Instant Notes-Analytical Chemistry

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Analytical Chemistry

ii Section K Lipid metabolism

The INSTANT NOTES series

Series editorB.D. HamesSchool of Biochemistry and Molecular Biology, University of Leeds, Leeds, UK

Animal BiologyBiochemistry 2nd editionChemistry for BiologistsDevelopmental BiologyEcology 2nd editionGeneticsImmunologyMicrobiologyMolecular Biology 2nd editionNeurosciencePlant BiologyPsychology

Forthcoming titlesBioinformatics

The INSTANT NOTES Chemistry seriesConsulting editor: Howard Stanbury

Analytical ChemistryInorganic ChemistryMedicinal ChemistryOrganic ChemistryPhysical Chemistry

Analytical Chemistry

D. Kealey

School of Biological and Chemical SciencesBirkbeck College, University of London, UK

andDepartment of Chemistry

University of Surrey, Guildford, UK

and

P. J. Haines

Oakland Analytical Services,Farnham, UK

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BIOS Scientic Publishers Limited, 2002

First published 2002 (ISBN 1 85996 189 4)

All rights reserved. No part of this book may be reproduced or transmitted, in any form orby any means, without permission.

A CIP catalogue record for this book is available from the British Library.

ISBN 1 85996 189 4

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Abbreviations viiPreface ix

Section A The nature and scope of analytical chemistry 1A1 Analytical chemistry, its functions and applications 1A2 Analytical problems and procedures 3A3 Analytical techniques and methods 5A4 Sampling and sample handling 10A5 Calibration and standards 15A6 Quality in analytical laboratories 18

Section B Assessment of data 21B1 Errors in analytical measurements 21B2 Assessment of accuracy and precision 26B3 Signicance testing 34B4 Calibration and linear regression 41B5 Quality control and chemometrics 49

Section C Analytical reactions in solution 55C1 Solution equilibria 55C2 Electrochemical reactions 61C3 Potentiometry 66C4 pH and its control 74C5 Titrimetry I: acidbase titrations 80C6 Complexation, solubility and redox equilibria 85C7 Titrimetry II: complexation, precipitation and redox

titrations 90C8 Gravimetry 95C9 Voltammetry and amperometry 98C10 Conductimetry 104

Section D Separation techniques 109D1 Solvent and solid-phase extraction 109D2 Principles of chromatography 119D3 Thin-layer chromatography 131D4 Gas chromatography: principles and instrumentation 137D5 Gas chromatography: procedures and applications 149D6 High-performance liquid chromatography: principles

and instrumentation 155D7 High-performance liquid chromatography: modes,

procedures and applications 166D8 Electrophoresis and electrochromatography: principles

and instrumentation 174D9 Electrophoresis and electrochromatography: modes,

procedures and applications 182

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CONTENTS

Section E Spectrometric techniques 189E1 Electromagnetic radiation and energy levels 189E2 Atomic and molecular spectrometry 195E3 Spectrometric instrumentation 201E4 Flame atomic emission spectrometry 206E5 Inductively coupled plasma spectrometry 209E6 X-ray emission spectrometry 214E7 Atomic absorption and atomic uorescence spectrometry 218E8 Ultraviolet and visible molecular spectrometry:

principles and instrumentation 223E9 Ultraviolet and visible molecular spectrometry:

applications 228E10 Infrared and Raman spectrometry: principles and

instrumentation 233E11 Infrared and Raman spectrometry: applications 242E12 Nuclear magnetic resonance spectrometry: principles

and instrumentation 248E13 Nuclear magnetic resonance spectrometry: interpretation

of proton and carbon-13 spectra 261E14 Mass spectrometry 270

Section F Combined techniques 283F1 Advantages of combined techniques 283F2 Sample identication using multiple spectrometric

techniques data 285F3 Gas chromatographymass spectrometry 294F4 Gas chromatographyinfrared spectrometry 298F5 Liquid chromatographymass spectrometry 302

Section G Thermal methods 305G1 Thermogravimetry 305G2 Differential thermal analysis and differential scanning

calorimetry 311G3 Thermomechanical analysis 316G4 Evolved gas analysis 320

Section H Sensors, automation and computing 323H1 Chemical sensors and biosensors 323H2 Automated procedures 328H3 Computer control and data collection 331H4 Data enhancement and databases 333

Further reading 337Index 339

vi Contents

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AAS atomic absorption spectrometryADC analog-to-digital converterAFS atomic uorescence spectrometryANOVA analysis of varianceATR attenuated total reectanceBPC bonded-phase chromatographyCC chiral chromatographyCGE capillary gel electrophoresisCI condence intervalCIEF capillary isoelectric focusingCL condence limitsCPU central processing unitCRM certied reference materialCZE capillary zone electrophoresisDAC digital-to-analog converterDAD diode array detectorDMA dynamic mechanical analysisDME dropping mercury electrodeDSC differential scanning calorimetryDTA differential thermal analysisDTG derivative thermogravimetryDVM digital voltmeterECD electron-capture detectorEDAX energy dispersive analysis

of X-raysEDTA ethylenediaminetetraacetic acidEGA evolved gas analysisFA factor analysisFAES ame atomic emission

spectometryFFT fast Fourier transformFID ame ionization detector

or free induction decayGC gas chromatographyGLC gas liquid chromatographyGSC gas solid chromatographyHATR horizontal attenuated total

reectanceHPLC high-performance liquid

chromatographyIC ion chromatographyICP inductively coupled plasmaICP-AES ICP-atomic emission spectrometryICP-OES ICP-optical emission spectrometry

ICP-MS ICP-mass spectrometryIEC ion-exchange chromatographyISE ion-selective electrodeLVDT linear variable differential

transformerMEKC micellar electrokinetic

chromatographyMIR multiple internal reectanceMS mass spectrometry NIR near infraredNMR nuclear-magnetic resonanceNPD nitrogen-phosphorus detectorPAH polycyclic aromatic hydrocarbonsPC paper chromatographyPCA principal component analysisPCR principal component regressionPDMS polydimethylsiloxanePLS partial least squaresQA quality assuranceQC quality controlRAM random access memoryRF radiofrequency RI refractive indexROM read only memoryRMM relative molecular massSCE saturated calomel electrodeSDS sodium dodecyl sulfateSDS-PAGE SDS-polyacrylamide gel

electrophoresisSE solvent extractionSEC size-exclusion chromatographySHE standard hydrogen electrodeSIM selected ion monitoringSPE solid phase extractionSPME solid phase microextractionSRM standard reference materialTCD thermal conductivity detectorTG thermogravimetryTIC total ion currentTISAB total ionic strength adjustment

bufferTLC thin-layer chromatography TMA thermomechanical analysis

ABBREVIATIONS

Analytical chemists and others in many disciplines frequently ask questions such as: What is thissubstance?; How concentrated is this solution?; What is the structure of this molecule? The answers tothese and many other similar questions are provided by the techniques and methods of analyticalchemistry. They are common to a wide range of activities, and the demand for analytical data of achemical nature is steadily growing. Geologists, biologists, environmental and materials scientists,physicists, pharmacists, clinicians and engineers may all nd it necessary to use or rely on some of thetechniques of analysis described in this book.

If we look back some forty or fty years, chemical analysis concentrated on perhaps three main areas:qualitative testing, quantitative determinations, particularly by classical techniques such as titrimetryand gravimetry, and structural analysis by procedures requiring laborious and time-consuming calcu-lations. The analytical chemist of today has an armoury of instrumental techniques, automated systemsand computers which enable analytical measurements to be made more easily, more quickly and moreaccurately.

However, pitfalls still exist for the unwary! Unless the analytical chemist has a thorough understand-ing of the principles, practice and limitations of each technique he/she employs, results may be inaccu-rate, ambiguous, misleading or invalid. From many years of stressing the importance of followingappropriate analytical procedures to a large number of students of widely differing abilities, backgroundsand degrees of enthusiasm, the authors have compiled an up-to-date, unied approach to the study ofanalytical chemistry and its applications. Surveys of the day-to-day operations of many industrial andother analytical laboratories in the UK, Europe and the USA have shown which techniques are the mostwidely used, and which are of such limited application that extensive coverage at this level would beinappropriate. The text therefore includes analytical techniques commonly used by most analyticallaboratories at this time. It is intended both to complement those on inorganic, organic and physicalchemistry in the Instant Notes series, and to offer to students in chemistry and other disciplines some guid-ance on the use of analytical techniques where they are relevant to their work. We have not given extendedaccounts of complex or more specialized analytical techniques, which might be studied beyond rst- andsecond-year courses. Nevertheless, the material should be useful as an overview of the subject for thosestudying at a more advanced level or working in analytical laboratories, and for revision purposes.

