Martin's Physical Pharmacy 6th.ed 2011 Dr.murtadha Alshareifi

PA RIOK J. SINKO

P1: Trim: 8.375in 10.875inLWBK401-fm-Dom LWW-Sinko-educational November 30, 2009 22:0

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MARTINS PHYSICAL PHARMACYAND PHARMACEUTICAL SCIENCES

Physical Chemical and Biopharmaceutical Principlesin the Pharmaceutical Sciences

S I X T H E D I T I O N

Editor

PATRICK J. SINKO, PhD, RPhProfessor II (Distinguished)

Parke-Davis Chair Professor in Pharmaceutics and Drug DeliveryErnest Mario School of Pharmacy

Rutgers, The State University of New JerseyPiscataway, New Jersey

Assistant Editor

YASHVEER SINGH, PhDAssistant Research Professor

Department of PharmaceuticsErnest Mario School of Pharmacy

Rutgers, The State University of New JerseyPiscataway, New Jersey

Dr. Murtadha Alshareifi e-Library

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Editor: David B. TroyProduct Manager: Meredith L. BrittainVendor Manager: Kevin JohnsonDesigner: Holly McLaughlinCompositor: Aptara, Inc.

Sixth Edition

Copyright c 2011, 2006 Lippincott Williams & Wilkins, a Wolters Kluwer business.351 West Camden StreetBaltimore, MD 21201

530 Walnut St.Philadelphia, PA 19106

Printed in China

All rights reserved. This book is protected by copyright. No part of this book may be reproducedor transmitted in any form or by any means, including photocopies or scanned-in or other electroniccopies, or utilized by any information storage and retrieval system without written permission fromthe copyright owner, except for brief quotations embodied in critical articles and reviews. Materialsappearing in this book prepared by individuals as part of their official duties as U.S. governmentemployees are not covered by the above-mentioned copyright. To request permission, please contactLippincott Williams & Wilkins at 530 Walnut Street, Philadelphia, PA 19106, via e-mail at [email protected], or via Website at lww.com (products and services).

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Library of Congress Cataloging-in-Publication Data

Martins physical pharmacy and pharmaceutical sciences : physicalchemical and biopharmaceutical principles in the pharmaceuticalsciences.6th ed. / editor, Patrick J. Sinko ; assistant editor,Yashveer Singh.

p. ; cm.Includes bibliographical references and index.ISBN 978-0-7817-9766-51. Pharmaceutical chemistry. 2. Chemistry, Physical and theoretical.

I. Martin, Alfred N. II. Sinko, Patrick J. III. Singh, Yashveer.IV. Title: Physical pharmacy and pharmaceutical sciences.[DNLM: 1. Chemistry, Pharmaceutical. 2. Chemistry, Physical. QV 744

M386 2011]RS403.M34 2011615.19dc22 2009046514

DISCLAIMER

Care has been taken to confirm the accuracy of the information present and to describe generallyaccepted practices. However, the authors, editors, and publisher are not responsible for errors oromissions or for any consequences from application of the information in this book and make nowarranty, expressed or implied, with respect to the currency, completeness, or accuracy of the contentsof the publication. Application of this information in a particular situation remains the professionalresponsibility of the practitioner; the clinical treatments described and recommended may not beconsidered absolute and universal recommendations.

The authors, editors, and publisher have exerted every effort to ensure that drug selection anddosage set forth in this text are in accordance with the current recommendations and practice at thetime of publication. However, in view of ongoing research, changes in government regulations, and theconstant flow of information relating to drug therapy and drug reactions, the reader is urged to checkthe package insert for each drug for any change in indications and dosage and for added warningsand precautions. This is particularly important when the recommended agent is a new or infrequentlyemployed drug.

Some drugs and medical devices presented in this publication have Food and Drug Administration(FDA) clearance for limited use in restricted research settings. It is the responsibility of the healthcare providers to ascertain the FDA status of each drug or device planned for use in their clinicalpractice.To purchase additional copies of this book, call our customer service department at (800) 638-3030or fax orders to (301) 223-2320. International customers should call (301) 223-2300.Visit Lippincott Williams & Wilkins on the Internet: at http://www.lww.com. Lippincott Williams &Wilkins customer service representatives are available from 8:30 am to 6:00 pm, EST.

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Dedicated to my parents Patricia and Patrick Sinko,my wife Renee, and my children Pat, Katie (and Maggie)

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DED ICAT IONALFRED N. MARTIN (19192003)

This fiftieth anniversary edition of Martins Physical Phar-macy and Pharmaceutical Sciences is dedicated to the mem-ory of Professor Alfred N. Martin, whose vision, creativity,dedication, and untiring effort and attention to detail led tothe publication of the first edition in 1960. Because of hisnational reputation as a leader and pioneer in the then emerg-ing specialty of physical pharmacy, I made the decision tojoin Professor Martins group of graduate students at Pur-due University in 1960 and had the opportunity to witnessthe excitement and the many accolades of colleagues fromfar and near that accompanied the publication of the firstedition of Physical Pharmacy. The completion of that workrepresented the culmination of countless hours of painstak-ing study, research, documentation, and revision on the partof Dr. Martin, many of his graduate students, and his wife,Mary, who typed the original manuscript. It also representedthe fruition of Professor Martins dream of a textbook thatwould revolutionize pharmaceutical education and research.Physical Pharmacy was for Professor Martin truly a labor oflove, and it remained so throughout his lifetime, as he workedunceasingly and with steadfast dedication on the subsequentrevisions of the book.

