Development and Validation of Analytical Methods

Edited by

Christopher M. Riley

Analytical Research and Development DuPont Merck Pharmaceutical Company Wilmington, DE 19880, USA and

Department of Pharmaceutical Chemistry University of Kansas, Lawrence, KS 66045, USA

Thomas W. Rosanske

Analytical Chemistry Department Hoechst Marion Roussel, Inc., Kansas City, MO 64134, USA

PERGAMON

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Copyright © 1996 Elsevier Science Ltd

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publisher.

First Edition 1996

Library of Congress Catalogue in Publication Data

Pharmaceutical and biomedial analysis: development and validation of analytical methods / edited by Christopher M. Riley and Thomas W. Rosanske. -- 1st ed.

p. cm. ~ (Progress in pharmaceutical and biomedial analysis; v. 3) Includes index.

1. Drugs-Analysis-Methodology. I. Riley, Christopher M. II. Rosanske, Thomas W. III. Series

[DNLM: 1. Chemistry, Pharmaceutical-methods. 2. Drugs-analysis.

W1 PR677LM v. 3 1996 / QV 25 P5353 1996]

RS189.P442 1996

DNLM/DLC

for Library of Congress 96-33800

British Library Cataloguing in Publication Data

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

ISBN 0 08 042792 8

Printed and Bound in Great Britain by Biddies (Printers) Ltd, Guildford

Table of Contents

List of contributors vii

Introduction ix

Part One: Basic Concepts 1

1. Assay Validation and Inter-laboratory Transfer

Eugene McGonigle 3

2. Statistical Parameters and Analytical Figures of Merit Christopher M. Riley 15

Part Two: Regulatory Considerations 73

3. Overview of Worldwide Regulations 75 Ian E. Davidson

4. Issues Related to United States v. Barr Laboratories Inc.

Cathy L. Burgess, esquire 101

5. Judge Wolin's Interpretations of Current Good Manufacturing Practice Issues Contained in the Court's Ruling in the United States v. Barr Laboratories

Richard J. Davis 119

Part Three: Specific Methods and Applications 133

6. Bulk Drug Substances and Finished Products

Paul K. Hovsepian 135

7. Dissolution Studies 169 Thomas W. Rosanske and Cynthia K. Brown

8. Robotics and Automated Workstations 185 Julie J. Tomlinson

9. Biotechnology Products 209 G. Susan Srivatsa

10. Biological Samples 249 Krzysztof A. Selinger

11. Analytical Methods for Cleaning Procedures 293 Thomas M. Rossi and Ralph R. Ryall

12. Computer Systems and Computer-aided Validation 303 Joseph G. Liscouski

Index 339

List of Contributors

Cynthia K. Brown, Department of Analytical Chemistry, Hoechst Marion Roussel Inc., Marion Park Drive, Kansas City, MO 64134, USA

Cathy L. Burgess, esquire, Winston & Strawn, 1400 L Street NW, Washington, DC 20005, USA

Ian E. Davidson, Department of Analytical Chemistry, Hoechst Marion Roussel Inc., Marion Park Drive, Kanas City, MO 64134, USA

Richard J. Davis, Formerly Mid-Atlantic Office of the Food and Drug Administration, Philadelphia, PA, USA; present address Quality Assurance/Regulatory Compliance, DuPont Merck Pharmaceutical Company, DuPont Merck Plaza, Wilmington, DE 19880-0722, USA

Paul K. Hovsepian, Analytical Research and Development, DuPont Merck Pharmaceutical Company, Wilmington, DE 19880, USA

Joseph G. Liscouski, Laboratory Automation Standards Foundation, PO Box 38, Groton, MA 01450, USA

Eugene McGonigle, Physical and Analytical Chemistry, Research and Development, Schering-Plough Research Institute, Kenilworth, NJ 07033, USA

Christopher M. Riley, Analytical Research and Development, DuPont Merck Pharmaceutical Company, Wilmington, DE 19880, USA; and Pharmaceutical Chemistry Department, University of Kansas, Lawrence, KS 66045, USA

Thomas W. Rosanske, Department of Analytical Chemistry, Hoechst Marion Roussel Inc., Marion Park Drive, Kanas City, MO 64134, USA

Thomas M. Rossi, Analytical Research and Development Department, The R.W. Johnson Pharmaceutical Research Institute, Raritan, NJ 08869, USA

Ralph R. Ryall, Analytical Research and Development Department, The R.W. Johnson Pharmaceutical Research Institute, Raritan, NJ 08869, USA