The layout of the book has been determined by the series format and by the requirements of the overall analytical process. Regardless of the discipline from which the need for chemical analysis arises,common questions must be asked:

How should a representative sample be obtained? What is to be determined and with what quantitative precision? What other components are present and will they interfere with the analytical measurements? How much material is available for analysis, and how many samples are to be analyzed? What instrumentation is to be used? How reliable is the data generated?

These and related questions are considered in Sections A and B.Most of the subsequent sections provide notes on the principles, instrumentation and applications of

both individual and groups of techniques. Where suitable supplementary texts exist, reference is madeto them, and some suggestions on consulting the primary literature are made.

We have assumed a background roughly equivalent to UK A-level chemistry or a US generalchemistry course. Some simplication of mathematical treatments has been made; for example, in thesections on statistics, and on the theoretical basis of the various techniques. However, the texts listedunder Further Reading give more comprehensive accounts and further examples of applications.

PREFACE

We should like to thank all who have contributed to the development of this text, especially the manyinstrument manufacturers who generously provided examples and illustrations, and in particular PerkinElmer Ltd. (UK) and Sherwood Scientic Ltd. (UK). We would like also to thank our colleagues whoallowed us to consult them freely and, not least, the many generations of our students who foundquestions and problems where we had thought there were none!

DKPJH

x Preface

Section A The nature and scope of analytical chemistry

A1 ANALYTICAL CHEMISTRY, ITSFUNCTIONS ANDAPPLICATIONS

Denition Analytical chemistry involves the application of a range of techniques andmethodologies to obtain and assess qualitative, quantitative and structuralinformation on the nature of matter.

Qualitative analysis is the identication of elements, species and/orcompounds present in a sample.

Quantitative analysis is the determination of the absolute or relative amountsof elements, species or compounds present in a sample.

Structural analysis is the determination of the spatial arrangement of atoms inan element or molecule or the identication of characteristic groups of atoms(functional groups).

An element, species or compound that is the subject of analysis is known as ananalyte.

The remainder of the material or sample of which the analyte(s) form(s) a partis known as the matrix.

Purpose The gathering and interpretation of qualitative, quantitative and structural infor-mation is essential to many aspects of human endeavor, both terrestrial andextra-terrestrial. The maintenance of, and improvement in, the quality of lifethroughout the world, and the management of resources rely heavily onthe information provided by chemical analysis. Manufacturing industries useanalytical data to monitor the quality of raw materials, intermediates and

Key Notes

Analytical chemistry is a scientic discipline used to study the chemicalcomposition, structure and behavior of matter.

The purpose of chemical analysis is to gather and interpret chemicalinformation that will be of value to society in a wide range of contexts.

Quality control in manufacturing industries, the monitoring of clinicaland environmental samples, the assaying of geological specimens, andthe support of fundamental and applied research are the principalapplications.

Related topics Analytical problems and Computer control and data procedures (A2) collection (H3)

Chemical sensors and biosensors Data enhancement and databases (H1) (H4)

Automated procedures (H2)

Scope andapplications

Denition

Purpose

nished products. Progress and research in many areas is dependent on estab-lishing the chemical composition of man-made or natural materials, and themonitoring of toxic substances in the environment is of ever increasing impor-tance. Studies of biological and other complex systems are supported by thecollection of large amounts of analytical data.

Analytical data are required in a wide range of disciplines and situations thatinclude not just chemistry and most other sciences, from biology to zoology, butthe arts, such as painting and sculpture, and archaeology. Space exploration andclinical diagnosis are two quite disparate areas in which analytical data is vital.Important areas of application include the following.

Quality control (QC). In many manufacturing industries, the chemicalcomposition of raw materials, intermediates and nished products needs to be monitored to ensure satisfactory quality and consistency. Virtually allconsumer products from automobiles to clothing, pharmaceuticals and food-stuffs, electrical goods, sports equipment and horticultural products rely, inpart, on chemical analysis. The food, pharmaceutical and water industries inparticular have stringent requirements backed by legislation for major compo-nents and permitted levels of impurities or contaminants. The electronicsindustry needs analyses at ultra-trace levels (parts per billion) in relation to themanufacture of semi-conductor materials. Automated, computer-controlledprocedures for process-stream analysis are employed in some industries.

Monitoring and control of pollutants. The presence of toxic heavy metals(e.g., lead, cadmium and mercury), organic chemicals (e.g., polychlorinatedbiphenyls and detergents) and vehicle exhaust gases (oxides of carbon,nitrogen and sulfur, and hydrocarbons) in the environment are health hazardsthat need to be monitored by sensitive and accurate methods of analysis, andremedial action taken. Major sources of pollution are gaseous, solid and liquidwastes that are discharged or dumped from industrial sites, and vehicleexhaust gases.

Clinical and biological studies. The levels of important nutrients, includingtrace metals (e.g., sodium, potassium, calcium and zinc), naturally producedchemicals, such as cholesterol, sugars and urea, and administered drugs in thebody uids of patients undergoing hospital treatment require monitoring.Speed of analysis is often a crucial factor and automated procedures have beendesigned for such analyses.

Geological assays. The commercial value of ores and minerals is determinedby the levels of particular metals, which must be accurately established.Highly accurate and reliable analytical procedures must be used for thispurpose, and referee laboratories are sometimes employed where disputesarise.

Fundamental and applied research. The chemical composition and structureof materials used in or developed during research programs in numerousdisciplines can be of signicance. Where new drugs or materials with potentialcommercial value are synthesized, a complete chemical characterization maybe required involving considerable analytical work. Combinatorial chemistryis an approach used in pharmaceutical research that generates very largenumbers of new compounds requiring conrmation of identity and structure.

Scope andapplications

2 Section A The nature and scope of analytical chemistry


A2 ANALYTICAL PROBLEMSAND PROCEDURES

The most important aspect of an analysis is to ensure that it will provide usefuland reliable data on the qualitative and/or quantitative composition of a materialor structural information about the individual compounds present. The analyt-ical chemist must often communicate with other scientists and nonscientists toestablish the amount and quality of the information required, the time-scale forthe work to be completed and any budgetary constraints. The most appropriateanalytical technique and method can then be selected from those available or newones devised and validated by the analysis of substances of known compositionand/or structure. It is essential for the analytical chemist to have an appreciationof the objectives of the analysis and an understanding of the capabilities of thevarious analytical techniques at his/her disposal without which the most appro-priate and cost-effective method cannot be selected or developed.

The stages or steps in an overall analytical procedure can be summarized asfollows.

Denition of the problem. Analytical information and level of accuracyrequired. Costs, timing, availability of laboratory instruments and facilities.

Choice of technique and method. Selection of the best technique for therequired analysis, such as chromatography, infrared spectrometry, titrimetry,thermogravimetry. Selection of the method (i.e. the detailed stepwise instruc-tions using the selected technique).

Sampling. Selection of a small sample of the material to be analyzed. Wherethis is heterogeneous, special procedures need to be used to ensure that agenuinely representative sample is obtained (Topic A4).

Analyticalprocedures

Analyticalproblems

Key Notes

Selecting or developing and validating appropriate methods of analysisto provide reliable data in a variety of contexts are the principal problemsfaced by analytical chemists.

Any chemical analysis can be broken down into a number of stages thatinclude a consideration of the purpose of the analysis, the quality of theresults required and the individual steps in the overall analyticalprocedure.

Related topics Analytical chemistry, its functions Automated procedures (H2)and applications (A1) Computer control and data

Sampling and sample handling collection (H3)(A4) Data enhancement and databases

Chemical sensors and biosensors (H4)(H1)

Analytical problems

Analyticalprocedures

Sample pre-treatment or conditioning. Conversion of the sample into a formsuitable for detecting or measuring the level of the analyte(s) by the selectedtechnique and method. This may involve dissolving it, converting theanalyte(s) into a specic chemical form or separating the analyte(s) from othercomponents of the sample (the sample matrix) that could interfere with detec-tion or quantitative measurements.

Qualitative analysis. Tests on the sample under specied and controlledconditions. Tests on reference materials for comparison. Interpretation of thetests.

Quantitative analysis. Preparation of standards containing known amountsof the analyte(s) or of pure reagents to be reacted with the analyte(s).Calibration of instruments to determine the responses to the standards undercontrolled conditions. Measurement of the instrumental response for eachsample under the same conditions as for the standards. All measurementsmay be replicated to improve the reliability of the data, but this has cost andtime implications. Calculation of results and statistical evaluation.