The publication of the first edition of Physical Pharmacygenerated broad excitement throughout the national and inter-national academic and industrial research communities inpharmacy and the pharmaceutical sciences. It was the worldsfirst textbook in the emerging discipline of physical pharmacyand has remained the gold standard textbook on the appli-cation of physical chemical principles in pharmacy and thepharmaceutical sciences. Physical Pharmacy, upon its publi-cation in 1960, provided great clarity and definition to a dis-cipline that had been widely discussed throughout the 1950sbut not fully understood or adopted. Alfred Martins Physi-cal Pharmacy had a profound effect in shaping the directionof research and education throughout the world of pharma-ceutical education and research in the pharmaceutical indus-try and academia. The publication of this book transformedpharmacy and pharmaceutical research from an essentiallyempirical mix of art and descriptive science to a quantita-tive application of fundamental physical and chemical scien-tific principles to pharmaceutical systems and dosage forms.Physical Pharmacy literally changed the direction, scope,

focus, and philosophy of pharmaceutical education during the1960s and the 1970s and paved the way for the specialty dis-ciplines of biopharmaceutics and pharmacokinetics which,along with physical pharmacy, were necessary underpinningsof a scientifically based clinical emphasis in the teaching ofpharmacy students, which is now pervasive throughout phar-maceutical education.

From the time of the initial publication of Physical Phar-macy to the present, this pivotal and classic book has beenwidely used both as a teaching textbook and as an indis-pensible reference for academic and industrial researchers inthe pharmaceutical sciences throughout the world. This sixthedition of Martins Physical Pharmacy and PharmaceuticalSciences serves as a most fitting tribute to the extraordinary,heroic, and inspired vision and dedication of Professor Mar-tin. That this book continues to be a valuable and widelyused textbook in schools and colleges of pharmacy through-out the world, and a valuable reference to pharmaceuticalscientists and researchers, is a most appropriate recognitionof the lifes work of Alfred Martin. All who have contributedto the thorough revision that has resulted in the publicationof the current edition have retained the original format andfundamental organization of basic principles and topics thatwere the hallmarks of Professor Martins classic first editionof this seminal book.

Professor Martin always demanded the best of himself, hisstudents, and his colleagues. The fact that the subsequent andcurrent editions of Martins Physical Pharmacy and Phar-maceutical Sciences have remained faithful to his vision ofscientific excellence as applied to understanding and apply-ing the principles underlying the pharmaceutical sciences isindeed a most appropriate tribute to Professor Martins mem-ory. It is in that spirit that this fiftieth anniversary edition isformally dedicated to the memory of that visionary and cre-ative pioneer in the discipline of physical pharmacy, AlfredN. Martin.

John L. Colaizzi, PhDRutgers, The State University of New Jersey

Piscataway, New JerseyNovember 2009

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PREFA C E

Ever since the First Edition of Martins Physical Pharmacywas published in 1960, Dr. Alfred Martins vision was to pro-vide a text that introduced pharmacy students to the applica-tion of physical chemical principles to the pharmaceutical sci-ences. This remains a primary objective of the Sixth Edition.Martins Physical Pharmacy has been used by generations ofpharmacy and pharmaceutical science graduate students for50 years and, while some topics change from time to time,the basic principles remain constant, and it is my hope thateach edition reflects the pharmaceutical sciences at that pointin time.

ORGANIZATION

As with prior editions, this edition represents an updating ofmost chapters, a significant expansion of others, and the addi-tion of new chapters in order to reflect the applications of thephysical chemical principles that are important to the Phar-maceutical Sciences today. As was true when Dr. Martin wasat the helm, this edition is a work in progress that reflectsthe many suggestions made by students and colleagues inacademia and industry. There are 23 chapters in the SixthEdition, as compared with 22 in the Fifth Edition. All chap-ters have been reformatted and updated in order to makethe material more accessible to students. Efforts were madeto shorten chapters in order to focus on the most importantsubjects taught in Pharmacy education today. Care has beentaken to present the information in layers from the basicto more in-depth discussions of topics. This approach allowsthe instructor to customize their course needs and focus theircourse and the students attention on the appropriate topicsand subtopics.

With the publication of the Sixth Edition, a Web-basedresource is also available for students and faculty members(see the Additional Resources section later in this preface).

FEATURES

Each chapter begins with a listing of Chapter Objectives thatintroduce information to be learned in the chapter. Key Con-cept Boxes highlight important concepts, and each ChapterSummary reinforces chapter content. In addition, illustra-tive Examples have been retained, updated, and expanded.Recommended Readings point out instructive additionalsources for possible reference. Practice Problems have been

moved to the Web (see the Additional Resources sectionlater in this preface).

