Additional guidance on evaluating uncertainty components, mainly of a practical nature, may be found in Annex F.
4.1.1 In most cases, a measurand Y is not measured directly, but is determined from N other quantities X_{1}, X_{2}, ..., X_{N} through a functional relationship f:
NOTE 1 For economy of notation, in this Guide the same symbol is used for the physical quantity (the measurand) and for the random variable (see 4.2.1) that represents the possible outcome of an observation of that quantity. When it is stated that X_{i} has a particular probability distribution, the symbol is used in the latter sense; it is assumed that the physical quantity itself can be characterized by an essentially unique value (see 1.2 and 3.1.3).
NOTE 2 In a series of observations, the kth observed value of X_{i} is denoted by X_{i,k}; hence if R denotes the resistance of a resistor, the kth observed value of the resistance is denoted by R_{k}.
NOTE 3 The estimate of X_{i} (strictly speaking, of its expectation) is denoted by x_{i}.
EXAMPLE If a potential difference V is applied to the terminals of a temperature-dependent resistor that has a resistance R_{0} at the defined temperature t_{0} and a linear temperature coefficient of resistance α, the power P (the measurand) dissipated by the resistor at the temperature t depends on V, R_{0}, α, and t according to
NOTE Other methods of measuring P would be modelled by different mathematical expressions.
4.1.2 The input quantities X_{1}, X_{2}, ..., X_{N} upon which the output quantity Y depends may themselves be viewed as measurands and may themselves depend on other quantities, including corrections and correction factors for systematic effects, thereby leading to a complicated functional relationship f that may never be written down explicitly. Further, f may be determined experimentally (see 5.1.4) or exist only as an algorithm that must be evaluated numerically. The function f as it appears in this Guide is to be interpreted in this broader context, in particular as that function which contains every quantity, including all corrections and correction factors, that can contribute a significant component of uncertainty to the measurement result.
Thus, if data indicate that f does not model the measurement to the degree imposed by the required accuracy of the measurement result, additional input quantities must be included in f to eliminate the inadequacy (see 3.4.2). This may require introducing an input quantity to reflect incomplete knowledge of a phenomenon that affects the measurand. In the example of 4.1.1, additional input quantities might be needed to account for a known nonuniform temperature distribution across the resistor, a possible nonlinear temperature coefficient of resistance, or a possible dependence of resistance on barometric pressure.
NOTE Nonetheless, Equation (1) may be as elementary as Y = X_{1} − X_{2}. This expression models, for example, the comparison of two determinations of the same quantity X.
4.1.3 The set of input quantities X_{1}, X_{2}, ..., X_{N} may be categorized as:
4.1.4 An estimate of the measurand Y, denoted by y, is obtained from Equation (1) using input estimates x_{1}, x_{2}, ..., x_{N} for the values of the N quantities X_{1}, X_{2}, ..., X_{N}, Thus the output estimate y, which is the result of the measurement, is given by
NOTE In some cases, the estimate y may be obtained from
That is, y is taken as the arithmetic mean or average (see 4.2.1) of n independent determinations Y_{k} of Y, each determination having the same uncertainty and each being based on a complete set of observed values of the N input quantities X_{i} obtained at the same time. This way of averaging, rather than y = f(X^{‾‾‾}_{1}, X^{‾‾‾}_{2}, ..., X^{‾‾‾}_{N}), where
is the arithmetic mean of the individual observations X_{i,k}, may be preferable when f is a nonlinear function of the input quantities X_{1}, X_{2}, ..., X_{N}, but the two approaches are identical if f is a linear function of the X_{i} (see H.2 and H.4).
4.1.5 The estimated standard deviation associated with the output estimate or measurement result y, termed combined standard uncertainty and denoted by u_{c}(y), is determined from the estimated standard deviation associated with each input estimate x_{i}, termed standard uncertainty and denoted by u(x_{i}) (see 3.3.5 and 3.3.6).