Preparation of report or certicate of analysis. This should include asummary of the analytical procedure, the results and their statistical assess-ment, and details of any problems encountered at any stage during theanalysis.

Review of the original problem. The results need to be discussed with regardto their signicance and their relevance in solving the original problem.Sometimes repeat analyses or new analyses may be undertaken.



A3 ANALYTICAL TECHNIQUESAND METHODS

There are numerous chemical or physico-chemical processes that can be used toprovide analytical information. The processes are related to a wide range ofatomic and molecular properties and phenomena that enable elements andcompounds to be detected and/or quantitatively measured under controlledconditions. The underlying processes dene the various analytical techniques.The more important of these are listed in Table 1, together with their suitability forqualitative, quantitative or structural analysis and the levels of analyte(s) in asample that can be measured.

Atomic and molecular spectrometry and chromatography, which togethercomprise the largest and most widely used groups of techniques, can be furthersubdivided according to their physico-chemical basis. Spectrometric techniquesmay involve either the emission or absorption of electromagnetic radiation overa very wide range of energies, and can provide qualitative, quantitative andstructural information for analytes from major components of a sample down to ultra-trace levels. The most important atomic and molecular spectrometrictechniques and their principal applications are listed in Table 2.

Chromatographic techniques provide the means of separating the compo-nents of mixtures and simultaneous qualitative and quantitative analysis, asrequired. The linking of chromatographic and spectrometric techniques, calledhyphenation, provides a powerful means of separating and identifyingunknown compounds (Section F). Electrophoresis is another separation tech-nique with similarities to chromatography that is particularly useful for theseparation of charged species. The principal separation techniques and theirapplications are listed in Table 3.

An analytical method consists of a detailed, stepwise list of instructions to befollowed in the qualitative, quantitative or structural analysis of a sample for oneor more analytes and using a specied technique. It will include a summary and

Analyticalmethods

Analyticaltechniques

Key Notes

Chemical or physico-chemical processes that provide the basis foranalytical measurements are described as techniques.

A method is a detailed set of instructions for a particular analysis using aspecied technique.

A process whereby an analytical method is checked for reliability interms of accuracy, reproducibility and robustness in relation to itsintended applications.

Related topic Quality in analytical laboratories (A6)

Analytical methods

Analyticaltechniques

Method validation


Table 1. Analytical techniques and principal applications

Technique Property measured Principal areas of application

Gravimetry Weight of pure analyte or compound Quantitative for major or minor of known stoichiometry components

Titrimetry Volume of standard reagent solution Quantitative for major or minor reacting with the analyte components

Atomic and molecular Wavelength and intensity of Qualitative, quantitative or structural spectrometry electromagnetic radiation emitted or for major down to trace level

absorbed by the analyte components

Mass spectrometry Mass of analyte or fragments of it Qualitative or structural for majordown to trace level componentsisotope ratios

Chromatography and Various physico-chemical properties Qualitative and quantitative electrophoresis of separated analytes separations of mixtures at major to

trace levels

Thermal analysis Chemical/physical changes in the Characterization of single or mixed analyte when heated or cooled major/minor components

Electrochemical analysis Electrical properties of the analyte Qualitative and quantitative for major in solution to trace level components

Radiochemical analysis Characteristic ionizing nuclear Qualitative and quantitative at major radiation emitted by the analyte to trace levels

Table 2. Spectrometric techniques and principal applications

Technique Basis Principal applications

Plasma emission spectrometry Atomic emission after excitation in high Determination of metals and some temperature gas plasma non-metals mainly at trace levels

Flame emission spectrometry Atomic emission after ame excitation Determination of alkali and alkalineearth metals

Atomic absorption spectrometry Atomic absorption after atomization Determination of trace metals and by ame or electrothermal means some non-metals

Atomic uorescence Atomic uorescence emission after Determination of mercury and spectrometry ame excitation hydrides of non-metals at trace

levels

X-ray emission spectrometry Atomic or atomic uorescence Determination of major and minor emission after excitation by electrons elemental components of or radiation metallurgical and geological samples

-spectrometry -ray emission after nuclear excitation Monitoring of radioactive elements inenvironmental samples

Ultraviolet/visible spectrometry Electronic molecular absorption in Quantitative determination of solution unsaturated organic compounds

Infrared spectrometry Vibrational molecular absorption Identication of organic compounds

Nuclear magnetic resonance Nuclear absorption (change of spin Identication and structural analysis spectrometry states) of organic compounds

Mass spectrometry Ionization and fragmentation of Identication and structural analysis molecules of organic compounds

lists of chemicals and reagents to be used, laboratory apparatus and glassware,and appropriate instrumentation. The quality and sources of chemicals,including solvents, and the required performance characteristics of instrumentswill also be specied as will the procedure for obtaining a representative sampleof the material to be analyzed. This is of crucial importance in obtaining mean-ingful results (Topic A4). The preparation or pre-treatment of the sample will befollowed by any necessary standardization of reagents and/or calibration ofinstruments under specied conditions (Topic A5). Qualitative tests for theanalyte(s) or quantitative measurements under the same conditions as those usedfor standards complete the practical part of the method. The remaining steps willbe concerned with data processing, computational methods for quantitativeanalysis and the formatting of the analytical report. The statistical assessment ofquantitative data is vital in establishing the reliability and value of the data, andthe use of various statistical parameters and tests is widespread (Section B).

Many standard analytical methods have been published as papers in analyt-ical journals and other scientic literature, and in textbook form. Collections bytrades associations representing, for example, the cosmetics, food, iron and steel,pharmaceutical, polymer plastics and paint, and water industries are available.Standards organizations and statutory authorities, instrument manufacturersapplications notes, the Royal Society of Chemistry and the US EnvironmentalProtection Agency are also valuable sources of standard methods. Often, labora-tories will develop their own in-house methods or adapt existing ones forspecic purposes. Method development forms a signicant part of the work ofmost analytical laboratories, and method validation and periodic revalidation isa necessity.

Selection of the most appropriate analytical method should take into accountthe following factors:

the purpose of the analysis, the required time scale and any cost constraints; the level of analyte(s) expected and the detection limit required; the nature of the sample, the amount available and the necessary sample

preparation procedure; the accuracy required for a quantitative analysis; the availability of reference materials, standards, chemicals and solvents,

instrumentation and any special facilities; possible interference with the detection or quantitative measurement of

the analyte(s) and the possible need for sample clean-up to avoid matrix interference;

A3 Analytical techniques and methods 7

Table 3. Separation techniques and principal applications

Technique Basis Principal applications

Thin-layer chromatography Qualitative analysis of mixturesDifferential rates of migration of

Gas chromatographyanalytes through a stationary phase

Quantitative and qualitative

by movement of a liquid or gaseous determination of volatile compounds

High-performance liquid mobile phase Quantitative and qualitative chromatography

determination of nonvolatilecompounds

Electrophoresis Differential rates of migration of Quantitative and qualitative analytes through a buffered medium determination of ionic compounds

the degree of selectivity available methods may be selective for a smallnumber of analytes or specic for only one;

quality control and safety factors.

Method validation Analytical methods must be shown to give reliable data, free from bias and suit-able for the intended use. Most methods are multi-step procedures, and theprocess of validation generally involves a stepwise approach in which optimizedexperimental parameters are tested for robustness (ruggedness), that is sensi-tivity to variations in the conditions, and sources of errors investigated.

A common approach is to start with the nal measurement stage, using cali-bration standards of known high purity for each analyte to establish the perfor-mance characteristics of the detection system (i.e. specicity, range, quantitativeresponse (linearity), sensitivity, stability and reproducibility). Robustness interms of temperature, humidity and pressure variations would be included atthis stage, and a statistical assessment made of the reproducibility of repeatedidentical measurements (replicates). The process is then extended backwards insequence through the preceding stages of the method, checking that the optimumconditions and performance established for the nal measurement on analytecalibration standards remain valid throughout. Where this is not the case, newconditions must be investigated by modication of the procedure and the processrepeated. A summary of this approach is shown in Figure 1 in the form of a owdiagram. At each stage, the results are assessed using appropriate statistical tests(Section B) and compared for consistency with those of the previous stage. Whereunacceptable variations arise, changes to the procedure are implemented and theassessment process repeated. The performance and robustness of the overallmethod are nally tested with eld trials in one or more routine analyticallaboratories before the method is considered to be fully validated.


Fig. 1. Flow chart for method validation.