SIGNIFICANT CHANGES FROM THE FIFTH EDITION

Important changes include new chapters on PharmaceuticalBiotechnology and Oral Solid Dosage Forms. Three chap-ters were rewritten de novo on the basis of the valuablefeedback received since the publication of the Fifth Edi-tion. These include Chapter 1 (Introduction), which isnow called Interpretive Tools; Chapter 20 (Biomaterials),which is now called Pharmaceutical Polymers; and Chap-ter 23 (Drug Delivery Systems), which is now calledDrug Delivery and Targeting.

ADDITIONAL RESOURCES

Martins Physical Pharmacy and Pharmaceutical Sciences,Sixth Edition, includes additional resources for both instruc-tors and students that are available on the books companionWeb site at thepoint.lww.com/Sinko6e.

InstructorsApproved adopting instructors will be given access to thefollowing additional resources:

Practice problems and answers to ascertain student under-standing.

StudentsStudents who have purchased Martins Physical Pharmacyand Pharmaceutical Sciences, Sixth Edition, have access tothe following additional resources:

A separate set of practice problems and answers to rein-force concepts learned in the text.

In addition, purchasers of the text can access the searchableFull Text Online by going to the Martins Physical Phar-macy and Pharmaceutical Sciences, Sixth Edition, Web siteat thePoint.lww.com/Sinko6e. See the inside front cover ofthis text for more details, including the passcode you willneed to gain access to the Web site.

Patrick SinkoPiscataway, New Jersey

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CONTR IBUTORS

GREGORY E. AMIDON, PhDResearch ProfessorDepartment of Pharmaceutical SciencesCollege of PharmacyUniversity of MichiganAnn Arbor, Michigan

CHARLES RUSSELL MIDDAUGH, PhDDistinguished ProfessorDepartment of Pharmaceutical ChemistryUniversity of KansasLawrence, Kansas

HOSSEIN OMIDIAN, PhDAssistant ProfessorDepartment of Pharmaceutical SciencesCollege of PharmacyNova Southeastern UniversityFt. Lauderdale, Florida

KINAM PARK, PhDShowalter Distinguished ProfessorDepartment of Biomedical EngineeringProfessor of PharmaceuticsDepartments of Biomedical Engineering and PharmaceuticsPurdue UniversityWest Lafayette, Indiana

TERUNA J. SIAHAAN, PhDProfessorDepartment of Pharmaceutical ChemistryUniversity of KansasLawrence, Kansas

YASHVEER SINGH, PhDAssistant Research ProfessorDepartment of PharmaceuticsErnest Mario School of PharmacyRutgers, The State University of New JerseyPiscataway, New Jersey

PATRICK J. SINKO, PhD, RPhProfessor II (Distinguished)Parke-Davis Chair Professor in Pharmaceutics and Drug DeliveryErnest Mario School of PharmacyRutgers, The State University of New JerseyPiscataway, New Jersey

HAIAN ZHENG, PhDAssistant ProfessorDepartment of Pharmaceutical SciencesAlbany College of Pharmacy and Health SciencesAlbany, New York

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ACKNOWLEDGMEN T S

The Sixth Edition reflects the hard work and dedication ofmany people. In particular, I acknowledge Drs. Gregory Ami-don (Ch 22), Russell Middaugh (Ch 21), Hamid Omidian(Chs 20 and 23), Kinam Park (Ch 20), Teruna Siahaan (Ch21), and Yashveer Singh (Ch 23) for their hard work in spear-heading the efforts to write new chapters or rewrite existingchapters de novo. In addition, Dr. Singh went beyond thecall of duty and took on the responsibilities of AssistantEditor during the proofing stages of the production of themanuscripts. Through his efforts, I hope that we have caughtmany of the minor errors from the fourth and fifth editions. Ialso thank HaiAn Zheng, who edited the online practice prob-lems for this edition, and Miss Xun Gong, who assisted him.

The figures and experimental data shown in Chapter 6were produced by Chris Olsen, Yuhong Zeng, WeiqiangCheng, Mangala Roshan Liyanage, Jaya Bhattacharyya,Jared Trefethen, Vidyashankara Iyer, Aaron Markham, JulianKissmann and Sangeeta Joshi of the Department of Pharma-ceutical Chemistry at the University of Kansas. The sectionon drying of biopharmaceuticals is based on a series of lec-tures and overheads presented by Dr. Pikal of the Universityof Connecticut in April of 2009 at the University of Kansas.

I would like to acknowledge Dr. Mayur Lodaya for his con-tributions to the continuous processing section of Chapter 22on Oral Dosage forms.

Numerous graduate students contributed in many waysto this edition, and I am always appreciative of their in-sights, criticisms, and suggestions. Thanks also to Mrs. AmyGrabowski for her invaluable assistance with coordinationefforts and support interactions with all contributors.

To all of the people at LWW who kept the project mov-ing forward with the highest level of professionalism, skill,and patience. In particular, to David Troy for supporting ourvision for this project and Meredith Brittain for her excep-tional eye for detail and her persistent efforts to keep us ontrack.

And to my wonderful wife, Renee, who deserves enor-mous credit for juggling her hectic professional life as apharmacist and her expert skill as the family organizer whilemaintaining a sense of calmness in what is an otherwisechaotic life.