4.1.6 Each input estimate x_{i} and its associated standard uncertainty u(x_{i}) are obtained from a distribution of possible values of the input quantity X_{i}. This probability distribution may be frequency based, that is, based on a series of observations X_{i,k} of X_{i}, or it may be an a priori distribution. Type A evaluations of standard uncertainty components are founded on frequency distributions while Type B evaluations are founded on a priori distributions. It must be recognized that in both cases the distributions are models that are used to represent the state of our knowledge.
4.2.1 In most cases, the best available estimate of the expectation or expected value μ_{q} of a quantity q that varies randomly [a random variable (C.2.2)], and for which n independent observations q_{k} have been obtained under the same conditions of measurement (see B.2.15), is the arithmetic mean or average q‾‾ (C.2.19) of the n observations:
Thus, for an input quantity X_{i} estimated from n independent repeated observations X_{i,k}, the arithmetic mean X^{‾‾‾}_{i} obtained from Equation (3) is used as the input estimate x_{i} in Equation (2) to determine the measurement result y; that is, x_{i} = X^{‾‾‾}_{i}. Those input estimates not evaluated from repeated observations must be obtained by other methods, such as those indicated in the second category of 4.1.3.
4.2.2 The individual observations q_{k} differ in value because of random variations in the influence quantities, or random effects (see 3.2.2). The experimental variance of the observations, which estimates the variance σ^{2} of the probability distribution of q, is given by
This estimate of variance and its positive square root s(q_{k}), termed the experimental standard deviation (B.2.17), characterize the variability of the observed values q_{k}, or more specifically, their dispersion about their mean q‾‾.
4.2.3 The best estimate of σ^{2}(q‾‾ ) = σ^{2} ⁄ n, the variance of the mean, is given by
The experimental variance of the mean s^{2}(q‾‾) and the experimental standard deviation of the mean s(q‾‾) (B.2.17, Note 2), equal to the positive square root of s^{2}(q‾‾), quantify how well q‾‾ estimates the expectation μ_{q} of q, and either may be used as a measure of the uncertainty of q‾‾.
Thus, for an input quantity X_{i} determined from n independent repeated observations X_{i,k}, the standard uncertainty u(x_{i}) of its estimate x_{i} = X^{‾‾‾}_{i} is u(x_{i}) = s(X^{‾‾‾}_{i}), with s^{2}(X^{‾‾‾}_{i}) calculated according to Equation (5). For convenience, u^{2}(x_{i}) = s^{2}(X^{‾‾‾}_{i}) and u(x_{i}) = s(X^{‾‾‾}_{i}) are sometimes called a Type A variance and a Type A standard uncertainty, respectively.
NOTE 1 The number of observations n should be large enough to ensure that q‾‾ provides a reliable estimate of the expectation μ_{q} of the random variable q and that s^{2}(q‾‾) provides a reliable estimate of the variance σ^{2}(q‾‾) = σ^{2}⁄n (see 4.3.2, note). The difference between s^{2}(q‾‾) and σ^{2}(q‾‾) must be considered when one constructs confidence intervals (see 6.2.2). In this case, if the probability distribution of q is a normal distribution (see 4.3.4), the difference is taken into account through the t‑distribution (see G.3.2).
NOTE 2 Although the variance s^{2}(q‾‾) is the more fundamental quantity, the standard deviation s(q‾‾) is more convenient in practice because it has the same dimension as q and a more easily comprehended value than that of the variance.
4.2.4 For a well‑characterized measurement under statistical control, a combined or pooled estimate of variance s^{2}_{p} (or a pooled experimental standard deviation s_{p}) that characterizes the measurement may be available. In such cases, when the value of a measurand q is determined from n independent observations, the experimental variance of the arithmetic mean q‾‾ of the observations is estimated better by s^{2}_{p} ⁄n than by s^{2}(q_{k}) ⁄n and the standard uncertainty is u = s_{p} ⁄ √n^{‾‾}. (See also the Note to H.3.6.)
