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Water is the most frequently and thoroughly characterised product due to the impact of the chemical composition of water of different sources or destinations on public health and on economy. The adequacy of water characterisation relies on measurement quality, which is a function of measurement traceability and uncertainty. In some analytical fields, target values of measurement performance parameters are set to ensure that the measurements quality is fit for the intended use. Nevertheless, frequently, these performance parameters do not allow the control of the magnitude of relevant components of measurement uncertainty. This work discusses the need for assessing fitness of the measurement for its intended use through the magnitude of uncertainty associated to the measurement value. The way this evaluation should be performed, when no guidelines are available, is also suggested. Target values of relevant performance parameters, results of interlaboratory tests, or the magnitude of trends of the measured quantity, are some types of information useful to define the maximum admissible uncertainty,

Water is the most frequently and thoroughly characterised product, in particular when intended for human consumption. Drinking water is monitored before and after treatment to guarantee high quality of this essential food. Urban and industrial wastewaters are also monitored before and after treatment to check treatment efficiency and ensure minimum impact of effluent discharges in the environment. The environment is subsequently monitored to check the adequacy of the regulation of contamination sources. Frequently, this involves the characterisation of surface or underground waters that due to their mobility can supply information about the state of the local environmental resources. Water of bathing and recreation areas is also controlled having in mind its possible impact on the user’s health. Water quality is also extremely important in industry; food, energy and transformation industry are some examples where water composition is critical.

Most commonly, water is monitored for checking compliance with relevant legislation or specification, considering a maximum and/or a minimum quantity defined as “target quantity value”. However, many relevant characterizations of water are performed without having a target quantity level defined. The quantification of a new contaminant, the assessment of variation of the eutrophic level of a lake in a dry season, or the determination of seawater acidification with global environmental changes are a few examples. In this work, the generic term “quantity” is preferred to specific examples such as, e.g., concentration, mass fraction, depletion rate or pH.

The quality of the information collected from water characterisation depends both on sample representativeness and measurement quality.

In most cases, the routine monitoring or enforcement of legislation of water quality involves the characterisation of a sample collected following a defined procedure without the representativeness of the sample being checked. This approach aims at saving resources needed for the sound characterisation of a large item. Nevertheless, whenever a large item needs to be characterised in detail, the representativeness of the sample must be checked or ensured through an adequate sampling protocol. In this case, the sampling stage is included in the measurement procedure.

Measurement fitness for the intended use depends on measurement results’ traceability and uncertainty. The latest edition of the international vocabulary of metrology (VIM) [

The quality of a measurement traceable to the selected reference is quantified by its respective “measurement uncertainty”. This is defined as the “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used” [

The latest edition of the VIM [

In some analytical fields, target values of measurement precision and mean analyte recovery of trueness tests are defined to guarantee acceptable quality of performed measurements. Nevertheless, depending on the way these performance parameters are defined or estimated, their control may or may not be enough to ensure acceptable uncertainty of measurement results.

If measurement repeatability is checked considering a target value, this will only prove that precision is fit for the intended use in the specific conditions under which the procedure was conducted. Usually, this conclusion cannot be extrapolated to different conditions; e.g., to a subsequent day. When precision components of the measurement uncertainty are estimated in repeatability conditions, additional uncertainty components must be considered to take the between days components into account.

For the purpose of the trueness test, if the mean analyte recovery is estimated through the analysis of spiked samples prepared from a stock solution used to produce calibrators, the uncertainty associated with the quantity value of the stock solution is not being tested. Therefore, if results of trueness test do not reflect relevant systematic effects, like the uncertainty of the quantity value of the reference solution used to prepare both spiked samples and calibrators, results of trueness tests must be combined with other uncertainty components in the uncertainty budget.

The assessment of the magnitude of measurement uncertainty in the analytical range is the last stage of validation of the measurement procedure that closes the previous preliminary assessments.