Patrick SinkoPiscataway, New Jersey

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CONTEN T S

1 INTERPRETIVE TOOLS 1

2 STATES OF MATTER 17

3 THERMODYNAMICS 54

4 DETERMINATION OF THE PHYSICAL PROPERTIESOF MOLECULES 77

5 NONELECTROLYTES 109

6 ELECTROLYTE SOLUTIONS 129

7 IONIC EQUILIBRIA 146

8 BUFFERED AND ISOTONIC SOLUTIONS 163

9 SOLUBILITY AND DISTRIBUTION PHENOMENA 182

10 COMPLEXATION AND PROTEIN BINDING 197

11 DIFFUSION 223

12 BIOPHARMACEUTICS 258

13 DRUG RELEASE AND DISSOLUTION 300

14 CHEMICAL KINETICS AND STABILITY 318

15 INTERFACIAL PHENOMENA 355

16 COLLOIDAL DISPERSIONS 386

17 COARSE DISPERSIONS 410

18 MICROMERITICS 442

19 RHEOLOGY 469

20 PHARMACEUTICAL POLYMERS 492

21 PHARMACEUTICAL BIOTECHNOLOGY 516

22 ORAL SOLID DOSAGE FORMS 563

23 DRUG DELIVERY AND TARGETING 594

Index 647

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1 Interpretive Tools Chapter Objectives At the conclusion of this chapter the student should be able to:

1. Understand the basic tools required to analyze and interpret data sets from the clinic, laboratory, or literature.

2. Describe the differences between classic dosage forms and modern drug delivery systems.

3. Use dimensional analysis. 4. Understand and apply the concept of significant figures. 5. Define determinant and indeterminant errors, precision, and accuracy. 6. Calculate the mean, median, and mode of a data set. 7. Understand the concept of variability. 8. Calculate standard deviation and coefficient of variation and understand when it is

appropriate to use these parameters. 9. Use graphic methods to determine the slope of lines. 10. Interpret slopes of lines and how they relate to absorption and elimination from the

body.

Introduction One of the earmarks of evidence-based medicine is that the practitioner should not just accept the conventional wisdom of his/her mentor. Evidence-based medicine uses the scientific method of using observations and literature searches to form a hypothesis as a basis for appropriate medical therapy. This process necessitates education in basic sciences and an understanding of basic scientific principles.1,2 Today more than ever before, the pharmacist and the pharmaceutical scientist are called upon to demonstrate a sound knowledge of biopharmaceutics, biochemistry, chemistry, pharmacology, physiology, and toxicology and an intimate understanding of the physical, chemical, and biopharmaceutical properties of medicinal products. Whether engaged in research and development, teaching, manufacturing, the practice of pharmacy, or any of the allied branches of the profession, the pharmacist must recognize the need to rely heavily on the basic sciences. This stems from the fact that pharmacy is an applied science, composed of principles and methods that have been culled from other disciplines. The pharmacist engaged in advanced studies must work at the boundaries between the various sciences and must keep abreast of advances in the physical, chemical, and biological fields in order to understand and contribute to the rapid developments in his or her profession. You are also expected to provide concise and practical interpretations of highly technical drug information to your patients and colleagues. With the abundance of information and misinformation that is freely and publicly available (e.g., on the Internet), having the tools and ability to provide meaningful interpretations of results is critical. Historically, physical pharmacy has been associated with the area of pharmacy that dealt with the quantitative and theoretical principles of physicochemical science as they applied to the practice of pharmacy. Physical pharmacy attempted to integrate the factual knowledge of pharmacy through the development of broad principles of its own, and it aided the pharmacist and the pharmaceutical scientist in their attempt to predict the solubility, stability, compatibility, and biologic action of drug products. Although this remains true today, the field has become even more highly integrated into the biomedical aspects of the practice of pharmacy. As such, the field is more broadly known today as the pharmaceutical sciences and the chapters that follow reflect the high degree of integration of the biological and physicalchemical aspects of the field. Developing new drugs and delivery systems and improving upon the various modes of administration are still the primary goals of the pharmaceutical scientist. A practicing pharmacist must also possess a thorough understanding of modern drug delivery systems as he or she advises patients on the best use of prescribed medicines. In the past, drug delivery focused nearly exclusively on pharmaceutical


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13

mindjouleunits

R

ExaHowpropthe elitersknowout t

One equiv

d the converes? Write dos. For the un

Table

Rule

All nonzerconsidered

Leading zesignificant

Trailing zecontainingare signific

The signifizeros in a ncontainingcan be amb

Zeros appebetween twdigits are s

ample 1-3 w many galloportions to soequation, thes, is set dowwn relations the indicated

may be concevalents such a

rsion factor: Iown the convnknown quan

1-1 Writin

ro digits ared significant

eros are not t

eros in a numg a decimal pcant

ficance of tranumber not

g a decimal pbiguous

earing anywwo nonzero significant

ns are equivolve this probe quantity den on the righin ratio form

d operations

erned about thas 1 pint = 473

f 1 cal equalersion factor

ntity, use anX

ng or Interp

Ex

t

9a

0a

mber point

91

ailing

point

Tndwt

where 61

valent to 2.0 lblem. In the mesired, X (galht side of the , such as 1 pyields the re

he apparent d3 mL. The qua

ls 4.184 jouler, being careX.