4.2.5 Often an estimate x_{i} of an input quantity X_{i} is obtained from a curve that has been fitted to experimental data by the method of least squares. The estimated variances and resulting standard uncertainties of the fitted parameters characterizing the curve and of any predicted points can usually be calculated by well-known statistical procedures (see H.3 and Reference [8]).
4.2.6 The degrees of freedom (C.2.31) v_{i} of u(x_{i}) (see G.3), equal to n − 1 in the simple case where x_{i} = X^{‾‾‾}_{i} and u(x_{i}) = s(X^{‾‾‾}_{i}) are calculated from n independent observations as in 4.2.1 and 4.2.3, should always be given when Type A evaluations of uncertainty components are documented.
4.2.7 If the random variations in the observations of an input quantity are correlated, for example, in time, the mean and experimental standard deviation of the mean as given in 4.2.1 and 4.2.3 may be inappropriate estimators (C.2.25) of the desired statistics (C.2.23). In such cases, the observations should be analysed by statistical methods specially designed to treat a series of correlated, randomly‑varying measurements.
NOTE Such specialized methods are used to treat measurements of frequency standards. However, it is possible that as one goes from short‑term measurements to long‑term measurements of other metrological quantities, the assumption of uncorrelated random variations may no longer be valid and the specialized methods could be used to treat these measurements as well. (See Reference [9], for example, for a detailed discussion of the Allan variance.)
4.2.8 The discussion of Type A evaluation of standard uncertainty in 4.2.1 to 4.2.7 is not meant to be exhaustive; there are many situations, some rather complex, that can be treated by statistical methods. An important example is the use of calibration designs, often based on the method of least squares, to evaluate the uncertainties arising from both short‑ and long‑term random variations in the results of comparisons of material artefacts of unknown values, such as gauge blocks and standards of mass, with reference standards of known values. In such comparatively simple measurement situations, components of uncertainty can frequently be evaluated by the statistical analysis of data obtained from designs consisting of nested sequences of measurements of the measurand for a number of different values of the quantities upon which it depends — a so‑called analysis of variance (see H.5).
NOTE At lower levels of the calibration chain, where reference standards are often assumed to be exactly known because they have been calibrated by a national or primary standards laboratory, the uncertainty of a calibration result may be a single Type A standard uncertainty evaluated from the pooled experimental standard deviation that characterizes the measurement.
4.3.1 For an estimate x_{i} of an input quantity X_{i} that has not been obtained from repeated observations, the associated estimated variance u^{2}(x_{i}) or the standard uncertainty u(x_{i}) is evaluated by scientific judgement based on all of the available information on the possible variability of X_{i}. The pool of information may include
For convenience, u^{2}(x_{i}) and u(x_{i}) evaluated in this way are sometimes called a Type B variance and a Type B standard uncertainty, respectively.
NOTE When x_{i} is obtained from an a priori distribution, the associated variance is appropriately written as u^{2}(X_{i}), but for simplicity, u^{2}(x_{i}) and u(x_{i}) are used throughout this Guide.
4.3.2 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 If the probability distribution of q in Note 1 to 4.2.3 is normal, then σ[s(q‾‾)] ⁄ σ(q‾‾), the standard deviation of s(q‾‾) relative to σ(q‾‾), is approximately [2(n − 1)]^{−1/2}. Thus, taking σ[s(q‾‾)] as the uncertainty of s(q‾‾), for n = 10 observations, the relative uncertainty in s(q‾‾) is 24 percent, while for n = 50 observations it is 10 percent. (Additional values are given in Table E.l in Annex E.)
4.3.3 If the estimate x_{i} is taken from a manufacturer's specification, calibration certificate, handbook, or other source and its quoted uncertainty is stated to be a particular multiple of a standard deviation, the standard uncertainty u(x_{i}) is simply the quoted value divided by the multiplier, and the estimated variance u^{2}(x_{i}) is the square of that quotient.