Krzysztof A. Seiinger, Clinical Pharmacology Department, Glaxo Wellcome Inc., Five Moore Drive, Research Triangle Park, NC 27709, USA

G. Susan Srivatsa, Development Analytical Chemistry, Isis Pharmaceuticals, 2292 Faraday Avenue, Carlsbad, CA 92008, USA

Julie J. Tomlinson, Bioanalytical Automation and Robotics, Clinical Pharmacology Department, Glaxo Wellcome Inc., Five Moore Drive, Research Triangle Park, NC 27709, USA

Introduction

The need to validate an analytical or bioanalytical method is encountered by analysts in the pharmaceutical industry on an almost daily basis, because adequately validated methods are a necessity for approvable regulatory filings. What constitutes a validated method, however, is subject to analyst interpretation because there is no universally accepted industry practice for assay validation. This book is intended to serve as a guide to the analyst in terms of the issues and parameters that must be considered in the development and validation of analytical methods.

The scope of this book was expanded considerably from our original intent of considering only the validation of analytical methods. As we began the process of planning the layout of the book, it became clear that factors such as method development, data acquisition systems and regulatory considerations were all closely interrelated with validation and that a text on validation would not be complete without rather detailed discussions of all these components.

The book is divided into three parts. Part One, comprising two chapters, looks at some of the basic concepts of method validation. Chapter 1 discusses the general concept of validation and its role in the process of transferring methods from laboratory to laboratory. Chapter 2 looks at some of the critical parameters included in a validation program and the various statistical treatments given to these parameters.

Part Two (Chapters 3, 4 and 5) of the book focuses on the regulatory perspective of analytical validation. Chapter 3 discusses in some detail how validation is treated by various regulatory agencies around the world, including the United States, Canada, the European Community, Australia and Japan. This chapter also discusses the International Conference on Harmonization (ICH) treatment of assay validation. Chapters 4 and 5 cover the issues and various perspectives of the recent United States vs. Barr Laboratories Inc. case involving the retesting of samples.

Part Three (Chapters 6 - 12) covers the development and validation of various analytical components of the pharmaceutical product development process. This part of the book contains specific chapters dedicated to bulk drug substances and finished products, dissolution testing, robotics and automated systems, biotechnology related products, materials presented in biological matrices, cleaning procedures, and data acquisition systems. Each chapter goes into some detail describing the critical development and related validation considerations for each topic.

This book is not intended to be a practical description of the analytical validation process, but more of a guide to the critical parameters and considerations that must be attended to in an analytical development program. Despite the existence of numerous guidelines including the recent attempts by the ICH, the practical part of assay validation will always remain, to a certain extent, a matter of the personal preference of the analyst or company. Nevertheless, this book brings together the perspectives of several experts having extensive experience in different capacities in the pharmaceutical industry in an attempt to bring some consistency to analytical method development and validation.

The initial concepts for this book were developed while one of us (CMR) was on sabbatical leave from the University of Kansas at Hoechst Marion Roussel in Kansas City. We wish to express our gratitude to Dr. Michael Baltezor (Senior Director, Analytical Chemistry, Hoechst Marion Roussel) for providing us with access to resources in his department and for the opportunity to collaborate on this book.

Christopher M. Riley Thomas W. Rosanske January 2, 1996

Part One: Basic Concepts

Chapter 1

Assay Validation and Inter-laboratory Transfer

Eugene McGonigle

1.1 Introduction

Although the process of transferring an analytical method from one laboratory to another uses some of the principles and components of validation, validation and method transfer should be regarded as two distinct processes. The purpose of this chapter is to summarize the relationship between method transfer and validation, as well as to introduce some of the terminology that will be discussed later in this book.

Validation may be viewed as the establishment of an experimental database that certifies an analytical method performs in the manner for which it was intended and is the responsibility of the method-development laboratory. Method transfer, on the other hand, is the introduction of a validated method into a designated laboratory so that it can be used in the same capacity for which it was originally developed. Ordinarily, the method-transfer process should be the responsibility of the designated laboratory that will use the previously validated method. However, successful method transfer relies upon close cooperation and communication between the two laboratories.

1.2 Functional definitions

At this point, the introduction of some functional definitions is necessary to distinguish the responsibilities of the laboratory that develops a method from those of the laboratory who will use the method. The "gap" between the two laboratories is bridged by the method-transfer process.