preting Sig

xample

98.513 has fand 3

0.00361 hasand 1

998.100 has1, 0, and 0

The numbernumbers likdecimal poiwritten as 1there are fiv

607.132 has1, 3, and 2

liters? It woumethod invollons), is placequation. Th

pint per 473 esult with its p

isregard for thantity of pints

es, how manful to expres

gnificant fig

five signific

s three signi

s six signific

r of significke 11,000 is int is missin1,000, it wo

ve significan

s six signific

uld be necesslving the ideced on the lehe right side mL, to give tproper units:

he rules of sigcan be measu

ny calories ars each quan

gures in Nu

cant figures

ificant figur

cant figures

cant figures uncertain b

ng. If the nuould be cleant figures

cant figures

sary to set upntity of units

eft and its equmust then bhe units of g

nificant figuresured as accura

re there in 3.ntity in its pro

umbers

s: 9, 8, 5, 1,

res: 3, 6,

s: 9, 9, 8,

in because a umber was ar that

s: 6, 0, 7,

p successiveon both sideuivalent, 2.0

be multiplied gallons. Carry

s in the ately as that o

00 oper

e es of

by ying

of


14

millilby deSigA sigstandSigntrailinsignimay P.5 glassestimmay suchthat meagivenExaHowThe the dbecaThuHowThe wheindicin an103 be tacont

Signof a digitsrepothat repoprecrulersignipartsThe 0.00pointmea4 mo

Wh

iters, so that wefinition, and s

gnificant gnificant figureds. The rules ificant figures ng zeros wherificant figures use a ruler, th

s tubing. If onemate the doub

yield the valuh as 27.4 cm isthe final figuren deviation of n in an official

ample 1-4 w Many Sig

two zeros imdecimal poinause it is nots, the value

w Many Sigquestion of

ether any or acate the magn unambiguosignifies thataken as signtains a total o

nificant figures value may be s introduced b

orting measurethe instrumen

orted. For examision as one m

r and a value oificant figures,s in 3000, wheabsolute mag053 mole/litert. These zerossurement. Wh

ole/liter, both it

Key Cohen Signific

we assume 1.0significant figuFigures

e is any digit ufor interpretinggive a sense

re they are usof a number ihe smallest su

e finds that thetful fraction, s

ue 27.6 or 27.2s encounterede is correct to f a single meas compendium

nificant Figmmediately font and are not needed to wcontains threnificant Figsignificant figall of the zerognitude of theous way, it ist the number

nificant. In theof four signif

are particularquestioned s

by calculationsements to a grnt used to makmple, a measumarked off in 0of, say, 27.46 is obviously t

ereas 27.4 impgnitude of a var as a relativelys are not signihen such a rests precision an

oncept cant Figure

00 pint is meaures need not

used to represeg significant fiof the accuraced merely to lnclude all cert

ubdivisions of

e tubing measay 0.4, and ex

2 cm, so that td in the literatuwithin about surement. How, it means 99.

gures in theollowing the

ot significant.write the numee significantgures in thegures in the os are meane number. His best to use r contains twe value 7.500icant figures

rly useful for inpecifically in cs carried out toreater precisioke the measureuring rule mar0.1 cm or mm. 0.02 cm witthe more precplies a precisioalue should noy small quantificant, howevesult is expressnd its magnitud

s do Not Ap

ant here. The qbe considered

ent a magnitugures and somcy of a numbeocate the dectain digits pluswhich are cen

sures slightly gxpress the numthe result is exure without fur 1 in the last dwever, when a.0 and not 98.

e Number 0decimal poin However, th

mber; if it wert figures.

e Number 75number 750t to be signifint: To expreexponential

wo significant 0 103, both.

ndicating the pcases when peo greater accu

on than the eqrement limits thrked off in cent. One may obtth the second.cise one. The non of only 2 pa

ot be confusedity because ther, and tell us

sed as 5.3 1de are readily

pply

quantities 1 gad in such case

ude or a quantme examples er. They includcimal point. Ans the first uncentimeters, to m

greater than 2mber as 27.4 xpressed as 2rther qualificatdecimal placea statement su9.

.00750? nt in the numhe zero followre not signific

500? 0 is ambiguoficant or wheess the signif

notation. Thfigures, and

h zeros are s

precision of a erforming calcuracy than thauipment supphe precision otimeter divisiotain a length o The latter rulnumber 27.46arts in 300.

d with its precishree zeros imm

nothing abou0-4 mole/liter,

y apparent.

allon and 1 litees.

ity in the placeare shown in

de all digits exnother way to sertain digit. Fomeasure the le

27 cm in lengthcm. A replicat7.4 0.2 cm. ion, the reade, which is meauch as not les

mber 0.00750wing the 5 is cant, it could

ous. One doether they are

ficant figures us, the expre the zeros in

significant, an

result. The proculations (e.g.,t of the originaorts. It is impo

of the resultingons will not proof 27.4 0.2 cer, yielding a rimplies a prec

sion. We consmediately followt the precisionor better as 5

er are also exa

e in which it Table 1-1.

xcept leading astate this is, th

or example, onength of a piec

h, it is proper tte measuremeWhen a value

er should assuant to signify tss than 99 is

0 merely locasignificant

d be omitted.