EXAMPLE A calibration certificate states that the mass of a stainless steel mass standard m_{S} 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 u(m_{S}) = (240 µg) ⁄3 = 80 µg. This corresponds to a relative standard uncertainty u(m_{S}) ⁄m_{S} of 80 ×10^{−9} (see 5.1.6). The estimated variance is u^{2}(m_{S}) = (80 µg)^{2} = 6,4 × 10^{−9} g^{2}.
NOTE In many cases, little or no information is provided about the individual components from which the quoted uncertainty has been obtained. This is generally unimportant for expressing uncertainty according to the practices of this Guide since all standard uncertainties are treated in the same way when the combined standard uncertainty of a measurement result is calculated (see Clause 5).
4.3.4 The quoted uncertainty of x_{i} is not necessarily given as a multiple of a standard deviation as in 4.3.3. Instead, one may find it stated that the quoted uncertainty defines an interval having a 90, 95, or 99 percent level of confidence (see 6.2.2). Unless otherwise indicated, one may assume that a normal distribution (C.2.14) was used to calculate the quoted uncertainty, and recover the standard uncertainty of x_{i} 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 (see also Table G.1 in Annex G).
NOTE 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 (see 7.2.3).
EXAMPLE A calibration certificate states that the resistance of a standard resistor R_{S} 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 u(R_{S}) = (129 µΩ) ⁄2,58 = 50 µΩ, which corresponds to a relative standard uncertainty u(R_{S}) ⁄ R_{S} of 5,0 × 10^{−6} (see 5.1.6). The estimated variance is u^{2}(R_{S}) = (50 µΩ)^{2} = 2,5 × 10^{−9} Ω^{2}.
4.3.5 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 X_{i} lies in the interval a_{−} to a_{+}” (in other words, the probability that X_{i} lies within this interval is 0,5 or 50 percent). If it can be assumed that the distribution of possible values of X_{i} is approximately normal, then the best estimate x_{i} of X_{i} can be taken to be the midpoint of the interval. Further, if the half‑width of the interval is denoted by a = (a_{+} − a_{−}) ⁄2, one can take u(x_{i}) = 1,48a, because for a normal distribution with expectation μ and standard deviation σ the interval μ ± σ ⁄1,48 encompasses approximately 50 percent of the distribution.
EXAMPLE 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 u^{2}(l) = (1,48 × 0,04 mm)^{2} = 3,5 × 10^{−3} mm^{2}.
4.3.6 Consider a case similar to that of 4.3.5 but where, based on the available information, one can state that “there is about a two out of three chance that the value of X_{i} lies in the interval a_{−} to a_{+}” (in other words, the probability that X_{i} lies within this interval is about 0,67). One can then reasonably take u(x_{i}) = a. because for a normal distribution with expectation μ and standard deviation σ the interval μ ± σ encompasses about 68,3 percent of the distribution.
NOTE It would give the value of u(x_{i}) 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 u(x_{i}) = a⁄ 0,96742 = 1,033a.
4.3.7 In other cases, it may be possible to estimate only bounds (upper and lower limits) for X_{i}, in particular, to state that “the probability that the value of X_{i}, lies within the interval a_{−} to a_{+} for all practical purposes is equal to one and the probability that X_{i} lies outside this interval is essentially zero”. If there is no speciflc knowledge about the possible values of X_{i} within the interval, one can only assume that it is equally probable for X_{i} to lie anywhere within it (a uniform or rectangular distribution of possible values — see 4.4.5 and Figure 2 a). Then x_{i}, the expectation or expected value of X_{i}, is the midpoint of the interval, x_{i} = (a_{−} + a_{+}) ⁄2, with associated variance
If the difference between the bounds, a_{+} − a_{−}, is denoted by 2a, then Equation (6) becomes
NOTE When a component of uncertainty determined in this manner contributes significantly to the uncertainty of a measurement result, it is prudent to obtain additional data for its further evaluation.