A3 Analytical techniques and methods 9

Step 1 Performance characteristics of detectorfor single analyte calibration standards

Step 2 Process repeated for mixed analytecalibration standards

Step 6 Field trials in routine laboratory withmore junior personnel to test ruggedness

Step 5 Analysis of 'spiked' simulated samplematrix. i.e. matrix with added known

amounts of analyte(s), to test recoveries

Step 3 Process repeated for analyte calibrationstandards with possible interferingsubstances and for reagent blanks

Step 4 Process repeated for analyte calibrationstandards with anticipated matrixcomponents to evaluate matrix

interference


A4 SAMPLING AND SAMPLEHANDLING

The importance of obtaining a representative sample for analysis cannot beoveremphasized. Without it, results may be meaningless or even grosslymisleading. Sampling is particularly crucial where a heterogeneous material is tobe analyzed. It is vital that the aims of the analysis are understood and an appro-priate sampling procedure adopted. In some situations, a sampling plan orstrategy may need to be devised so as to optimize the value of the analyticalinformation collected. This is necessary particularly where environmentalsamples of soil, water or the atmosphere are to be collected or a complex indus-trial process is to be monitored. Legal requirements may also determine asampling strategy, particularly in the food and drug industries. A small sampletaken for analysis is described as a laboratory sample. Where duplicate analysesor several different analyses are required, the laboratory sample will be dividedinto sub-samples which should have identical compositions.

Homogeneous materials (e.g., single or mixed solvents or solutions and mostgases) generally present no particular sampling problem as the composition ofany small laboratory sample taken from a larger volume will be representative ofthe bulk solution. Heterogeneous materials have to be homogenized prior toobtaining a laboratory sample if an average or bulk composition is required.Conversely, where analyte levels in different parts of the material are to be

Representativesample

Key Notes

A representative sample is one that truly reects the composition of thematerial to be analyzed within the context of a dened analyticalproblem.

Due to varying periods of time that may elapse between samplecollection and analysis, storage conditions must be such as to avoidundesirable losses, contamination or other changes that could affect theresults of the analysis.

Preliminary treatment of a sample is sometimes necessary before it is in asuitable form for analysis by the chosen technique and method. This mayinvolve a separation or concentration of the analytes or the removal ofmatrix components that would otherwise interfere with the analysis.

Samples generally need to be brought into a form suitable formeasurements to be made under controlled conditions. This may involvedissolution, grinding, fabricating into a specic size and shape,pelletizing or mounting in a sample holder.

Related topic Analytical problems and procedures (A2)

Representativesample

Sample storage

Sample pre-treatment

Sample preparation

measured, they may need to be physically separated before laboratory samplesare taken. This is known as selective sampling. Typical examples of hetero-geneous materials where selective sampling may be necessary include:

surface waters such as streams, rivers, reservoirs and seawater, where theconcentrations of trace metals or organic compounds in solution and in sedi-ments or suspended particulate matter may each be of importance;

materials stored in bulk, such as grain, edible oils, or industrial organic chem-icals, where physical segregation (stratication) or other effects may lead tovariations in chemical composition throughout the bulk;

ores, minerals and alloys, where information about the distribution of a partic-ular metal or compound is sought;

laboratory, industrial or urban atmospheres where the concentrations of toxicvapors and fumes may be localized or vary with time.

Obtaining a laboratory sample to establish an average analyte level in a highlyheterogeneous material can be a lengthy procedure. For example, sampling alarge shipment of an ore or mineral, where the economic cost needs to bedetermined by a very accurate assay, is typically approached in the followingmanner.

(i) Relatively large pieces are randomly selected from different parts of theshipment.

(ii) The pieces are crushed, ground to coarse granules and thoroughly mixed.(iii) A repeated coning and quartering process, with additional grinding to

reduce particle size, is used until a laboratory-sized sample is obtained.This involves creating a conical heap of the material, dividing it into fourequal portions, discarding two diagonally opposite portions and forming anew conical heap from the remaining two quarters. The process is thenrepeated as necessary (Fig. 1).

A4 Sampling and sample handling 11

2

4

1 3

2

4

1 3

2

4

1 3

Fig. 1. A diagrammatic representation of coning and quartering (quarters 1 and 3, or 2 and 4 are discarded each time).

The distribution of toxic heavy metals or organic compounds in a land rede-velopment site presents a different problem. Here, to economize on the numberof analyses, a grid is superimposed on the site dividing it up into approximatelyone- to ve-metre squares. From each of these, samples of soil will be taken atseveral specied depths. A three-dimensional representation of the distributionof each analyte over the whole site can then be produced, and any localized highconcentrations, or hot spots, can be investigated by taking further, more closely-spaced, samples. Individual samples may need to be ground, coned andquartered as part of the sampling strategy.

Repeated sampling over a period of time is a common requirement. Examplesinclude the continuous monitoring of a process stream in a manufacturing plantand the frequent sampling of patients body uids for changes in the levels ofdrugs, metabolites, sugars or enzymes, etc., during hospital treatment. Studies ofseasonal variations in the levels of pesticide, herbicide and fertilizer residues insoils and surface waters, or the continuous monitoring of drinking water suppliesare two further examples.

Having obtained a representative sample, it must be labeled and stored underappropriate conditions. Sample identication through proper labeling, increas-ingly done by using bar codes and optical readers under computer control, is anessential feature of sample handling.

Sample storage Samples often have to be collected from places remote from the analytical labora-tory and several days or weeks may elapse before they are received by the labo-ratory and analyzed. Furthermore, the workload of many laboratories is such thatincoming samples are stored for a period of time prior to analysis. In bothinstances, sample containers and storage conditions (e.g., temperature, humidity,light levels and exposure to the atmosphere) must be controlled such that nosignicant changes occur that could affect the validity of the analytical data. Thefollowing effects during storage should be considered:

increases in temperature leading to the loss of volatile analytes, thermal orbiological degradation, or increased chemical reactivity;

decreases in temperature that lead to the formation of deposits or the precipi-tation of analytes with low solubilities;

changes in humidity that affect the moisture content of hygroscopic solids andliquids or induce hydrolysis reactions;

UV radiation, particularly from direct sunlight, that induces photochemicalreactions, photodecomposition or polymerization;

air-induced oxidation; physical separation of the sample into layers of different density or changes in

crystallinity.

In addition, containers may leak or allow contaminants to enter.A particular problem associated with samples having very low (trace and

ultra-trace) levels of analytes in solution is the possibility of losses by adsorp-tion onto the walls of the container or contamination by substances beingleached from the container by the sample solvent. Trace metals may be depletedby adsorption or ion-exchange processes if stored in glass containers, whilstsodium, potassium, boron and silicates can be leached from the glass into thesample solution. Plastic containers should always be used for such samples.


Conversely, sample solutions containing organic solvents and other organicliquids should be stored in glass containers because the base plastic or additivessuch as plasticizers and antioxidants may be leached from the walls of plasticcontainers.

Samples arriving in an analytical laboratory come in a very wide assortment ofsizes, conditions and physical forms and can contain analytes from majorconstituents down to ultra-trace levels. They can have a variable moisture contentand the matrix components of samples submitted for determinations of the sameanalyte(s) may also vary widely. A preliminary, or pre-treatment, is often used tocondition them in readiness for the application of a specic method of analysis orto pre-concentrate (enrich) analytes present at very low levels. Examples of pre-treatments are:

drying at 100C to 120C to eliminate the effect of a variable moisture content; weighing before and after drying enables the water content to be calculated or

it can be established by thermogravimetric analysis (Topic G1); separating the analytes into groups with common characteristics by dis-

tillation, ltration, centrifugation, solvent or solid phase extraction (TopicD1);

removing or reducing the level of matrix components that are known to causeinterference with measurements of the analytes;

concentrating the analytes if they are below the concentration range of theanalytical method to be used by evaporation, distillation, co-precipitation, ionexchange, solvent or solid phase extraction or electrolysis.

Sample clean-up in relation to matrix interference and to protect special-ized analytical equipment such as chromatographic columns and detectionsystems from high levels of matrix components is widely practised using solidphase extraction (SPE) cartridges (Topic D1). Substances such as lipids, fats,proteins, pigments, polymeric and tarry substances are particularly detri-mental.

A laboratory sample generally needs to be prepared for analytical measurementby treatment with reagents that convert the analyte(s) into an appropriate chem-ical form for the selected technique and method, although in some instances it isexamined directly as received or mounted in a sample holder for surfaceanalysis. If the material is readily soluble in aqueous or organic solvents, a simpledissolution step may sufce. However, many samples need rst to be decom-posed to release the analyte(s) and facilitate specic reactions in solution. Samplesolutions may need to be diluted or concentrated by enrichment so that analytesare in an optimum concentration range for the method. The stabilization of solu-tions with respect to pH, ionic strength and solvent composition, and the removalor masking of interfering matrix components not accounted for in any pre-treat-ment may also be necessary. An internal standard for reference purposes inquantitative analysis (Topic A5 and Section B) is sometimes added before adjust-ment to the nal prescribed volume. Some common methods of decompositionand dissolution are given in Table 1.