es not know e simply useds of such a vaession 7.5

n 7500 are nond the numb

oper interpreta, when spurioual data) or wheortant to remeg value that is oduce as greacm with the firsresult with fou

ecision of abou

sider the numbw the decima

n of the .3 (0.1) 10

act

and he ne ce of

to ent e ume the

ate

d to alue

ot to er

ation us en mber

t a st ur ut 2

ber l

-


15

Since significant figure rules are based upon estimations derived from statistical rules for handling probability distributions, they apply only to measuredvalues. The concept of significant figures does not pertain to values that are known to be exact. For example, integer counts (e.g., the number of tablets dispensed in a prescription bottle); legally defined conversions such as 1 pint = 473 mL; constants that are defined arbitrarily (e.g., a centimeter is 0.01 m); scalar operations such as doubling or halving; and mathematical constants, such as and e. However, physical constants such as Avogadro's number have a limited number of significant figures since the values for these constants are derived from measurements.

Example 1-5 The following example is used to illustrate excessive precision. If a faucet is turned on and 100 mL of water flows from the spigot in 31.47 sec, what is the average volumetric flow rate? By dividing the volume by time using a calculator, we get a rate of 3.177629488401652 mL/sec. Directly stating the uncertainty is the simplest way to indicate the precision of any result. Indicating the flow rate as 3.177 0.061 mL/sec is one way to accomplish this. This is particularly appropriate when the uncertainty itself is important and precisely known. If the degree of precision in the answer is not important, it is acceptable to express trailing digits that are not known exactly, for example, 3.1776 mL/sec. If the precision of the result is not known you must be careful in how you report the value. Otherwise, you may overstate the accuracy or diminish the precision of the result.

In dealing with experimental data, certain rules pertain to the figures that enter into the computations:

1. In rejecting superfluous figures, increase by 1 the last figure retained if the following figure rejected is 5 or greater. Do not alter the last figure if the rejected figure has a value of less than 5.

2. Thus, if the value 13.2764 is to be rounded off to four significant figures, it is written as 13.28. The value 13.2744 is rounded off to 13.27.

3. In addition or subtraction include only as many figures to the right of the decimal point as there are present in the number with the least such figures. Thus, in adding 442.78, 58.4, and 2.684, obtain the sum and then round off the result so that it contains only one figure following the decimal point:

This figure is rounded off to 503.9.

Rule 2 of course cannot apply to the weights and volumes of ingredients in the monograph of a pharmaceutical preparation. The minimum weight or volume of each ingredient in a pharmaceutical formula or a prescription

P.6

should be large enough that the error introduced is no greater than, say, 5 in 100 (5%), using the weighing and measuring apparatus at hand. Accuracy and precision in prescription compounding are discussed in some detail by Brecht.5

4. In multiplication or division, the rule commonly used is to retain the same number of significant figures in the result as appears in the value with the least number of significant figures. In multiplying 2.67 and 3.2, the result is recorded as 8.5 rather than as 8.544. A better rule here is to retain in the result the number of figures that produces a percentage error no greater than that in the value with the largest percentage uncertainty.


16

Remare uinforan ethat the cDaThe consrelatdepemea

whicequasay y

Becacons

Henccons

then

It is fnota

whic+ logway As wbackneedaccoThe of labconc

5. In the usefigures inmagnitudtheoreticalogarithmcover of trarely use

6. If the resthe rules

member, signifiused to prevenrmation than yxperiment in tlooks somethi

collection of thta Typesscientist is co

solidate and inionship betweendence of onsurable quant

ch is read y vaation as followy1 and x1, y2 a

ause the ratio stant, or, in ge

ce, it is a simpstant, k. To su

frequently destion

ch is read y isg (a/x). The fuwithout specif

we begin to laykground informded. In 1946, Sording to a rulefirst two, interboratory meas

centrations, we

e of logarithmn the mantissade of the numbal pharmacy u

m table yields sthis book. Theed today. ult is to be usejust given. Th

icant figures ant the loss of p

you actually kntriplicate (in oting like 4.351

he data. It simps

ntinually attemnterpret experieen two quantine property, thtity, the indepe

aries directly aws. If y is propoand x2,, are

of any y to itsneral

ple matter to cmmarize, if

sirable to show

some functionctional notatifying the actuay the foundatiomation about thStevens definee.4 He proposrvals and ratiosurements foreights). Only r

s for multiplicaa as there are ber and accordusually requiresufficient prece calculator is

ed in further che final result i

are not meant precision whenow. Error andher words, yo0.076. This dply means tha

mpting to relatmental data. Tities that are ce dependent vendent variabl

as x or y is dortional to x inproportional. T

s correspondin

change a propo

w the relations

n of x. That ision in equational equation byon for the interhe types of daed measuremsed a classifics, are categornearly all of t

ratio or interva

ation and divisin the originaldingly is not s

e no more thanision for our wmore conveni

calculations, reis then rounde

to be a perfecn rounding nu

d uncertainty aou repeat the edoes not meanat the outcome

te phenomenaThe problem fchanging at a variable, y, onle x, is expres

irectly proport general, thenThus,

ng x is equal to

ortionality to a

ship between x

s, y may be eqn (1-5) merelyy which they arpretation of data that you wi

ment as the ascation system rized as continhe data that a

al measureme

sion, retain the numbers. Theignificant. Becn three signific

work. Such a taient, however,

etain at least oed off to the la

ct representatiumbers. They are not the samexperiment thrn that you made is naturally s

a and establishfrequently resocertain rate or

n the change osed mathema

tional to x. A n all pairs of sp

o any other ra

an equality by

x and y by the

qual, for examy signifies thatre connected.