EXAMPLE 1 A handbook gives the value of the coefficient of linear thermal expansion of pure copper at 20 °C, α_{20}(Cu), as 16,52 × 10^{−6} °C^{−1} and simply states that “the error in this value should not exceed 0,40 × 10^{−6} °C^{−1}”. Based on this limited information, it is not unreasonable to assume that the value of α_{20}(Cu) lies with equal probability in the interval 16,12 × 10^{−6} °C^{−1} to 16,92 × 10^{−6} °C^{−1}, and that it is very unlikely that α_{20}(Cu) lies outside this interval. The variance of this symmetric rectangular distribution of possible values of α_{20}(Cu) of half‑width a = 0,40 × 10^{−6} °C^{−1} is then, from Equation (7), u^{2}(α_{20}) = (0,40 ×10^{−6} °C^{−1})^{2} ⁄3 = 53,3 × 10^{−15} °C^{−2}, and the standard uncertainty is u(α_{20}) = (0,40 × 10^{−6} °C^{−1})⁄ √3^{‾‾} = 0,23 × 10^{−6} °C^{−1}.
EXAMPLE 2 A manufacturer's specifications for a digital voltmeter state that “between one and two years after the instrument is calibrated, its accuracy on the 1 V range is 14 × 10^{−6} times the reading plus 2 × 10^{−6} times the range”. Consider that the instrument is used 20 months after calibration to measure on its 1 V range a potential difference V, and the arithmetic mean of a number of independent repeated observations of V is found to be V^{‾‾‾} = 0,928 571 V with a Type A standard uncertainty u(V^{‾‾‾}) = 12 µV. One can obtain the standard uncertainty associated with the manufacturer's specifications from a Type B evaluation by assuming that the stated accuracy provides symmetric bounds to an additive correction to V^{‾‾‾}, ΔV^{‾‾‾}, of expectation equal to zero and with equal probability of lying anywhere within the bounds. The half‑width a of the symmetric rectangular distribution of possible values of ΔV^{‾‾‾} is then a = (14 × 10^{−6}) × (0,928 571 V) + (2 × 10^{−6}) × (1 V) = 15 µV, and from Equation (7), u^{2}(ΔV^{‾‾‾}) = 75 µV^{2} and u(ΔV^{‾‾‾}) = 8,7 µV. The estimate of the value of the measurand V, for simplicity denoted by the same symbol V, is given by V = V^{‾‾‾} + ΔV^{‾‾‾} = 0,928 571 V. One can obtain the combined standard uncertainty of this estimate by combining the 12 µV Type A standard uncertainty of V^{‾‾‾} with the 8,7 µV Type B standard uncertainty of ΔV^{‾‾‾}. The general method for combining standard uncertainty components is given in Clause 5, with this particular example treated in 5.1.5.
4.3.8 In 4.3.7, the upper and lower bounds a_{+} and a_{−} for the input quantity X_{i} may not be symmetric with respect to its best estimate x_{i}; more specifically, if the lower bound is written as a_{−} = x_{i} − b_{−} and the upper bound as a_{+} = x_{i} − b_{+}, then b_{−} ≠ b_{+}. Since in this case x_{i} (assumed to be the expectation of X_{i}) is not at the centre of the interval a_{−} to a_{+}, the probability distribution of X_{i} 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
EXAMPLE If in Example 1 of 4.3.7 the value of the coefficient is given in the handbook as α_{20}(Cu) = 16,52 × 10^{−6} °C^{−1} and it is stated that “the smallest possible value is 16,40 × 10^{−6} °C^{−1} and the largest possible value is 16,92 × 10^{−6} °C^{−1}”, then b_{−} = 0,12 × 10^{−6} °C^{−1}, b_{+} = 0,40 × 10^{−6} °C^{−1}, and, from Equation (8), u(α_{20}) = 0,15 × 10^{−6} °C^{−1}.
NOTE 1 In many practical measurement situations where the bounds are asymmetric, it may be appropriate to apply a correction to the estimate x_{i} of magnitude (b_{+} − b_{−}) ⁄2 so that the new estimate x′_{i} of X_{i} is at the midpoint of the bounds: x′_{i} = (a_{−} + a_{+})⁄2. This reduces the situation to the case of 4.3.7, with new values b′_{+} = b′_{−} = (b_{+} + b_{−}) ⁄2 = (a_{+} − a_{−}) ⁄2 = a.