In a pharmaceutical-industrial setting, the Analytical Research and Development (ARD) group usually provides validated analytical methods. Consistent with this, when a method is submitted to regulatory agency, a Method Validation Report will be provided. The method and the supporting data in the report are reviewed by both manufacturing and control reviewing chemists and one or more validation laboratories at the agency. The assessment by the validation laboratory is referred to as the "Validation of the Analytical Procedure". While use of the term validation in all the above situations is correct, it is not necessarily correct to state that a method used by the Quality Control (QC) Department to release commercial goods has been validated by that department. It may be correct if the QC Department developed the procedure, or if, for some reason, they choose to repeat the original method validation experiments.

Although the ARD group normally develops the original method, the real answer to the questions "who validates analytical methods" and "who transfers them" depends upon the circumstances. For example, analytical methods can either be developed by or transferred to: the laboratories, QC laboratories or third parties such as contract laboratories, depending upon a company's organizational structure and their objectives at a particular point in time. At most pharmaceutical companies the ARD laboratories are responsible for the development of analytical methods and their charter is to provide appropriate test methods, specifications and stability data to support all domestic and international registrations of new pharmaceutical entities and related dosage forms. Clearly, analytical methods for newly registered products require validation reports. However, it is equally probable that the QC Department may modify existing methods or develop improved procedures as new technology emerges. For example, the QC department may undertake developing a more rapid and efficient chromatographic separation rendering a particular analysis more cost effective.

Analytical methods may also be transferred to either an ARD laboratory or a QC laboratory from outside sources. This is likely to occur more frequently in the future as companies increase their activity in the area of licensing and acquisition of new products or, due to limited internal resources, sub-contract an increasingly larger fraction of their method development activities to contract laboratories. These possibilities reinforce the necessary distinction between method validation and method transfer since both ARD and QC laboratories can be involved in both processes. The experimental rationale and Standard Operating Procedures (SOPs) for method validation and method transfer are different in both their objectives and their utilization1.

1.2 Essential principles of method transfer

There are five essential principles, which if followed, will ensure successful method transfer: documentation, communication, acceptance criteria, implementation, and method modification and revalidation.

1.2.1 Documentation

The first essential principle of successful method transfer requires that the development laboratory provide the designated laboratory that will utilize the method with three essential elements, which demonstrate collectively that an analytical method is fully validated and ready for transfer. These include:

a) a written procedure, which contains a detailed, step-by-step description of the manipulations, specific reagents, equipment, instrument settings and other critical parameters. Each step in the procedure should be an explicit instruction allowing only one possible interpretation. This element of detail, from which there should be no essential deviation, most distinguishes method transfer from method validation or, the more general category under which method transfer is sometime placed, "technology transfer". The procedure must have a unique, cataloged identification so there can be no doubt that the correct procedure has been provided and received.

b) a method-validation report (see also sec. 1.3), which should include both the experimental design and the data that justify the

1 Editors' note: Although this chapter is concerned mainly with the validation and transfer of methods developed for the characterization of the drug substance and related pharmaceutical products, much of the discussion is equally applicable to validated methods for the analysis of drugs and metabolites in biological samples.

conclusion that the analytical method, as written, performs as intended; and c) system suitability criteria, which define the minimum acceptable performance criteria prior to each analysis.

1.2.2 Communication

As part of the FDA's Pre-approval Inspection (PAI) process, more and more emphasis is being placed on verifying that the official (sic) QC method is consistent with that included in the New Drug Application (NDA) dossier. Therefore, the second principle of successful method transfer is that ARD and QC staffs should meet before transfer to discuss all relevant, practical aspects of the method, particularly the manipulative steps. Ideally, these discussions should be initiated before validation is complete. In this fashion, it may be possible to incorporate any desirable modifications into the ARD validation report prior to registration. If such modifications constitute an equivalent alternative, early interdepartmental communications allow time to generate the data necessary to justify the alternative or to prepare a rationale why additional data are unnecessary. This also allows for preparing an Alternative Method Validation Report to be written and included in the registration package. Typical examples of equivalent alternatives include: changes in sample preparation automation requirements and the availability, suitability or cost-effectiveness of required reagents and equipment.

1.2.3 Acceptance criteria

The third principle requires that the designated laboratory be responsible for issuing and following SOPs that define their criteria for accepting an analytical method. Data generated in accordance with those SOPs form the basis for the Method Transfer Report, which should issue from the designated laboratory. This is important because the designated laboratory must ultimately assume responsibility for data and conclusions resulting from use of the method. As with most SOPs, it is both probable and acceptable for laboratories to have unique criteria for defining a method as acceptable. Some examples where different acceptance criteria may be used are: different statistical approaches for data evaluation, differences in the desired linear range, and different schemes for evaluating operator-to-operator or day-to-day variability.