Samplepreparation

Sample pre-treatment

A4 Sampling and sample handling 13

Table 1. Some methods for sample decomposition and dissolution

Method of attack Type of sample

Heated with concentrated mineral Geological, metallurgicalacids (HCl, HNO3, aqua regia) or strong alkali, including microwave digestion

Fusion with ux (Na2O2, Na2CO3, Geological, refractory materialsLiBO2, KHSO4, KOH)

Heated with HF and H2SO4 or HClO4 Silicates where SiO2 is not the analyte

Acid leaching with HNO3 Soils and sediments

Dry oxidation by heating in a furnace Organic materials with inorganic analytesor wet oxidation by boiling with concentrated H2SO4 and HNO3 or HClO4



A5 CALIBRATION ANDSTANDARDS

Calibration With the exception of absolute methods of analysis that involve chemical reac-tions of known stoichiometry (e.g., gravimetric and titrimetric determinations), acalibration or standardization procedure is required to establish the relationbetween a measured physico-chemical response to an analyte and the amount orconcentration of the analyte producing the response. Techniques and methodswhere calibration is necessary are frequently instrumental, and the detectorresponse is in the form of an electrical signal. An important consideration is theeffect of matrix components on the analyte detector signal, which may besupressed or enhanced, this being known as the matrix effect. When this isknown to occur, matrix matching of the calibration standards to simulate thegross composition expected in the samples is essential (i.e. matrix componentsare added to all the analyte standards in the same amounts as are expected in thesamples).

There are several methods of calibration, the choice of the most suitabledepending on the characteristics of the analytical technique to be employed, thenature of the sample and the level of analyte(s) expected. These include:

External standardization. A series of at least four calibration standardscontaining known amounts or concentrations of the analyte and matrixcomponents, if required, is either prepared from laboratory chemicals of guar-anteed purity (AnalaR or an equivalent grade) or purchased as a concentratedstandard ready to use. The response of the detection system is recorded foreach standard under specied and stable conditions and additionally for ablank, sometimes called a reagent blank (a standard prepared in an identical

Key Notes

Calibration or standardization is the process of establishing the responseof a detection or measurement system to known amounts orconcentrations of an analyte under specied conditions, or thecomparison of a measured quantity with a reference value.

A chemical standard is a material or substance of very high purityand/or known composition that is used to standardize a reagent orcalibrate an instrument.

A reference material is a material or substance, one or more properties ofwhich are sufciently homogeneous and well established for it to be usedfor the calibration of apparatus, the assessment of a measurement methodor for assigning values to materials.

Related topic Calibration and linear regression (B4)

Calibration

Chemical standard

Reference material

fashion to the other standards but omitting the analyte). The data is eitherplotted as a calibration graph or used to calculate a factor to convert detectorresponses measured for the analyte in samples into corresponding masses orconcentrations (Topic B4).

Standard addition. Internal standardization.

The last two methods of calibration are described in Topic B4.Instruments and apparatus used for analytical work must be correctly main-

tained and calibrated against reference values to ensure that measurements areaccurate and reliable. Performance should be checked regularly and records keptso that any deterioration can be quickly detected and remedied. Microcomputerand microprocessor controlled instrumentation often has built-in performancechecks that are automatically initiated each time an instrument is turned on.Some examples of instrument or apparatus calibration are

manual calibration of an electronic balance with certied weights; calibration of volumetric glassware by weighing volumes of pure water; calibration of the wavelength and absorbance scales of spectrophotometers

with certied emission or absorption characteristics; calibration of temperature scales and electrical voltage or current readouts

with certied measurement equipment.

Materials or substances suitable for use as chemical standards are generallysingle compounds or elements. They must be of known composition, and highpurity and stability. Many are available commercially under the name AnalaR.Primary standards, which are used principally in titrimetry (Section C) tostandardize a reagent (titrant) (i.e. to establish its exact concentration) must beinternationally recognized and should full the following requirements:

be easy to obtain and preserve in a high state of purity and of known chemicalcomposition;

be non-hygroscopic and stable in air allowing accurate weighing; have impurities not normally exceeding 0.02% by weight; be readily soluble in water or another suitable solvent; react rapidly with an analyte in solution; other than pure elements, to have a high relative molar mass to minimize

weighing errors.

Primary standards are used directly in titrimetric methods or to standardizesolutions of secondary or working standards (i.e. materials or substances that donot fulll all of the above criteria, that are to be used subsequently as the titrant ina particular method). Chemical standards are also used as reagents to effectreactions with analytes before completing the analysis by techniques other thantitrimetry.

Some approved primary standards for titrimetric analysis are given in Table 1.

Reference materials are used to demonstrate the accuracy, reliability and com-parability of analytical results. A certied or standard reference material (CRMor SRM) is a reference material, the values of one or more properties of whichhave been certied by a technically valid procedure and accompanied by a trace-able certicate or other documentation issued by a certifying body such as the

Referencematerial

Chemicalstandard


Bureau of Analytical Standards. CRMs or SRMs are produced in various formsand for different purposes and they may contain one or more certied compo-nents, such as

pure substances or solutions for calibration or identication; materials of known matrix composition to facilitate comparisons of analytical

data; materials with approximately known matrix composition and specied

components.

They have a number of principal uses, including

validation of new methods of analysis; standardization/calibration of other reference materials; conrmation of the validity of standardized methods; support of quality control and quality assurance schemes.

A5 Calibration and standards 17

Table 1. Some primary standards used in titrimetric analysis

Type of titration Primary standard

Acid-base Sodium carbonate, Na2CO3Sodium tetraborate, Na2B4O7.10H2OPotassium hydrogen phthalate, KH(C8H4O4)Benzoic acid, C6H5COOH

Redox Potassium dichromate, K2Cr2O7Potassium iodate, KIO3Sodium oxalate, Na2C2O4

Precipitation (silver halide) Silver nitrate, AgNO3Sodium chloride, NaCl

Complexometric (EDTA) Zinc, ZnMagnesium, MgEDTA (disodium salt), C10H14N2O8Na2


A6 QUALITY IN ANALYTICALLABORATORIES

Quality control Analytical data must be of demonstrably high quality to ensure condence in theresults. Quality control (QC) comprises a system of planned activities in ananalytical laboratory whereby analytical methods are monitored at every stage toverify compliance with validated procedures and to take steps to eliminate thecauses of unsatisfactory performance. Results are considered to be of sufcientlyhigh quality if

they meet the specic requirements of the requested analytical work withinthe context of a dened problem;

there is condence in their validity; the work is cost effective.

To implement a QC system, a complete understanding of the chemistry andoperations of the analytical method and the likely sources and magnitudes oferrors at each stage is essential. The use of reference materials (Topic A5) duringmethod validation (Topic A3) ensures that results are traceable to certiedsources. QC processes should include:

checks on the accuracy and precision of the data using statistical tests (SectionB);

detailed records of calibration, raw data, results and instrument performance; observations on the nature and behavior of the sample and unsatisfactory

aspects of the methodology; control charts to determine system control for instrumentation and repeat

analyses (Topic B5);

Key Notes

Quality control (QC) is the process of ensuring that the operationaltechniques and activities used in an analytical laboratory provide resultssuitable for the intended purpose.

Quality assurance (QA) is the combination of planned and systematicactions necessary to provide adequate condence that the process ofquality control satises specied requirements.

This is a system whereby the quality control and quality assuranceprocedures adopted by a laboratory are evaluated by inspection andaccredited by an independent body.

Related topics Analytical techniques and Quality control and chemometricsmethods (A3) (B5)

Quality control

Quality assurance

Accreditationsystem

provision of full documentation and traceability of results to recognizedreference materials through recorded identication;

maintenance and calibration of instrumentation to manufacturers specica-tions;

management and control of laboratory chemicals and other materials includingchecks on quality;

adequate training of laboratory personnel to ensure understanding and competence;

external verication of results wherever possible; accreditation of the laboratory by an independent organization.