NOTE 2 Based on the principle of maximum entropy, the probability density function in the asymmetric case may be shown to be p(X_{i}) = A exp[− λ(X_{i} − x_{i})], with A = [b_{−} exp(λb_{−}) + b_{+} exp(− λb_{+})]^{−1} and λ = {exp[λ(b_{−} + b_{+})] − 1} ⁄{b_{−} exp[λ(b_{−} + b_{+})] + b_{+}}. This leads to the variance u^{2}(x_{i}) = b_{+}b_{−} − (b_{+} − b_{−})⁄ λ; for b_{+} > b_{−}, λ > 0 and for b_{+} < b_{−}, λ < 0.
4.3.9 In 4.3.7, because there was no specific knowledge about the possible values of X_{i} within its estimated bounds a_{−} to a_{+}, one could only assume that it was equally probable for X_{i} 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 2aβ, where 0 ≤ β ≤ 1. As β → 1, this trapezoidal distribution approaches the rectangular distribution of 4.3.7, while for β = 0, it is a triangular distribution (see 4.4.6 and Figure 2 b). Assuming such a trapezoidal distribution for X_{i}, one finds that the expectation of X_{i} is x_{i} = (a_{−} + a_{+})⁄2 and its associated variance is
NOTE 1 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 X_{i} can be assumed to be approximately normally distributed rather than there being no specific knowledge about X_{i} between the bounds as in 4.3.7, then u^{2}(x_{i}) = a^{2}⁄ 9. By comparison, the variance of a symmetric rectangular distribution of half‑width a is a^{2} ⁄3 [Equation (7)] and that of a symmetric triangular distribution of half‑width a is a^{2} ⁄6 [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.
NOTE 2 The trapezoidal distribution is equivalent to the convolution of two rectangular distributions [10], one with a half‑width a_{1} equal to the mean half‑width of the trapezoid, a_{1} = a(1 + β) ⁄2, the other with a half‑width a_{2} equal to the mean width of one of the triangular portions of the trapezoid, a_{2} = a(1 − β) ⁄2. The variance of the distribution is u^{2} = a^{2}_{1} ⁄3 + a^{2}_{2} ⁄3. The convolved distribution can be interpreted as a rectangular distribution whose width 2a_{1} has itself an uncertainty represented by a rectangular distribution of width 2a_{2} and models the fact that the bounds on an input quantity are not exactly known. But even if a_{2} is as large as 30 percent of a_{1}, u exceeds a_{1} ⁄√3^{‾‾} by less than 5 percent.
4.3.10 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.
4.3.11 The discussion of Type B evaluation of standard uncertainty in 4.3.3 to 4.3.9 is meant only to be indicative. Further, evaluations of uncertainty should be based on quantitative data to the maximum extent possible, as emphasized in 3.4.1 and 3.4.2.
4.4.1 Figure 1 represents the estimation of the value of an input quantity X_{i} and the evaluation of the uncertainty of that estimate from the unknown distribution of possible measured values of X_{i}, or probability distribution of X_{i}, that is sampled by means of repeated observations.
a)
b)
4.4.2 In Figure 1 a), it is assumed that the input quantity X_{i} is a temperature t and that its unknown distribution is a normal distribution with expectation μ_{t} = 100 °C and standard deviation σ = 1,5 °C. Its probability density function (see C.2.14) is then
NOTE The definition of a probability density function p(z) requires that the relation ∫p(z)dz = 1 is satisfied.
4.4.3 Figure 1 b) shows a histogram of n = 20 repeated observations t_{k} of the temperature t that are assumed to have been taken randomly from the distribution of Figure 1 a). To obtain the histogram, the 20 observations or samples, whose values are given in Table 1, are grouped into intervals 1 °C wide. (Preparation of a histogram is, of course, not required for the statistical analysis of the data.)