1.2.4 Implementating the method as written and validated

This principle is the most important. The designated laboratory must follow the procedure, as written, to ensure that the method is supported by the database to be included in the Method Validation Report. Adherence to this principle will preclude the need to add additional data at the time of method transfer or, even worse, repeating the validation experiments. This again emphasizes the importance of effective communications between ARD and QC staffs.

1.2.5 Method modification and revalidation

Finally, if significant modifications to a method are incorporated at the time of transfer, revalidation may be necessary to ensure that the modifications have not invalidated previous, conclusive data in the Method Validation Report. Obviously, not all changes to the method require revalidation. Mention was made in sec. 1.2.2 of the potential advantages of discussing/incorporating modifications to an analytical method before method transfer. With the possible exception of automation, the examples of method modification given above represent method revisions and, therefore, may require revalidation of the method and, more importantly, a new and unique catalogue revision . The latter is important to ensure that there is no confusion over which version of the method has been used or is being employed for product analysis or which version is associated with a particular Method Validation or Method Transfer Report.

The following are some examples of modifications and subsequent aspects of method development that require revalidation:

a) automation The most common example of automation is the use of robotics (see also Chapter 7). If the robot repeats, identically, all manipulations of the previous manual method, only the precision experiments need be repeated. Since this might occur in a laboratory after method transfer, a statistical comparison of the precision of the manual and automated methods may be conducted on the same population of a representative product. Obviously, this assumes that the robotic system has been appropriately calibrated.

b) sample preparation One example of a change in sample preparation would be the desire to use whole tablets instead of ground-tablet composites. Another example might be a modification in which the solvent:solvent or solvent:solid ratios are changed in the extraction step. In these examples, accuracy, precision, linearity and range may be affected so the validation of these parameters should be repeated using both synthetic preparations of the drug product components and representative product samples.

c) dilutions If the solvent ratios remain identical to those in the original method and the analytical concentrations are also the same, only the precision is likely to be affected and those experiments should be repeated. Analysis of a representative product would be adequate for this purpose. In contrast, if the solvent aliquots are substantially different or the analytical solvent ratios differ in the dilution step(s), linearity and precision should be reassessed. Synthetic preparations of the drug product components should be utilized to re-establish comparable analytical parameters. Testing a representative product is probably not necessary but most laboratories tend to include this. If the proposed dilution change effects a different analytical concentration the analyst should also consider re-evaluating the specificity and the resolution (for chromatographic methods).

d) alternative chromatographic columns Substituting an alternative chromatographic column always raises the possibility of a change in specificity and resolution as well as the quantitative aspects of the method. Therefore, modifications of this type require that all method validation parameters to be reassessed, that is specificity(resolution), linearity, range, accuracy, precision and limit of quantitation (LOQ).

1.3 Method Validation Report

The preceding sections emphasize the importance of distinguishing between method validation and method transfer. Method validation establishes the scientific qualification of a specific analytical method and details of this are included in the Method

Validation Report. Transfer of a validated method is governed by the SOP established by the designated laboratory, which defines their acceptable performance criteria. Thus, the Method Validation Report is a pivotal document for any regulatory submission because it forms the basis for scientific qualification of the method. Consequently, it is appropriate to review certain aspects of the Method Validation Report, which relate directly to the method-transfer process and thereby qualify acceptable performance in the designated laboratory. Readers are referred to the other chapters in this book for more complete definitions and descriptions of the essential elements of assay validation.

1.3.1 Specificity (selectivity)

For a stability indicating assay of an intact drug, specificity (or selectivity)2 defines the ability of the method to measure the analyte to the exclusion of relevant components, which might interfere. Experiments to establish method specificity include evaluating formulation matrix components (e.g. a placebo) and any known related compounds such as synthesis-related impurities and degradation products. Other less relevant compounds such as metabolites or isomers, which might help to define the limits of a method's resolution may also be evaluated. A similar assessment is repeated after stressing the drug to accelerate degradation under the influence of heat, light, oxidation, and acid and base hydrolysis.