Quality assurance The overall management of an analytical laboratory should include the provisionof evidence and assurances that appropriate QC procedures for laboratory activ-ities are being correctly implemented. Quality assurance (QA) is a managerialresponsibility that is designed to ensure that this is the case and to generatecondence in the analytical results. Part of QA is to build condence through thelaboratory participating in interlaboratory studies where several laboratoriesanalyze one or more identical homogeneous materials under specied condi-tions. Prociency testing is a particular type of study to assess the performanceof a laboratory or analyst relative to others, whilst method performance studiesand certication studies are undertaken to check a particular analytical methodor reference material respectively. The results of such studies and their statisticalassessment enable the performances of individual participating laboratories to bedemonstrated and any deciencies in methodology and the training of personnelto be addressed.

Because of differences in the interpretation of the term quality, which can bedened as tness for purpose, QC and QA systems adopted by analyical labora-tories in different industries and elds of activity can vary widely. For thisreason, dened quality standards have been introduced by a number of organi-zations throughout the world. Laboratories can design and implement their ownquality systems and apply to be inspected and accredited by the organization forthe standard most appropriate to their activity. A number of organizations thatoffer accreditation suitable for analytical laboratories and their correspondingquality standards are given in Table 1.

Table 1. Accreditation organizations and their quality standards

Name of accreditation organization Quality standard

Organization for Economic Co-operation Good Laboratory Practice (GLP)and Development (OECD)

The International Organization for ISO 9000 series of quality standardsStandardization (ISO) ISO Guide 25 general requirements for

competence of calibration and testing laboratories

European Committee for Standardization EN 29000 series(CEN) EN 45000 series

British Standards Institution (BSI) BS 5750 quality standardBS 7500 series

National Measurement Accreditation NAMASService (NAMAS)

Accreditationsystem

A6 Quality in analytical laboratories 19

Section B Assessment of data

B1 ERRORS IN ANALYTICALMEASUREMENTS

The causes of measurement errors are numerous and their magnitudes are vari-able. This leads to uncertainties in reported results. However, measurementerrors can be minimized and some types eliminated altogether by careful exper-imental design and control. Their effects can be assessed by the application ofstatistical methods of data analysis and chemometrics (Topic B5). Gross errorsmay arise from faulty equipment or bad laboratory practice; proper equipmentmaintenance and appropriate training and supervision of personnel shouldeliminate these.

Nevertheless, whether it is reading a burette or thermometer, weighing asample or timing events, or monitoring an electrical signal or liquid ow, therewill always be inherent variations in the measured parameter if readings arerepeated a number of times under the same conditions. In addition, errors maygo undetected if the true or accepted value is not known for comparisonpurposes.

Errors must be controlled and assessed so that valid analytical measurementscan be made and reported. The reliability of such data must be demonstrated sothat an end-user can have an acceptable degree of condence in the results ofan analysis.

Measurementerrors

Key Notes

All measurement processes are subject to measurement errors that affectnumerical data and which arise from a variety of sources.

An absolute error is the numerical difference between a measured valueand a true or accepted value. A relative error is the absolute error dividedby the true or accepted value.

Also known as systematic errors, or bias, these generally arise fromdeterminate or identiable sources causing measured values to differfrom a true or accepted value.

Also known as random errors, these arise from a variety of uncontrolledsources and cause small random variations in a measured quantity whenthe measurement is repeated a number of times.

Where several different measurements are combined to compute anoverall analytical result, the errors associated with each individualmeasurement contribute to a total or accumulated error.

Related topic Assessment of accuracy and precision (B2)

Measurement errors

Absolute andrelative errors

Determinate errors

Accumulated errors

Indeterminate errors

The absolute error, EA, in a measurement or result, xM, is given by the equation

EA = xM - xT

where xT is the true or accepted value. Examples are shown in Figure 1 where a200 mg aspirin standard has been analyzed a number of times. The absoluteerrors range from -4 mg to +10 mg.

The relative error, ER, in a measurement or result, xM, is given by the equation

ER = (xM - xT)/xT

Often, ER is expressed as a percentage relative error, 100ER. Thus, for the aspirinresults shown in Figure 1, the relative error ranges from -2% to +5%. Relativeerrors are particularly useful for comparing results of differing magnitude.

Absolute andrelative errors

22 Section B Assessment of data

Aspirin (mg)195 200 205 210

Absolute error (EA; mg)

Relative error (ER; %)

5 0 5 10

2.5 0 2.5 5

Fig. 1. Absolute and relative errors in the analysis of an aspirin standard.

There are three basic sources of determinate or systematic errors that lead to abias in measured values or results:

the analyst or operator; the equipment (apparatus and instrumentation) and the laboratory environ-

ment; the method or procedure.

It should be possible to eliminate errors of this type by careful observation andrecord keeping, equipment maintenance and training of laboratory personnel.

Operator errors can arise through carelessness, insufcient training, illness ordisability. Equipment errors include substandard volumetric glassware, faultyor worn mechanical components, incorrect electrical signals and a poor orinsufciently controlled laboratory environment. Method or procedural errorsare caused by inadequate method validation, the application of a method tosamples or concentration levels for which it is not suitable or unexpected varia-tions in sample characteristics that affect measurements. Determinate errors thatlead to a higher value or result than a true or accepted one are said to show apositive bias; those leading to a lower value or result are said to show a nega-tive bias. Particularly large errors are described as gross errors; these should beeasily apparent and readily eliminated.

Determinateerrors

Determinate errors can be proportional to the size of sample taken foranalysis. If so, they will have the same effect on the magnitude of a resultregardless of the size of the sample, and their presence can thus be difcult todetect. For example, copper(II) can be determined by titration after reaction withpotassium iodide to release iodine according to the equation

2Cu2+ + 4I- 2CuI + I2

However, the reaction is not specic to copper(II), and any iron(III) present inthe sample will react in the same way. Results for the determination of copper inan alloy containing 20%, but which also contained 0.2% of iron are shown inFigure 2 for a range of sample sizes. The same absolute error of +0.2% or relativeerror of 1% (i.e. a positive bias) occurs regardless of sample size, due to thepresence of the iron. This type of error may go undetected unless theconstituents of the sample and the chemistry of the method are known.

B1 Errors in analytical measurements 23

Sample size (g)

0

21

20

190.1 0.2 0.3 0.4 0.5 0.6 0.7

Cop

per

foun

d (

%)

True value

Positivebias

Fig. 2. Effect of a proportional error on the determination of copper by titration in thepresence of iron.

Constant determinate errors are independent of sample size, and thereforebecome less signicant as the sample size is increased. For example, where avisual indicator is employed in a volumetric procedure, a small amount oftitrant is required to change the color at the end-point, even in a blank solution(i.e. when the solution contains none of the species to be determined). Thisindicator blank (Topic C5) is the same regardless of the size of the titer whenthe species being determined is present. The relative error, therefore, decreaseswith the magnitude of the titer, as shown graphically in Figure 3. Thus, for anindicator blank of 0.02 cm3, the relative error for a 1 cm3 titer is 2%, but this fallsto only 0.08% for a 25 cm3 titer.

Known also as random errors, these arise from random uctuations inmeasured quantities, which always occur even under closely controlled condi-tions. It is impossible to eliminate them entirely, but they can be minimized bycareful experimental design and control. Environmental factors such as temper-ature, pressure and humidity, and electrical properties such as current, voltageand resistance are all susceptible to small continuous and random variationsdescribed as noise. These contribute to the overall indeterminate error in any

Indeterminateerrors

physical or physico-chemical measurement, but no one specic source can beidentied.

A series of measurements made under the same prescribed conditions andrepresented graphically is known as a frequency distribution. The frequency ofoccurrence of each experimental value is plotted as a function of the magnitudeof the error or deviation from the average or mean value. For analytical data,the values are often distributed symmetrically about the mean value, the mostcommon being the normal error or Gaussian distribution curve. The curve(Fig. 4) shows that

small errors are more probable than large ones, positive and negative errors are equally probable, and the maximum of the curve corresponds to the mean value.

The normal error curve is the basis of a number of statistical tests that can beapplied to analytical data to assess the effects of indeterminate errors, to comparevalues and to establish levels of condence in results (Topics B2 and B3).


2.5

2

1.5

1

0.5

00 10 20 30

Size of titer (cm3)

Rel

ativ

e er

ror

(%)

Fig. 3. Effect of a constant error on titers of differing magnitudes.

Freq

uenc

y of

occ

urre

nce

of e

ach

dev

iatio

n

0 +

Deviation from mean,

Fig. 4. The normal error or Gaussian distribution curve.

Errors are associated with every measurement made in an analytical procedure,and these will be aggregated in the nal calculated result. The accumulation orpropagation of errors is treated similarly for both determinate (systematic) andindeterminate (random) errors.