In most measurements of chemical quantities in water, target measurement uncertainty is not defined even when the target quantity value is set. This situation forces analysts to define the target measurement uncertainty themselves. This work suggests a strategy for defining the target uncertainty in water analysis, when no complete guidance is available, using information from different sources. This information is presented from the most adequate to the less likely to produce consensual target uncertainties.

This manuscript intends to be a support for any research or compliance assessment based on quantitative measurements in water.

Guidance about the acceptable magnitude of measurement uncertainty should start with consultation of the document—typically a legislation or a technical specification—defining the target quantity value.

In some fields, target values of measurement performance parameters, like limit of detection, precision and mean analyte recovery, are defined instead of the target measurement uncertainty. In those cases, if such performance parameters reflect the most relevant random and systematic effects, they can be converted into a target measurement uncertainty.

The measurement quality requirements can also be deduced from the quantity value beyond which a wrong compliance decision should be unlikely.

When no target values are defined, measurement quality requirements can be defined from the way laboratory competence is checked in, e.g., proficiency tests. International proficiency tests are preferred to national ones if measurements are to be compared at an international level.

If none of the previous references are available, information from a different analytical problem pertaining to the analyte, the matrix or the test purpose, may be used for defining the target measurement uncertainty. The equivalence of the process of defining target quantity values can justify the extrapolation of the target measurement uncertainty from one analytical problem to another. In any such cases, arguments must be presented in a convincing way.

When target values of measurement performance parameters are available for a specific quantity value and adequacy of measurements at other quantity values need to be assessed, a prediction model of the variation of measurement performance with the quantity can be used to define the requirements at those ranges.

If measurement performance does not meet the defined requirements, the measurement procedure should be revised or changed to satisfy the quality requirements.

This section details the way to proceed when different types of references or data are available for defining the target measurement uncertainty. The types of data presented range from the most adequate sources for target measurement uncertainty to the sources less likely to become consensual.

If legislation or specification defining target quantity values also define the target measurement uncertainty, compliance assessment should be supported on measurements fulfilling the defined quality requirements. In most of these cases, target measurement uncertainty is defined for measurements next to the target quantity value.

Since the absolute uncertainty

Expected trend of the variation of the absolute

The relative target uncertainty should be used up to two times above the target quantity value and the absolute target uncertainty up to two times below the target quantity value. This range of quantity values is the relevant one for compliance assessment. If target measurement uncertainty needs to be defined for a wider range, see

In some legislation or technical specifications, target values of measurement performance parameters related to relevant uncertainty components are defined. Maximum limit of detection and/or quantification, maximum range of results of duplicate measurements or maximum coefficient of variance of results of replicate measurements obtained under different precision conditions, and maximum absolute mean error, are some examples.

Before checking measurement performance, analysts must verify how terms are defined in the reference, since sometimes performance parameters are presented using terminology different from the one presented in the International Vocabulary of Metrology (VIM) [

If measurement performance parameters for which target values are defined reflect relevant random and systematic effects, this can be used alone to estimate the target measurement uncertainty.

Measurement performance parameters for which defined target values can be used to estimate the target measurement uncertainty.

Performance |
Description |
---|---|

Limit of |
The Limit of Detection (LD) can be estimated in repeatability or intermediate precision conditions. For instrumental methods of analysis requiring daily calibration of the instrumentation, LD estimated in repeatability conditions is only applicable to the daily run. The LD estimated from the precision of measurements of different calibrations can be applicable to a larger time scale. At this quantity level, the measurement coefficient of variance is 33% or 30% if LD is calculated by multiplying precision standard deviation by 3 or 3.3, respectively. Since absolute precision is constant in a narrow quantity range, the precision estimated at LD can be used to estimate precision between LD and the Limit of Quantification (LQ) (3 or 3.3 times larger than LD). Only seldom relevant systematic effects affect measurements at this quantity range [LD, LQ[. Convergent (“[#” or “#]”) or divergent (“]#” or “#[”) brackets indicate the inclusion or exclusion of number “#” in the interval, respectively.^{tg}^{tg}^{tg}^{tg}^{tg} |