1.3.2 Chromatographic parameters/system suitability

For chromatographic methods a number of quantitative features should be defined. These parameters are ultimately used as the minimum standards of performance in system suitability tests. The resolution of a crucial pair (or pairs) of peaks in the chromatogram defines minimum separation requirement(s). Thus, the minimum resolution factor in the system suitability tests is generally used in conjunction with the column efficiency (number of theoretical plates) and the tailing factor.

2 See Chapter 2 for a detailed discussion of the differences between selectivity and specificity

1.3.3 Linearity

Linearity defines the analytical response as a function of analyte concentration and range prescribes a region over which acceptable linearity, precision and accuracy are achieved. While many analysts prefer to use a broad range, quantitative measurements for pharmaceutical assays (c.f. bioanalytical methods) are made over a range that minimally encompasses 80%, 100% and 120% of the analytical concentration prescribed in the method, recognizing this can be different than product "label claim". Either a sample of the drug substance or the reference standard may be used for these experiments.

1.3.4 Accuracy (recovery)

Recovery of the analyte of interest from a given matrix can be used as a measure of the accuracy or the bias of the method. The same range of concentrations as employed in the linearity studies is used. That is, the linearity experiment is repeated in the presence of matrix constituents; however, incorporation of impurities and degradation products may also be appropriate. It is important to note that, for the purposes of this discussion, recovery experiments of this type (while commonly employed) are limited to the detection of positive matrix interferences or adsorption effects and are not true "recovery" experiments. This is especially true of analytical methods for solid dosage forms. Furthermore, the method of additions can mask positive or negative bias and should only be used as a last resort.

1.3.5 Precision

Precision quantifies the variability of an analytical result as a function of operator, method manipulations and day-to-day environment. Statistical analysis of data generated to demonstrate assay precision is essential. For efficiency, analysts can use both the linearity and the recovery data for the statistical assessment. In addition, it is prudent to include an additional tier of comparative analyses of the sample of a representative product, usually a minimum of ten replicates. All the data to this point will have been generated by the development laboratory chemists. From the perspective of the designated laboratory, these validation data should be viewed as intra-laboratory data, even though more than one development chemist or more than one development laboratory may have been involved in generating the precision data.

1.4 The Inter-laboratory Qualification (ILQ) Process

Clearly the method-transfer process should involve more than simply the designated laboratory obtaining the "expected" result after analyzing a sample of representative product. This alone will not assure consistent performance of the method over time and may actually mask erroneous results arising from compensating errors. Furthermore, analysts in the pharmaceutical industry must be prepared answer the question: "Why do you always accept good results without challenge and question only what you conclude to be incorrect assays?" A good QC laboratory must be able to recognize rapidly, and with confidence, a method that is out of control, independent of the assay result, before a product is released. There are a number of ways to achieve this level of performance, begining with a sound method-transfer protocol.

Each laboratory involved in the method transfer process should define, independently, an experimental protocol to be followed for every method transferred. However, it is obvious that the most efficient way is to take advantage of the scientific database already established and included in the Method Validation Report. First, the designated laboratory should confirm the linearity and recovery for the analyte alone and in the presence of the known product components. Designing the experimental protocol for the ILQ so that it resembles, as much as possible, that which was carried out by the development laboratory provides two advantages:

a) it allows comparison of resultant raw data and calculated results with those already in the Method Validation Report; and b) assuming the analytical method remains essentially unchanged during transfer, the continuity of experimental design and resultant data allow the validation and the transfer reports to be reviewed by a regulatory agency as a complementary package.

Both the method development laboratory and the designated laboratory should test a common sample population that should be representative of the intended product. Comparison of these data provide an additional level of intra-laboratory information as well as forming the basis of the inter-laboratory qualification (ILQ) process. This simple experimental design generally allows bias or imprecision to be traced to an instrument in one of the laboratories, the method itself or a specific operator, the reason being that both laboratories will have analyzed both "analytically prepared" and "representative product" samples. Bias or imprecision associated with the assay of the former are clearly method related. Anomalous results associated only with latter usually indicate a problem with the conduct of the method in the designated laboratory, which under normal circumstances can be corrected by incorporating additional instructions into the method or by further training.

The results and conclusions of these experiments are summarized in the Method Transfer Report. Since the overall objective of this report should be documentation that the method is acceptable, it is the responsibility of the designated laboratory. The Method Transfer Report should remain in the files of the designated laboratory along with the Method Validation Report to support subsequent audits.