data using desill encounter inssignment of nthat is widely

nuous variableare normally conts can have

e same numbee characteristicause calculatcant figures, aable is found o, and tables of

one digit more st significant f

on of uncertaialso help you me. For examree times), youde an error in statistical.

h generalizatioolves into a ser in a particulaor alteration ofatically as

proportionalitypecific values

tio of y and x,

introducing a

e use of the mo

mple, to 2x, to 2

y and x are re criptive statistn the pharmacnumbers to objused today to

es. These wouollected in theunits of measu

er of significanic signifies ontions involved a four-place on the inside bf logarithms a

than suggestfigure.

inty. Instead, tavoid stating ple, if you peru will get a vathe experimen

ons with whichearch for the ar manner. Thef another

y is changed tof y and x,

the ratios are

proportionalit

ore general

27x2, or to 0.0elated in some

tics, some ceutical sciencbjects or evento define data tyuld include rese laboratory (eurement, and

nt ly the in

back re

ted in

they more

rform lue nt or

h to

e

o an

e

ty

0051 e

ces is ts ypes. sults .g.,


17

these variables are quantitative in nature. In other words, if you were given a set of interval data you would be able to calculate the exact differences between the different values. This makes this type of data quantitative. Since the interval between measurements can be very small, we can also say that the data are continuous. Another laboratory example of interval data measures is temperature. Think of the gradations on a common thermometer (in Celsius or Fahrenheit scale)they are typically spaced apart by 1 degree with minor gradations at the 1/10th degree. The intervals could become even smaller; however, because of the physical limitations of common thermometers, smaller gradations are not possible since they cannot be read accurately. Of course, with digital thermometers the gradations (or intervals) could be much smaller but then the precision of the thermometer may become questionable. Another temperature scale that will be used in various sections of this text is the Kelvin scale, a thermodynamic temperature scale. By international agreement, P.7 the Kelvin and Celsius scales are related through the definition of absolute zero (in other words, 0 K = -273.15C). Since the thermodynamic temperature is measured relative to absolute zero, the Kelvin scale is considered a ratio measurement. This also holds true for other physical quantities such as length or mass. The third common data type in the pharmaceutical sciences is ordinal scale measurements. Ordinal measurements represent the rank order of what is being measured. Ordinals are more subjective than interval or ratio measurements. The final type of measurement is called nominal data. In this type of measurement, there is no order or sequence of the observations. They are merely assigned different groupings such as by name, make, or some similar characteristic. For example, you may have three groups of tablets: white tablets, red tablets, and yellow tablets. The only way to associate the various tablets is by their color. In clinical research, variables measured at a nominal level include sex, marital status, or race. There are a variety of ways to classify data types and the student is referred to texts devoted to statistics such as those listed in the recommended readings at the end of this chapter.6,7 Error and Describing Variability If one is to maintain a high degree of accuracy in the compounding of prescriptions, the manufacture of products on a large scale, or the analysis of clinical or laboratory research results, one must know how to locate and eliminate constant and accidental errors as far as possible. Pharmacists must recognize, however, that just as they cannot hope to produce a perfect pharmaceutical product, neither can they make an absolute measurement. In addition to the inescapable imperfections in mechanical apparatus and the slight impurities that are always present in chemicals, perfect accuracy is impossible because of the inability of the operator to make a measurement or estimate a quantity to a degree finer than the smallest division of the instrument scale. Error may be defined as a deviation from the absolute value or from the true average of a large number of results. Two types of errors are recognized: determinate(constant) and indeterminate (random or accidental). Determinate Errors Determinate or constant errors are those that, although sometimes unsuspected, can be avoided or determined and corrected once they are uncovered. They are usually present in each measurement and affect all observations of a series in the same way. Examples of determinate errors are those inherent in the particular method used, errors in the calibration and the operation of the measuring instruments, impurities in the reagents and drugs, and biased personal errors that, for example, might recur consistently in the reading of a meniscus, in pouring and mixing, in weighing operations, in matching colors, and in making calculations. The change of volume of solutions with temperature, although not constant, is a systematic error that can also be determined and accounted for once the coefficient of expansion is known. Determinate errors can be reduced in analytic work by using a calibrated apparatus, using blanks and controls, using several different analytic procedures and apparatus, eliminating impurities, and carrying out the experiment under varying conditions. In pharmaceutical manufacturing, determinate errors can