Limit of |
The calculation of the Limit of Quantification (LQ) is similar to that involved to estimate LD where the multiplying factor of the standard deviation of measurement precision is 10 instead of 3 or 3.3. At this quantity level, systematic effects can be relevant. |

Limit of |
A target limit of quantification, ^{tg}^{tg}^{tg}^{tg}^{tg} |

Range of |
Whenever a target range of results of duplicate measurements is defined, the respective confidence level and involved precision conditions should be checked. If the confidence level is not reported, a value of 95% should be considered. Since the repeatability or intermediate precision limits [^{tg}^{tg} |

Coefficient of |
If a target coefficient of variance is defined without specifying the precision conditions considered (typically repeatability or intermediate precision conditions), it can be assumed that the more informative intermediate precision is reported. Many references of measurement performance parameters do not use terminology of the latest, or even previous, VIM [ |

Mean error | Measurement error, _{true}_{true}_{m} |

The target limit of detection (^{tg}^{tg}^{tg}

LD can be estimated as 3 or 3.3 times the standard deviation of measurement results obtained under repeatability or intermediate precision conditions. Therefore, ^{tg}^{tg}^{tg}^{tg}^{tg}

The ^{tg}^{tg}^{tg}^{tg}^{tg}^{tg}^{tg}

If ^{tg}^{tg}^{tg}^{tg}

If only target values of the measurement repeatability are defined, analysts can assume reproducibility standard deviation is 3/2 times larger than the repeatability standard deviation, as proposed in a SANCO document [

For measurements performed above 5LQ, ^{tg}

The target absolute mean error

Some references define a tolerance for the mean analyte recovery observed from the analysis of spiked samples or other reference material. This tolerance can be converted into the relative mean error allowing for estimation of

The target measurement uncertainty

Measurement error,

The mean error _{m}_{m}

The target standard uncertainty (

In some analytical fields, the target quantity values are defined without guidelines on the quality of measurements performed to check compliance with these levels. If a single minimum or maximum quantity value is defined, measurement quality should be assessed, at least, at this level. If a compliance interval of the quantity value is defined, measurement performance within and next to this interval should be assessed. The following section describes the way to define the target uncertainty for the assessment of the compliance of an item with a maximum or minimum quantity value.

If the compliance rule defines a maximum (^{max}^{min}^{tg}

If together with the target quantity value and the decision rule of compliance assessment [

(i) Maximum target quantity value ^{tg}^{tr}

If the following conditions for deciding product compliance are applicable (C1 to C3), the target standard uncertainty ^{tg}

(C1) Measurement uncertainty: Measurement results are associated with a normal distribution and the absolute measurement uncertainty is approximately constant in a relevant range next to the target quantity value;

(C2) Decision rule: The product is considered not compliant if the measured quantity value, ^{tg}^{tg}_{1}

(C3) Decision risk: A minimum threshold quantity, ^{tr}^{tr}^{tg}

The standard uncertainty ^{tr}^{tq}^{tg}_{1}u^{tr}

If the previously defined C2 condition, regarding the decision rule, is changed to “compliance is decided without taking measurement uncertainty into account”, a larger target measurement uncertainty is set [Equation (4)] (

The standard uncertainty ^{tr}^{tg}^{tg}^{tr}

(ii) Minimum target quantity value ^{tg}^{tr}

If a minimum, instead of a maximum, quantity value is defined, and together with this a maximum threshold quantity, ^{tr}^{tg}^{tg}

The standard uncertainty ^{tr}^{tq}^{tg}_{1}u^{tr}

The standard uncertainty ^{tr}^{tg}^{tg}^{tr}

(iii) Maximum target quantity value ^{tg}^{tr}

The measurement quality requirement can focus on the chance of measurement of a compliant product producing a measured quantity value above a maximum threshold quantity, ^{tr}^{tg}

The standard uncertainty ^{tg}^{tr}

(iv) Minimum target quantity value ^{tg}^{tr}

If a minimum target quantity, ^{tg}^{tr}^{tg}

The standard uncertainty ^{tg}^{tr}

The criteria of ^{tr}^{tr}^{tg}

If reference documents of the target quantity value of the analysed items do not define measurement quality requirements, this information can be obtained from interlaboratory performance data or from how measurements are assessed in proficiency tests or other interlaboratory comparisons.

