Step 4. Calculating the Combined Uncertainty

8. Step 4. Calculating the Combined Uncertainty

8.1. Standard uncertainties

appropriate to assume a triangular distribution, with a standard deviation of a/ √ 6 (see Appendix

8.1.1. Before combination, all uncertainty

E).

contributions must be expressed as standard uncertainties, that is, as standard deviations. This

EXAMPLE

may involve conversion from some other measure

A 10 ml Grade A volumetric flask is certified to

of dispersion. The following rules give some

within ±0.2 ml, but routine in-house checks

guidance for converting an uncertainty

show that extreme values are rare. The standard

component to a standard deviation.

uncertainty is 0.2/ √ 6 ≈

0.08 ml.

8.1.6. Where an estimate is to be made on the evaluated experimentally from the dispersion of

8.1.2. Where the uncertainty component was

basis of judgement, it may be possible to estimate repeated measurements, it can readily be

the component directly as a standard deviation. If expressed as a standard deviation. For the

this is not possible then an estimate should be contribution to uncertainty in single

made of the maximum deviation which could measurements, the standard uncertainty is simply

reasonably occur in practice (excluding simple the observed standard deviation; for results

mistakes). If a smaller value is considered subjected to averaging, the standard deviation

substantially more likely, this estimate should be of the mean [B.24] is used.

treated as descriptive of a triangular distribution.

8.1.3. Where an uncertainty estimate is derived If there are no grounds for believing that a small from previous results and data, it may already be

error is more likely than a large error, the expressed as a standard deviation. However

estimate should be treated as characterising a where a confidence interval is given with a level

rectangular distribution.

of confidence, (in the form ±a at p % ) then divide

8.1.7. Conversion factors for the most commonly the value a by the appropriate percentage point of

used distribution functions are given in Appendix the Normal distribution for the level of

E.1.

confidence given to calculate the standard deviation.

8.2. Combined standard uncertainty

EXAMPLE

8.2.1. Following the estimation of individual or

A specification states that a balance reading is

groups of components of uncertainty and

within ±0.2 mg with 95 % confidence. From

expressing them as standard uncertainties, the

standard tables of percentage points on the

next stage is to calculate the combined standard

normal distribution, a 95% confidence interval

uncertainty using one of the procedures described

is calculated using a value of 1.96 σ . Using this

below.

figure gives a standard uncertainty of (0.2/1.96) ≈ 0.1.

8.2.2. The general relationship between the combined standard uncertainty u c (y) of a value y

8.1.4. If limits of ±a are given without a and the uncertainty of the independent parameters confidence level and there is reason to expect that

x 1 ,x 2 , ...x n on which it depends is extreme values are likely, it is normally appropriate to assume a rectangular distribution, * u

c (y(x 1 ,x 2 ,.. .)) =

with a standard deviation of a/ √ 3 (see Appendix

E). where y(x 1 ,x 2 ,.. ) is a function of several

EXAMPLE

parameters x 1 ,x 2 ..., c i is a sensitivity coefficient evaluated as c i = ∂ y/ ∂ x i , the partial differential of y

A 10 ml Grade A volumetric flask is certified to

with respect to x

i and u(y,x i ) denotes the uncertainty in y arising from the uncertainty in x .

within ±0.2 ml. The standard uncertainty is

0.2/ √ 3 ≈ 0.12 ml.

Each variable's contribution u(y,x i ) is just the

8.1.5. If limits of ±a are given without a

confidence level, but there is reason to expect that

* The ISO Guide uses the shorter form u i (y) instead of

extreme values are unlikely, it is normally

u (y,x i )

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Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty

square of the associated uncertainty expressed as that this method, or another appropriate computer-

a standard deviation multiplied by the square of based method, be used for all but the simplest the relevant sensitivity coefficient. These

cases.

sensitivity coefficients describe how the value of

8.2.6. In some cases, the expressions for

y varies with changes in the parameters x 1 ,x 2 etc.

combining uncertainties reduce to much simpler

N OTE : Sensitivity coefficients may also be evaluated

forms. Two simple rules for combining standard

directly by experiment; this is particularly

uncertainties are given here.

valuable where no reliable mathematical description of the relationship exists.

Rule 1

8.2.3. Where variables are not independent, the For models involving only a sum or difference of relationship is more complex:

quantities, e.g. y=(p+q+r+...), the combined standard uncertainty u c (y) is given by u y x

i , j ... )) =

u ( y ( p , q .) = u ( p ) 2 + u ( q ) 2 + .....

Rule 2

where u(x i ,x k ) is the covariance between x i and x k and c i and c k are the sensitivity coefficients as

For models involving only a product or quotient, described and evaluated in 8.2.2. The covariance

e.g. y=(p × q × r × ...) or y= p / (q × r × ...) , the is related to the correlation coefficient r ik by

combined standard uncertainty u c (y) is given by u (x i ,x k ) = u(x i ) ⋅ u (x k ) ⋅ r ik

where -1 ≤ r ik ≤ 1.

 p 

 q 

8.2.4. These general procedures apply whether where (u(p)/p) etc. are the uncertainties in the the uncertainties are related to single parameters,

parameters, expressed as relative standard grouped parameters or to the method as a whole.

deviations.