Interval t_{1} ≤ t < t_{2} | Temperature | |
---|---|---|
t_{1} ⁄°C | t_{2} ⁄°C | t ⁄°C |
94,5 | 95,5 | — |
95,5 | 96,5 | — |
96,5 | 97,5 | 96,90 |
97,5 | 98,5 | 98,18; 98,25 |
98,5 | 99,5 | 98,61; 99,03; 99,49 |
99,5 | 100,5 | 99,56; 99,74; 99,89; 100,07; 100,33; 100,42 |
100,5 | 101,5 | 100,68; 100,95; 101,11; 101,20 |
101,5 | 102,5 | 101,57; 101,84; 102,36 |
102,5 | 103,5 | 102,72 |
103,5 | 104,5 | — |
104,5 | 105,5 | — |
The arithmetic mean or average t^{‾‾} of the n = 20 observations calculated according to Equation (3) is t^{‾‾} = 100,145 °C ≈ 100,14 °C and is assumed to be the best estimate of the expectation μ_{t} of t based on the available data. The experimental standard deviation s(t_{k}) calculated from Equation (4) is s(t_{k}) = 1,489 °C ≈ 1,49 °C, and the experimental standard deviation of the mean s(t^{‾‾}) calculated from Equation (5), which is the standard uncertainty u(t^{‾‾}) of the mean t^{‾‾}, is u(t^{‾‾}) = s(t^{‾‾}) = s(t_{k}) ⁄√20^{‾‾‾} = 0,333 °C ≈ 0,33 °C. (For further calculations, it is likely that all of the digits would be retained.)
NOTE Although the data in Table 1 are not implausible considering the widespread use of high‑resolution digital electronic thermometers, they are for illustrative purposes and should not necessarily be interpreted as describing a real measurement.
a)
b)
4.4.4 Figure 2 represents the estimation of the value of an input quantity X_{i} and the evaluation of the uncertainty of that estimate from an a priori distribution of possible values of X_{i}, or probability distribution of X_{i}, based on all of the available information. For both cases shown, the input quantity is again assumed to be a temperature t,
4.4.5 For the case illustrated in Figure 2 a), it is assumed that little information is available about the input quantity t and that all one can do is suppose that t is described by a symmetric, rectangular a priori probability distribution of lower bound a_{−} = 96 °C, upper bound a_{+} = 104 °C, and thus half‑width a = (a_{+} − a_{−}) ⁄2 = 4 °C (see 4.3.7). The probability density function of t is then
As indicated in 4.3.7, the best estimate of t is its expectation μ_{t} = (a_{+} + a_{−})⁄2 = 100 °C, which follows from C.3.1. The standard uncertainty of this estimate is u(μ_{t}) = a⁄ √3^{‾‾} ≈ 2,3 °C, which follows from C.3.2 [see Equation (7)].
4.4.6 For the case illustrated in Figure 2 b), it is assumed that the available information concerning t is less limited and that t can be described by a symmetric, triangular a priori probability distribution of the same lower bound a_{−} = 96 °C, the same upper bound a_{+} = 104 °C, and thus the same half‑width a = (a_{+} − a_{−}) ⁄2 = 4 °C as in 4.4.5 (see 4.3.9). The probability density function of t is then
As indicated in 4.3.9, the expectation of t is μ_{t} = (a_{+} + a_{−})⁄2 = 100 °C, which follows from C.3.1. The standard uncertainty of this estimate is u(μ_{t}) = a⁄√6^{‾‾} ≈ 1,6 °C, which follows from C.3.2 [see Equation 9 b)].
The above value, u(μ_{t}) = 1,6 °C, may be compared with u(μ_{t}) = 2,3 °C obtained in 4.4.5 from a rectangular distribution of the same 8 °C width; with σ = 1,5 °C of the normal distribution of Figure 1 a) whose − 2,58σ to + 2,58σ width, which encompasses 99 percent of the distribution, is nearly 8 °C; and with u(t^{‾‾}) = 0,33 °C obtained in 4.4.3 from 20 observations assumed to have been taken randomly from the same normal distribution.