In most analytical fields, performance in proficiency tests is evaluated by calculating z-scores estimated by the ratio between the measurement error and a reference standard deviation [Equation (7)].
_{i}_{true}_{true}_{true}

The reference standard deviation of the proficiency test σ can be used to define the target standard uncertainty (

This strategy is preferable to the use of target intervals of the quantity value to define the target measurement uncertainty (see end of

Whenever the standard deviation of the measurement reproducibility _{R}_{R}_{R}_{R}_{tg}^{tg}_{tg}

For rational measurements, where bias attributed to the physical-chemical principles of the procedure δ can be significant, if _{R}^{tg},^{tg}_{R}^{tg}

Some producers of Certified Reference Materials (CRM) present, together with the certified value with respective uncertainty, the proposed maximum absolute error of single measurements performed in the routine analysis of the CRM. This information, frequently presented as an internal between the maximum _{+}_{−}^{tg}

The uncertainty associated with the certified value is inadequate to this assessment since reflects a much larger effort in the characterisation of the material than in single routine measurements.

If the uncertainty associated with the certified value _{CRM}

Many important measurements are performed without having a target value of the measured quantity. The monitoring of a new contaminant in surface water, the study of the depletion of an endocrine disruptor after wastewater treatment and the composition of thermal water are just some examples. In these cases, measurement quality should be adequate to detect predicted trends or differences of items to be distinguished. The expected rate of variation of the analysed parameter in the studied time scale or the expected heterogeneity of studied property in items to be compared should be adequately studied by developed measurement procedure. Measurement standard uncertainty should be, at least, 4.24 times smaller than trends or differences needed to be distinguished, to be fit for the intended use. For instance, if methylparaben depletion in wastewater of more than 10% needs to be detected, measurement procedure should be developed to ensure the determination of the depletion rate with a standard uncertainty not larger than 2.4% (

The 4.24 factor is estimated from the equation used to check the compatibility of measurement results for a confidence level of 99% [_{A}_{A}_{A}_{B}_{B}_{B}_{i}_{i}_{i}_{AB} = ǀ_{A}_{B}_{d}_{d}_{d}_{A})^{2} + (_{B})^{2}]^{½}} for a confidence level of 99% and the degrees of freedom of _{d}

If _{A}_{B}_{d}_{A}_{B}_{A}_{B}_{A}_{B}

Therefore, the target standard uncertainty ^{tg} allowed to distinguish a minimum range, ^{min}_{A}_{B}^{min}^{min}_{d}_{d}

Many relevant analytical measurements are performed where no target values of measurement performance parameters are supplied, or proficiency tests or other comparisons were or are regularly promoted. In these cases, target uncertainty can be defined considering target values of performance parameters of related measurements.

The target quantity values and target measurement uncertainties are defined allowing for the impact of the measured quantity on the managed interests, ranging from individual or public health matters to issues relevant to the economy. If similarity or relations between the managed interests are identified, target measurement uncertainty defined for one “analyte/matrix/measurement goal” combination can be used to define target measurement uncertainty in other analytical problem. This extrapolation is more straightforward for more similar or more related analytical problems.

When clear differences in the demand of the control of two quantities are observed, this can be used to justify the adopted proportion between the respective target measurement uncertainties. For instance, the target uncertainty of measurements of lead in drinking water should be smaller than the one associated with measurements of lead in wastewaters.

This extrapolation is less obvious for different parameters but it is also possible. For instance, the relative uncertainty associated with measurements of hardness in drinking water can be larger than the one associated with measurements of chloride or sodium for which target quantity values are defined in Directive 98/83/EC [

Any transference of the target measurement uncertainty should be clearly justified. Consecutive transference of information from one analytical problem to another should be avoided since it tends to become less consensual.