Determinate (systematic) errors can be either positive or negative, hence somecancellation of errors is likely in computing an overall determinate error, and insome instances this may be zero. The overall error is calculated using one of twoalternative expressions, that is

where only a linear combination of individual measurements is required tocompute the result, the overall absolute determinate error, ET, is given by

ET = E1 + E2 + E3 + .

E1 and E2 etc., being the absolute determinate errors in the individualmeasurements taking sign into account

where a multiplicative expression is required to compute the result, theoverall relative determinate error, ETR, is given by

ETR = E1R + E2R + E3R + .

E1R and E2R etc., being the relative determinate errors in the individual measure-ments taking sign into account.

The accumulated effect of indeterminate (random) errors is computed bycombining statistical parameters for each measurement (Topic B2).

Accumulatederrors

B1 Errors in analytical measurements 25


B2 ASSESSMENT OF ACCURACYAND PRECISION

These two characteristics of numerical data are the most important and the mostfrequently confused. It is vital to understand the difference between them, andthis is best illustrated diagrammatically as in Figure 1. Four analysts have each performed a set of ve titrations for which the correct titer is known to be20.00 cm3. The titers have been plotted on a linear scale, and inspection revealsthe following:

the average titers for analysts B and D are very close to 20.00 cm3 - these twosets are therefore said to have good accuracy;

the average titers for analysts A and C are well above and below 20.00 cm3

respectively - these are therefore said to have poor accuracy; the ve titers for analyst A and the ve for analyst D are very close to one

another within each set these two sets therefore both show good precision; the ve titers for analyst B and the ve for analyst C are spread widely

within each set - these two sets therefore both show poor precision.

Accuracy andprecision

Key Notes

Accuracy is the closeness of an experimental measurement or result tothe true or accepted value. Precision is the closeness of agreementbetween replicated measurements or results obtained under the sameprescribed conditions.

The standard deviation of a set of values is a statistic based on the normalerror (Gaussian) curve and used as a measure of precision.

Relative standard deviation (coefcient of variation) is the standarddeviation expressed as a percentage of the measured value.

A standard deviation can be calculated for two or more sets of data bypooling the values to give a more reliable measure of precision.

This is the square of the standard deviation, which is used in somestatistical tests.

An estimate of the overall precision of an analytical procedure can bemade by combining the precisions of individual measurements.

This is the range of values around an experimental result within whichthe true or accepted value is expected to lie with a dened level ofprobability.

Related topic Errors in analytical measurements (B1)

Accuracy andprecision

Standard deviation

Variance

Relative standarddeviation

Pooled standarddeviation

Overall precision

Condence interval

It should be noted that good precision does not necessarily produce goodaccuracy (analyst A) and poor precision does not necessarily produce pooraccuracy (analyst B). However, condence in the analytical procedure and theresults is greater when good precision can be demonstrated (analyst D).

Accuracy is generally the more important characteristic of quantitative data tobe assessed, although consistency, as measured by precision, is of particularconcern in some circumstances. Trueness is a term associated with accuracy,which describes the closeness of agreement between the average of a largenumber of results and a true or accepted reference value. The degree of accuracyrequired depends on the context of the analytical problem; results must be shownto be t for the purpose for which they are intended. For example, one result maybe satisfactory if it is within 10% of a true or accepted value whilst it may benecessary for another to be within 0.5%. By repeating an analysis a number oftimes and computing an average value for the result, the level of accuracy will beimproved, provided that no systematic error (bias) has occurred. Accuracycannot be established with certainty where a true or accepted value is not known,as is often the case. However, statistical tests indicating the accuracy of a resultwith a given probability are widely used (vide infra).

Precision, which is a measure of the variability or dispersion within a set ofreplicated values or results obtained under the same prescribed conditions, canbe assessed in several ways. The spread or range (i.e. the difference between thehighest and lowest value) is sometimes used, but the most popular method is toestimate the standard deviation of the data (vide infra). The precision of resultsobtained within one working session is known as repeatability or within-runprecision. The precision of results obtained over a series of working sessions isknown as reproducibility or between-runs precision. It is sometimes necessaryto separate the contributions made to the overall precision by within-run and

B2 Assessment of accuracy and precision 27

Correctresult

A

B

C

D19.70 20.00

Titer (cm3)

20.30

Fig. 1. Plots of titration data to distinguish accuracy and precision.

between-runs variability. It may also be important to establish the precision ofindividual steps in an analysis.

This is the most widely used measure of precision and is a parameter of thenormal error or Gaussian curve (Topic B1, Fig. 4). Figure 2 shows two curves forthe frequency distribution of two theoretical sets of data, each having an innitenumber of values and known as a statistical population.

Standarddeviation


Deviation from mean

Freq

uenc

y of

occ

urre

nce

of e

ach

dev

iatio

n sd = s2

sd = s1

s1 > s2

m +

Fig. 2. Normal error or Gaussian curves for the frequency distributions of two statisticalpopulations with differing spreads.

The maximum in each curve corresponds to the population mean, which forthese examples has the same value, m. However, the spread of values for thetwo sets is quite different, and this is reected in the half-widths of the twocurves at the points of inection, which, by denition, is the population stan-dard deviation, s. As s2 is much less than s1, the precision of the second set ismuch better than that of the rst. The abscissa scale can be calibrated inabsolute units or, more commonly, as positive and negative deviations from themean, m.

In general, the smaller the spread of values or deviations, the smaller thevalue of s and hence the better the precision. In practice, the true values of mand s can never be known because they relate to a population of innite size.However, an assumption is made that a small number of experimental values ora statistical sample drawn from a statistical population is also distributednormally or approximately so. The experimental mean, x

_

, of a set of values x1,x2, x3,.xn is therefore considered to be an estimate of the true or populationmean, m, and the experimental standard deviation, s, is an estimate of the trueor population standard deviation, s.

A useful property of the normal error curve is that, regardless of the magni-tude of m and s, the area under the curve within dened limits on either side ofm (usually expressed in multiples of s) is a constant proportion of the totalarea. Expressed as a percentage of the total area, this indicates that a particularpercentage of the population will be found between those limits.

Thus, approximately 68% of the area, and therefore of the population, will be

found within 1s of the mean, approximately 95% will be found within 2s andapproximately 99.7% within 3s. More practically convenient levels, as shownin Figure 3, are those corresponding to 90%, 95% and 99% of the population,which are dened by 1.64s, 1.96s and 2.58s respectively. Many statisticaltests are based on these probability levels.

The value of the population standard deviation, s, is given by the formula

s = (1)where xi represents any individual value in the population and N is the totalnumber of values, strictly innite. The summation symbol, S, is used to showthat the numerator of the equation is the sum for i = 1 to i = N of the squares ofthe deviations of the individual x values from the population mean, m. For verylarge sets of data (e.g., when N >50), it may be justiable to use this formula asthe difference between s and s will then be negligible. However, most analyticaldata consists of sets of values of less than ten and often as small as three.Therefore, a modied formula is used to calculate an estimated standarddeviation, s, to replace s, and using an experimental mean, x

_

, to replace thepopulation mean, m:

s = (2)i=N

i=1(xi x

_)2

N 1

i=N

i=1(xi m)2

N


4s 3s 2s 1s 0 1s 2s 3s 4s

Rel

ativ

efre

que

ncy

(y/N

)

1.29s +1.29s

80%

4s 3s 2s 1s 0 1s 2s 3s 4s

Rel

ativ

efre

que

ncy

(y/N

)

1.64s +1.64s

90%

4s 3s 2s 1s 0 1s 2s 3s 4s

Rel

ativ

efre

que

ncy

(y/N

)

1.96s +1.96s

95%

4s 3s 2s 1s 0 1s 2s 3s 4s

Rel

ativ

efre

que

ncy

(y/N

)

2.58s +2.58s

99%

Fig. 3. Proportions of a population within dened limits of the mean.

Note that N in the denominator is replaced by N - 1, which is known as thenumber of degrees of freedom and is dened as the number of independentdeviations (xi x

_

) used to calculate s. For single sets of data, this is always oneless than the number in the set because when N - 1 deviations are known the last

one can be deduced as, taking sign into account, i=N

i=1(xi x

_

) must be zero (see

Example 1 below).In summary, the calculation of an estimated standard deviation, s, for a small

number of values involves the following steps:

calculation of an experimental mean; calculation of the deviations of individual xi values from the mean; squaring the deviations and summing them; dividing by the number of degrees of freedom, N - 1, and taking the square root of the result.

Note that if N were used in the denominator, the calculated value of s would bean underestimate of s.

Estimated standard deviations are easily obtained using a calculator thatincorporates statistical function keys or with one of the many computer soft-ware packages. It is, however, useful to be able to perform a stepwise arithmeticcalculation, and an example using the set of ve replicate titers by analyst A(Fig. 1) is shown below.