1.5 Conclusions

Method validation and method transfer are distinct processes. Method validation certifies that the method performs in the manner for which it was developed and is the responsibility of the method development laboratory. Experimental rationale, the supporting database and conclusions are summarized in the Method Validation Report. Method transfer certifies that an analytical method is acceptable to the designated laboratory once it has generated data demonstrating the method performs in the same capacity for which it was validated. Accordingly, method transfer criteria should be based on the SOPs, which are unique to the designated laboratory. While selected in accordance with minimum scientific criteria, method transfer SOPs should be consistent with the unique requirements of the individual laboratories, which may not (and need not) be identical in all laboratories. Finally, supporting data and conclusions for method transfer should be documented in a Method Transfer Report prepared by and retained in the designated laboratory.

Chapter 2

Statistical Parameters and Analytical Figures of Merit

Christopher M. Riley

2.1 Introduction

Validation of an analytical method is primarily concerned with the identification of the sources and the subsequent quantification of the potential errors in the method. Although basic assay validation uses relatively simple statistical approaches, Miller and Miller [1]

claim that " many highly competent scientists are woefully ignorant of even the most elementary statistical methods. It is even more astonishing that analytical chemists, who practise one of the most quantitative of all sciences are no more immune than others to this dangerous, but entirely curable, affliction." Thus, no text on the subject of validation would be complete without a discussion of the sources and types of error that may occur in the course of an analytical determination. Readers wishing to learn more about the application of statistics to analytical chemistry are referred to several excellent texts on the subject [2-5], The recently revised edition of "Statistics for Analytical Chemistry" by Miller and Miller [2] is highly recommended. Three types of error may be encountered in an analytical measurement: gross, systematic and random; and the analyst should be able to distinguish between each type.

2.1.1 Gross errors

Gross errors are simply defined as errors that are so serious that they require termination of the experiment. Such errors might include loss or contamination of a sample, omission of critical reagents, or instrument failure. Occasional gross errors are inevitable and ordinarily they are easily recognized. However, the recent legal action brought by the Food and Drug Administration (FDA) against Barr Laboratories, which focused on Barr Laboratories' procedures for retesting samples that were out of specification, highlights the importance of defining what constitutes a failure, as well as the procedures for failure investigations and for retesting of samples. The case of the United States versus Barr Laboratories [6] will be discussed in more detail in Chapter 4.

2.1.2 Systematic errors

Systematic errors (or determinate errors) affect the accuracy of an analytical measurement causing all the results to be in error in the same sense [7], The origin of systematic errors can usually be traced to errors of the analyst, instrumental errors (i.e. errors in calibration of the instrument), errors introduced by the reagent (e.g. side reactions or incomplete reactions) or some combination of the above. Systematic errors can bias the results so that they will be all too high or all too low. The systematic error associated with an analytical measurement can be defined by the bias, which may be positive or negative, or the accuracy, which may range between 0 (completely inaccurate) and 100 (completely accurate).

Accuracy :

where:

and n. is the true mean (or the population mean), x is the mean or the average of n measurements made on a sample taken to be representative of the total population and xi is the value of the ith measurement.

2.1.3 Random errors

Random errors (or indeterminate errors) result from uncontrolled variables in the measurement conditions, which cause individual results or measurements to fall on either side of the true mean, |a.. Random errors can be reduced but can never be eliminated. However, they are amenable to statistical analysis. If the individual results are normally distributed about the population mean, and there are no systematic errors (c.f. Sec. 2.1.2), the probability (yi) of obtaining a value xi is given by:

aV2n

where a is the standard deviation of the population (eq. 2.5). The larger the value of the standard deviation, the greater the spread of the data about the mean (Fig. 2.1).

Figure 2.1

Normal distribution of experimental data with different values for the means (|i) and the standard deviations (a) plotted according to eq. 2.4. Key: n.a=Wb<Hc; oa=oc<ob x- n

Figure 2.1

Normal distribution of experimental data with different values for the means (|i) and the standard deviations (a) plotted according to eq. 2.4. Key: n.a=Wb<Hc; oa=oc<ob i=n _ g I (xi-x)

It is possible for the distribution of data to be asymmetrically distributed about the mean as shown in Fig. 2.2. Nevertheless the methods for the analysis of data are essentially the same, irrespective of whether they are normally or asymmetrically distributed. Equation 2.4 predicts and Fig. 2.3 shows that approximately 68% of the values lie within ±1 a of the mean, approximately 95% of the values lie within ±2g of the mean and approximately 99.7% of the values lie within ±3o of the mean. Such information allows the description of confidence intervals about the sample mean.

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