The definition of the target measurement uncertainty should balance the need to ensure acceptance, by an individual or the community, with the feasibility of the target measurement uncertainty being reached considering the state-of-the-art of measurement procedures. Often, people without analytical or metrological background in the field of measurements tend to request unrealistically low measurement uncertainty. In these cases, it is the analyst’s responsibility to make it clear why the proposed target measurement uncertainty is adequate and/or possible.

If a target measurement uncertainty is, or can be, defined for only some quantity values and measurement adequacy must be assessed in a range of quantities, the expected variation of the measurement uncertainty with the quantity can be used to define target uncertainty values for the analytical scope.

The absolute measurement uncertainty

Expected trends of the variation of (

The relative measurement uncertainty

These frequent trends suggest that a target relative measurement uncertainty set at a quantity value is feasible above that level, and a target absolute measurement uncertainty is applicable down to five times below this level (

(^{tg}^{tg}

Since target measurement uncertainty is particularly important next to the target quantity value, any extrapolation to this critical value should involve a reflexion of the adequacy of the measurement requirements.

The comparison of estimated and target measurement uncertainties should be performed keeping in mind the uncertainty associated with uncertainty estimation itself [

If the estimated measurement uncertainty ^{tg}^{tg}^{tg}

Since most results of measurements in chemistry are, at least, approximately normally distributed, the estimated ^{tg}^{tg}^{tg}^{tg}^{tg}^{tg}^{tg}^{tg}/2

The comparison of ^{tg}^{tg}_{tg}

If ^{tg}

At the end of measurement procedure validation, when measurement procedure is checked to be fit for intended use, prediction models of the measurement uncertainty estimated for a relevant quantity range are compared with the target measurement uncertainty ^{tg}^{max}

The target measurement uncertainty ^{tg}^{max}

If comparison between estimated and target measurement uncertainties proves measurements are not fit for the intended use in relevant quantity values or ranges, measurement uncertainty must be reduced. Measurement uncertainty can be reduced if relevant uncertainty components can be minimized. The so called bottom-up approach [

The standard deviation of the intermediate precision _{IP}_{r}_{BD}_{IP}_{r}^{2} + (_{BD}^{2}]^{½}}. The _{BD}_{IP}_{r}_{BD}

The standard deviation of the intermediate precision, adequately estimating relevant random effects (_{ra}_{IP}

(i) If results of unknown samples are estimated by the mean of _{PI(1)}

(ii) If replicated measurements are performed under intermediate precision conditions, the standard deviation of the intermediate precision of this mean of the _{IP(2)}_{IP}

(iii) Both equations can be combined if the reported result is the mean of the

This strategy to improve precision—in many cases, a major uncertainty component—can be particularly useful for measurements next to the target quantity value. In that case, the extra experimental effort will be performed only when strictly needed.

The following sections present examples of the definition of the target uncertainty in the previously described scenarios.

European legislation defines target quantity values of several chemical parameters in water intended for the abstraction of drinking water, environmental water, bathing water, and urban and industrial wastewater [

In the European Union, the monitoring of drinking water quality must be supported by measurements and measurement procedures fulfilling the requirements presented in Council Directive 98/83/EC [

For the determination of Cd in drinking water, the parametric value is 5 μg L^{−1}, and τ and π are 0.5 μg L^{−1} (τ = π = 10%·5 μg L^{−1}).