Example 1

xi/cm3 (xi x) (xi x)2

20.16 -0.04 1.6 10-3

20.22 +0.02 4 10-4

20.18 -0.02 4 10-4

20.20 0.00 020.24 +0.04 1.6 10-3

101.00 4 10-3

x_

20.20

s 4 4103 = 0.032 cm3

The relative standard deviation, RSD or sr, is also known as the coefcient ofvariation, CV. It is a measure of relative precision and is normally expressed asa percentage of the mean value or result

sr (s/x_) 100 (3)

It is an example of a relative error (Topic B1) and is particularly useful forcomparisons between sets of data of differing magnitude or units, and in calcu-lating accumulated (propagated) errors. The RSD for the data in Example 1 isgiven below.

Example 2

sr 02.00.3220

100 = 0.16%

Relativestandarddeviation


Where replicate samples are analyzed on a number of occasions under the sameprescribed conditions, an improved estimate of the standard deviation can beobtained by pooling the data from the individual sets. A general formula for thepooled standard deviation, spooled, is given by the expression

spooled = (4)where N1, N2, N3Nk are the numbers of results in each of the k sets, and x

_1, x

_2,

x_

3, . . . x_

k, are the means for each of the k sets.

The square of the standard deviation, s2, or estimated standard deviation, s2, isused in a number of statistical computations and tests, such as for calculatingaccumulated (propagated) errors (Topic B1 and below) or when comparing theprecisions of two sets of data (Topic B3).

Overall precision Random errors accumulated within an analytical procedure contribute to theoverall precision. Where the calculated result is derived by the addition orsubtraction of the individual values, the overall precision can be found bysumming the variances of all the measurements so as to provide an estimate ofthe overall standard deviation, i.e.

soverall = s12 + s22 + s32 + . . .ExampleIn a titrimetric procedure, the buret must be read twice, and the error associatedwith each reading must be taken into account in estimating the overall preci-sion. If the reading error has an estimated standard deviation of 0.02 cm3, thenthe overall estimated standard deviation of the titration is given by

soverall = 0.022+ 0.022 0.028 cm3Note that this is less than twice the estimated standard deviation of a singlereading. The overall standard deviation of weighing by difference is estimatedin the same way.

If the calculated result is derived from a multiplicative expression, the overallrelative precision is found by summing the squares of the relative standard devia-tions of all the measurements, i.e.

sr(overall) = s2r1 + s2r2 + s2r3 + . . .

The true or accepted mean of a set of experimental results is generallyunknown except where a certied reference material is being checked oranalyzed for calibration purposes. In all other cases, an estimate of the accu-racy of the experimental mean, x

_, must be made. This can be done by dening a

range of values on either side of x_

within which the true mean, m, is expected tolie with a dened level of probability. This range, which ideally should be asnarrow as possible, is based on the standard deviation and is known as the

Condenceinterval

Variance

i=N1

i=1(xi x

_1)2 +

i=N2

i=1(xi x

_2)2 +

i=N3

i=1(xi x

_3)2 + ... +

i=Nk

i=1(xi x

_k)2

i=k

i=1Ni = k

Pooled standarddeviation


condence interval, CI, and the upper and lower limits of the range as con-dence limits, CL. Condence limits can be calculated using the standard devia-tion, s, if it is known, or the estimated standard deviation, s, for the data. Ineither case, a probability level must be dened, otherwise the test is of novalue.

When the standard deviation is already known from past history, the con-dence limits are given by the equation

CL(m) = x_

(5)

where z is a statistical factor related to the probability level required, usually90%, 95% or 99%. The values of z for these levels are 1.64, 1.96 and 2.58, respec-tively, and correspond to the multiples of the standard deviation shown inFigure 3.

Where an estimated standard deviation is to be used, s is replaced by s,which must rst be calculated from the current data. The condence limits arethen given by the equation

CL(m) = x_

(6)

where z is replaced by an alternative statistical factor, t, also related to the prob-ability level but in addition determined by the number of degrees of freedomfor the set of data, i.e. one less than the number of results. It should be notedthat (i) the condence interval is inversely proportional to N, and (ii) thehigher the selected probability level, the greater the condence interval becomesas both z and t increase. A probability level of 100 percent is meaningless, as thecondence limits would then have to be .

The following examples demonstrate the calculation of condence limitsusing each of the two formulae.

Example 3The chloride content of water samples has been determined a very large numberof times using a particular method, and the standard deviation found to be7 ppm. Further analysis of a particular sample gave experimental values of350 ppm for a single determination, for the mean of two replicates and for themean of four replicates. Using equation (5), and at the 95% probability level, z = 1.96 and the condence limits are:

1 determinations CL(m) = 350 = 350 14 ppm


4 determinations CL(m) = 350 = 350 7 ppm7

Example 4The same chloride analysis as in Example 3, but using a new method for whichthe standard deviation was not known, gave the following replicate results,mean and estimated standard deviation:

1.96 7

4

1.96 7

2

1.96 7

1

tsN

zsN


Chloride/ppm Mean Estimated standard deviation

346 351.67 ppm 6.66 ppm359350

Using equation (6), and at the 95% probability level, t = 4.3 for two degrees offreedom, and the condence limits are:


The wider limits given by equation (6) when the standard deviation is estimatedwith only three results reects the much greater uncertainty associated with thisvalue, which in turn affects the condence in the degree of accuracy. To demon-strate good accuracy, the condence interval, CI, should be as small as possibleand increasing the number of replicates will clearly achieve this. However, dueto the N term in the denominator, to reduce the interval by, say, a factor oftwo requires an increase in the number of replicates by a factor of four as shownby Example 3. Unfortunately, the law of diminishing returns applies here, so ifthe CI is to be halved again, the number of replicates must be increased fromfour to sixteen. Similarly, in Example 4, the number of replicates would have to be increased from three to twelve to halve the CI, which would represent anunacceptable amount of time and money for most analytical laboratories.

4.3 6.66

3



B3 SIGNIFICANCE TESTING

Signicance tests involve a comparison between a calculated experimentalfactor and a tabulated factor determined by the number of values in the set(s) ofexperimental data and a selected probability level that the conclusion is correct.They are used for several purposes, such as:

to check individual values in a set of data for the presence of determinateerrors (bias);

to compare the precision of two or more sets of data using their variances; to compare the means of two or more sets of data with one another or with

known values to establish levels of accuracy.

Tests are based on a null hypothesis - an assumption that there is no signi-cant difference between the values being compared. The hypothesis is acceptedif the calculated experimental factor is less than the corresponding tabulatedfactor, otherwise it is rejected and there is said to be a signicant differencebetween the values at the selected probability level. The conclusion shouldalways be stated clearly and unambiguously.

Probability levels of 90%, 95% and 99% are generally considered appropriatefor most purposes, but it should be remembered that there are also corre-sponding 10%, 5% or 1% probabilities, respectively, of the opposite conclusionbeing valid. For example, if a test indicates that the null hypothesis is correctand that there is no signicant difference between two values at the 95% proba-bility level, it also allows the possibility that there is a signicant difference atthe 5% level.

Signicancetests

Key Notes

These are statistical tests used to compare individual values or sets ofvalues for signicant differences.

A measurement or result that appears to differ signicantly from othersin the same set of replicates is described as an outlier.

The Q-test is used to determine whether to reject or retain a suspectedoutlier.

The F-test enables the precisions of two sets of data to be compared usingtheir variances.

The t-test is used to compare two experimental means, an experimentalmean with a known value or sets of pairs of experimental values.

F-tests can be applied to several sets of data to assess and comparedifferent sources of variability.

Related topic Assessment of accuracy and precision (B2)

Signicance tests

Outliers

Q-test

F-test

t-test

Analysis of variance

Separate tabular values for some signicance test factors have been compiledfor what are described as one-tailed and two-tailed tests. The exact purpose ofthe comparison that is to be made determines which table to use.

The one-tailed test is used EITHER to establish whether one experimentalvalue is signicantly greater than the other OR the other way around.

The two-tailed test is used to establish whether there is a signicant differ-ence between the two values being compared, whether one is higher orlower than the other not being specied.

The two-tailed test is by far the most widely used. Examples are given below.

Outliers Inspection of a set of replicate measurements or results may reveal that one ormore is considerably higher or lower than the remainder and appears to be outsidethe range expected from the inherent effects of indeterminate (random) errorsalone. Such values are termed outliers, or suspect values, because it is possiblethat they may have a bias due to a determinate error. On occasions, the source oferror may

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