, including that emphasizing the approximate nature of the uncertainty evaluation process and the impracticality of trying to distinguish between intervals having levels of confidence that differ by one or two percent, one may do the following:
⎯ adopt k= 2 and assume that U= 2uc(y) defines an interval having a level of confidence of approximately 95 percent;
or, for more critical applications,
⎯ adopt k = 3 and assume that U= 3uc(y) defines an interval having a level of confidence of approximately 99 percent.
Although this approach should be suitable for many practical measurements, its applicability to any particular measurement will depend on how close k= 2 must be to t95(veff) or k= 3 must be to t99(veff); that is, on how close the level of confidence of the interval defined by U= 2uc(y) or U= 3uc(y) must be to 95 percent or 99 percent, respectively. Although for veff= 11, k= 2 and k = 3 underestimate t95(11) and t99(11) by only about 10 percent and 4 percent, respectively (see Table G.2), this may not be acceptable in some cases. Further, for all values of veff somewhat larger than 13, k = 3 produces an interval having a level of confidence larger than 99 percent. (See Table G.2, which also shows that for veff→
∞
the levels of confidence of the intervals produced by k= 2 and k= 3 are 95,45 percent and 99,73 percent, respectively). Thus, in practice, the size of veff and what is required of the expanded uncertainty will determine whether this approach can be used.Table G.2 — Value of tp(v) from the t-distribution for degrees of freedom v that defines an interval −tp(v) to +tp(v) that encompasses the fraction p of the distribution Degrees of
freedom Fraction p in percent
v 68,27a) 90 95 95,45a) 99 99,73a) 1 1,84 6,31 12,71 13,97 63,66 235,80 2 1,32 2,92 4,30 4,53 9,92 19,21 3 1,20 2,35 3,18 3,31 5,84 9,22 4 1,14 2,13 2,78 2,87 4,60 6,62 5 1,11 2,02 2,57 2,65 4,03 5,51 6 1,09 1,94 2,45 2,52 3,71 4,90 7 1,08 1,89 2,36 2,43 3,50 4,53 8 1,07 1,86 2,31 2,37 3,36 4,28 9 1,06 1,83 2,26 2,32 3,25 4,09 10 1,05 1,81 2,23 2,28 3,17 3,96 11 1,05 1,80 2,20 2,25 3,11 3,85 12 1,04 1,78 2,18 2,23 3,05 3,76 13 1,04 1,77 2,16 2,21 3,01 3,69 14 1,04 1,76 2,14 2,20 2,98 3,64 15 1,03 1,75 2,13 2,18 2,95 3,59 16 1,03 1,75 2,12 2,17 2,92 3,54 17 1,03 1,74 2,11 2,16 2,90 3,51 18 1,03 1,73 2,10 2,15 2,88 3,48 19 1,03 1,73 2,09 2,14 2,86 3,45 20 1,03 1,72 2,09 2,13 2,85 3,42 25 1,02 1,71 2,06 2,11 2,79 3,33 30 1,02 1,70 2,04 2,09 2,75 3,27 35 1,01 1,70 2,03 2,07 2,72 3,23 40 1,01 1,68 2,02 2,06 2,70 3,20 45 1,01 1,68 2,01 2,06 2,69 3,18 50 1,01 1,68 2,01 2,05 2,68 3,16 100 1,005 1,660 1,984 2,025 2,626 3,077 ∞ 1,000 1,645 1,960 2,000 2,576 3,000
a) For a quantity z described by a normal distribution with expectation µz and standard deviation σ, the interval
µz ± kσ encompasses p = 68,27 percent, 95,45 percent and 99,73 percent of the distribution for k = 1, 2 and 3,
Annex H
Examples
This annex gives six examples, H.1 to H.6, which are worked out in considerable detail in order to illustrate the basic principles presented in this Guide for evaluating and expressing uncertainty in measurement. Together with the examples included in the main text and in some of the other annexes, they should enable the users of this Guide to put these principles into practice in their own work.
Because the examples are for illustrative purposes, they have by necessity been simplified. Moreover, because they and the numerical data used in them have been chosen mainly to demonstrate the principles of this Guide, neither they nor the data should necessarily be interpreted as describing real measurements. While the data are used as given, in order to prevent rounding errors, more digits are retained in intermediate calculations than are usually shown. Thus the stated result of a calculation involving several quantities may differ slightly from the result implied by the numerical values given in the text for these quantities.
It is pointed out in earlier portions of this Guide that classifying the methods used to evaluate components of uncertainty as Type A or Type B is for convenience only; it is not required for the determination of the combined standard uncertainty or expanded uncertainty of a measurement result because all uncertainty components, however they are evaluated, are treated in the same way (see 3.3.4, 5.1.2, and E.3.7). Thus, in the examples, the method used to evaluate a particular component of uncertainty is not specifically identified as to its type. However, it will be clear from the discussion whether a component is obtained from a Type A or a Type B evaluation.