According to Equation (1):

For instance, if the standard uncertainty ^{−1}, measurement is not fit for the intended use since ^{−1} (

In the European Union, the quality of bathing water is regulated by Directive 76/160/EEC [^{tg}^{tg}^{'tg}

In 2007, the Institute for Reference Materials and Measurements (IRMM) of the European Commission promoted a proficiency test, for National Reference Laboratories appointed by the Member States, for the determination of Cd, Pb and Hg in mineral water [^{'tg}

The relative standard deviation of the reproducibility of measurements of 300 µg L^{−1} of copper in wastewaters, by electrothermal atomic absorption spectrometry following the SMEWW3113B (SMEWW-Standard Methods for the Examination of Water & Wastewater) procedure, is 14% [

The certificate of the reference material “EnviroMAT EU-H-1” of the mass concentration of metals in wastewater [^{−1} for a confidence level of 95% and the tolerance of the measured quantity value estimated from single measurements is (0.73 ± 0.08) mg L^{−1} for the same confidence level. Therefore, the producer of this reference material suggests a target expanded measurement uncertainty of 0.079 mg L^{−1} estimated by Equation (19) (

If the variability of the uncertainty estimation process is also considered, the expanded uncertainty of measurements of Pb in wastewater are adequate if smaller than 0.092 mg L^{−1}. This criterion is defined considering measurement quality requirements defined by the reference material producer. This target uncertainty can be applied in a quantity range of 0.15–3.6 mg L^{−1} (see ^{−1} is acceptable.

Regardless of the defined ^{tg}

The optimisation of a wastewater treatment scheme, by changing conditions in a pilot plant, is supported by the percentage reduction of the chemical oxygen demand (COD) with the treatment. If variations in COD reduction of 5% are considered relevant, the determination of the COD reduction should be affected by a standard uncertainty not larger than 1.18% (

The hardness of drinking water—measured as the combined concentration of calcium and magnesium ions reported in mass concentration of CaCO_{3}—is determined due to its impact on the organoleptic quality of water, performance of detergents and production of incrustation in heating water devices. Nevertheless, no target quantity values are defined for the mass concentration of calcium, magnesium or hardness in Directive 98/83/EC [^{−1}) and the respective target performance parameters indicate a relative target expanded uncertainty of 14% next to the target quantity value (see

If the variability of the measurement uncertainty estimation process is considered and uncertainty is estimated with a high number of degrees of freedom, the quantified relative expanded uncertainty can reach up to 32%.

Various interlaboratory trials of the determination of copper in wastewater by electrothermal atomic absorption spectrometry (ETAAS), following SMEWW3113B procedure [

Relative standard deviation of measurement reproducibility,

Mass concentration of Cu (μg L^{−1}) |
||||

Wastewater 1 | Wastewater 2 | Wastewater 3 | Maximum | |

234 | 21 | - | 26 | 26 |

300 | - | 14 | - | 14 |

1670 | - | 13 | - | 13 |

_{R}^{tg}^{'tg}

Predictive worst case model of the variation of _{R}^{tg}

Mass concentration of copper (μg L^{−1}) |
_{R}^{tg}^{−1}) |
^{'tg} |
---|---|---|

[2.02, 10.1[ | 7.5 | - |

[10.1, 234[ | - | 74 |

[234, 300[ | - | 26 |

[300, 1670[ | - | 14 |

[1670, (…)[ | - | 13 |

Assuming measurement uncertainty is estimated with a high number of degrees of freedom, the maximum admissible absolute, ^{max}^{'max}^{tg}^{'tg}

Maximum admissible absolute ^{max}^{'max}

Mass concentration of copper (μg L^{−1}) |
^{max}^{−1}) |
^{'max} |
---|---|---|

[2.02, 10.1[ | 8.7 | - |

[10.1, 234[ | - | 86 |

[234, 300[ | - | 30 |

[300, 1670[ | - | 16 |

[1670, (…)[ | - | 15 |

For instance, if the mass concentration of copper in a wastewater sample determined in a routine laboratory, using procedure SMEWW3113B, is (277 ± 95) mg L^{−1} (for a coverage factor of 2 defined for 95% confidence level from a high number of degrees of freedom), it can be concluded the measurement is fit for the intended use since it presents a relative standard uncertainty ^{'}^{'}^{'max}

The fitness of a measurement for its intended use is demonstrated by the defined measurement traceability and magnitude of the measurement uncertainty. Therefore, the validation of measurements of water for different uses require observation of a target uncertainty (

The authors declare no conflict of interest.