However, when an uncertainty contribution is associated with the whole procedure, it is usually

N OTE Subtraction is treated in the same manner as

expressed as an effect on the final result. In such

addition, and division in the same way as

cases, or when the uncertainty on a parameter is

multiplication.

expressed directly in terms of its effect on y, the

8.2.7. For the purposes of combining uncertainty sensitivity coefficient ∂ y / ∂ x i is equal to 1.0.

components, it is most convenient to break the original mathematical model down to expressions

EXAMPLE

which consist solely of operations covered by one

A result of 22 mg l -1 shows a measured standard

of the rules above. For example, the expression

deviation of 4.1 mg l -1 . The standard uncertainty u(y) associated with precision under

these conditions is 4.1 mg l . The implicit

model for the measurement, neglecting other

should be broken down to the two elements (o+p)

factors for clarity, is

and (q+r). The interim uncertainties for each of

y = (Calculated result) + ε

these can then be calculated using rule 1 above;

where ε represents the effect of random

these interim uncertainties can then be combined

variation under the conditions of measurement.

using rule 2 to give the combined standard

∂ y / ∂ε is accordingly 1.0

uncertainty.

8.2.8. The following examples illustrate the use of sensitivity coefficient is equal to one, and for the

8.2.5. Except for the case above, when the

the above rules:

special cases given in Rule 1 and Rule 2 below,

EXAMPLE 1

the general procedure, requiring the generation of partial differentials or the numerical equivalent

y = (p-q+r) The values are p=5 . 02, q=6 . 45 and

must be employed. Appendix E gives details of a

r =9 . 04 with standard uncertainties u(p)=0 . 13,

numerical method, suggested by Kragten [H.12], u (q)=0 . 05 and u(r)= 0 . 22. which makes effective use of spreadsheet

y = 5.02 - 6.45 + 9.04 = 7.61

software to provide a combined standard uncertainty from input standard uncertainties and

a known measurement model. It is recommended

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Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty

EXAMPLE 2

• Any knowledge of the number of values used

y = (op/qr). The values are o=2.46, p=4.32,

to estimate random effects (see 8.3.3 below).

q =6.38 and r=2.99, with standard uncertainties

8.3.3. For most purposes it is recommended that k

of u(o)=0.02, u(p)=0.13, u(q)=0.11 and u(r)=

is set to 2. However, this value of k may be

insufficient where the combined uncertainty is

y =( 2.46 × 4.32 ) / (6.38 ×

based on statistical observations with relatively

2 2 few degrees of freedom (less than about six). The

choice of k then depends on the effective number

u ( y ) = 0 . 56 × 

of degrees of freedom.

8.3.4. Where the combined standard uncertainty

is dominated by a single contribution with fewer

⇒ u(y) = 0.56 × 0.043 = 0.024

than six degrees of freedom, it is recommended that k be set equal to the two-tailed value of

8.2.9. There are many instances in which the Student’s t for the number of degrees of freedom magnitudes of components of uncertainty vary

associated with that contribution, and for the with the level of analyte. For example,

level of confidence required (normally 95%). uncertainties in recovery may be smaller for high

Table 1 (page 28) gives a short list of values for levels of material, or spectroscopic signals may

vary randomly on a scale approximately proportional to intensity (constant coefficient of

EXAMPLE:

variation). In such cases, it is important to take

A combined standard uncertainty for a weighing

account of the changes in the combined standard

operation is formed from contributions

uncertainty with level of analyte. Approaches

u cal =0.01 mg arising from calibration

include:

uncertainty and s obs =0.08 mg based on the

• standard deviation of five repeated Restricting the specified procedure or

observations. The combined standard

uncertainty estimate to a small range of

2 uncertainty u 2 c is equal to 0 . 01 + 0 . 08 analyte concentrations.

=0.081 mg. This is clearly dominated by the

• Providing an uncertainty estimate in the form

repeatability contribution s obs , which is based on

of a relative standard deviation.

five observations, giving 5-1=4 degrees of freedom. k is accordingly based on Student’s t.

• Explicitly calculating the dependence and

The two-tailed value of t for four degrees of

recalculating the uncertainty for a given

freedom and 95% confidence is, from tables,

result.

2.8; k is accordingly set to 2.8 and the expanded uncertainty U=2.8 × 0.081=0.23 mg.

Appendix E4 gives additional information on these approaches.

8.3.5. The Guide [H.2] gives additional guidance on choosing k where a small number of

8.3. Expanded uncertainty

measurements is used to estimate large random effects, and should be referred to when estimating

8.3.1. The final stage is to multiply the combined degrees of freedom where several contributions standard uncertainty by the chosen coverage

are significant.

factor in order to obtain an expanded uncertainty. The expanded uncertainty is required to provide

8.3.6. Where the distributions concerned are an interval which may be expected to encompass

normal, a coverage factor of 2 (or chosen

a large fraction of the distribution of values which according to paragraphs 8.3.3.-8.3.5. using a level could reasonably be attributed to the measurand.

of confidence of 95%) gives an interval containing approximately 95% of the distribution

8.3.2. In choosing a value for the coverage factor of values. It is not recommended that this interval k , a number of issues should be considered. These

is taken to imply a 95% confidence interval include:

without a knowledge of the distribution • The level of confidence required

concerned.

• Any knowledge of the underlying distributions

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Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty

Table 1: Student’s t for 95% confidence (2-tailed)

Degrees of freedom

νν t

Page 28

Quantifying Uncertainty Reporting Uncertainty