18

be ewith indetIndIndeWhescattpattepattea nuIndeall mThosvariabelondetetemppresappaindetthe ptruly PrePrecagreof thDeteP.8 The indetinacc

liminated by cother workersterminate erro

determinaterminate erro

en one fires a ntered around tern on the targern around an mber of capsuterminate erro

measurementsse errors that ations involvedng to the classrminate errorsperature bathssure where inaratus can alsterminate, canpart of the worindeterminate

ecision acision is a meaeement betweee results, and

erminate or co

techniques usterminate errocuracies will b

calibrating the s. Adequate coors can have aate Errorors occur by anumber of bulthis central poget. Likewise, average or ce

ules with a druors cannot be . arise from rand in reading ins of determinas. These errors and ovens, tdicated. Careo reduce pseun thus be deterker. Only erroe.

and Accuasure of the agen the data an the measuremnstant errors a

sed in analyzinors, will be conbe discussed la

weights and oorrections for dany significancrs ccident or chalets at a targe

oint. The greatin a chemical entral value, kug, and the finallowed for or

ndom fluctuatiostruments are

ate errors and s may be reduhe use of buffin reading fra

udoaccidental ermined and coors that result

uracy greement amond the true valment of the praffect the accu

ng the precisionsidered first, ater.

other apparatudeterminate ece.

ance, and theyet, some may hter the skill of tanalysis, the

known as the mnished productr corrected be

ons in tempere not to be con are often calluced by controfers, and the mactions of units errors. Variaborrected by cafrom pure ran

ong the valuesue. Indetermin

recision is accuracy of data.

on of results, wand the detec

us and by cheerrors must be

y vary from onhit the bull's ethe marksmanresults of a semean. Randomts will show a cause of the n

ature or other nsidered accided pseudoaccolling conditionmaintenance os on graduatesble determinatareful analysisdom fluctuatio

s in a group ofnate or chanc

complished be

which in turn sction and elimi

cking calculatmade before

ne measuremeye, whereas on, the less scaeries of tests wm errors will adefinite variat

natural fluctuat

external factodental or randocidental or varns through theof constant hums, balances, ate errors, althos and refinemeons in nature a

f data, whereae errors influest by statistica

supply a measnation of dete

tions and resuthe estimatio

ent to the nextothers will be attered will be will yield a ranalso occur in fition in weight.

ations that occ

ors and from tom. Instead, triable e use of constmidity and

and other ough seemingent of techniquare considere

as accuracy isence the precisal means.

sure of the erminate errors

ults n of

t.

the dom lling

ur in

he hey

tant

gly ue on d

s the sion

s or


19

Fig.

Indeequaresuaxis,in Figreprethe usampDesSincsectitimesout obasicsampalwatabustatis(e.g.how of cthe CeCentindicmeathe sThe meathe a

in whnummeashorExaA neusinbalatimeafter

. 1-1. The n

terminate or cally probable, lts having vari, one obtains agure 1-1. If theesented exactuniverse or pople is that portscriptivee the typical pion will focus os in later chapone of the manc features of aple and the m

ays sufficient tolar descriptionstics is summ, the dose strebig the value entral tendenc

variation amontral Tentral tendency c

cation of the avsurements (thsymbol (mu)arithmetic mesurements anarithmetic mea

hich stands ber of values.surements N

rthand notationample 1-6 ew student hng a 1-mL pipance in a weies and take tr 10 repeats?

normal curve

chance errors and larger errious errors aloa bell-shaped e distribution otly by the curv

opulation. Whetion of the pop

e Statisticpharmacy studon introducingpters. The studny outstandinga data set colleeasures. Howo understand n to perform a

mary statisticsength of indiviis and the var

cy (e.g., whatong a group o

ndency: Mcan be describverage value

he universe or ). ean [X with barnd dividing thean for a small

for the sum o [X with bar abis increased. Rn describing th

as just joinedpettor and is ghing boat. The average. ? The densit

e for the dis

obey the lawsors being less

ong the verticacurve, knownof results folloe for an infinit

ereas the popupulation used cs dent has sufficg (or reintroducdent who requg texts that haected from an

wever, viewingthe behavior oquantitative a

s. These are sdual tablets inriability amongt is the averag

of values). Mean, Mebed using a suin the data setpopulation) is

r above] is obttotal by the ngroup of value

of, Xi is the ithbove] is an esRemember, thhe various rela

d the lab andasked to witTo determineWhat is the y of water is

stribution of

s of probabilitys probable thaal axis againstn as a normal fows the normate number of oulation is the win the analysis

cient exposurecing) some of

uires additionaave been publi experimental the individuaof the data. Tyanalysis of theingle numbersn a batch of 10g the values. Tge?), while the

edian, Moummary statist. The theoret

s known as the

tained by addumber N of thes is expresse

h individual mestimate of anhe equationsationships tha

d is being trathdraw 1 mL e her pipettinaverage volu1 g/mL.

f indetermin

y, both positivean smaller onet the magnitudfrequency dist

al probability laobservations, wwhole of the cas.

e to descriptivethe key conce

al background ished.6,7 Desc study. They gl data and tabypically, a grap

e data set. Thes that summar0,000 tablets),The first of thee second refer

ode stic (the meanical mean for e universe or p

ing together thhe measuremeed as

easurement ond approaches used in all oft define some

ained to pipetof water from

ng skill, she iume of water

nate errors.

e and negativees. If one plotsde of the errorstribution curveaw, the deviatiwhich constituategory under

e statistics in oepts that will bin statistics iscriptive statistgive summarieles of results aphic analysis ie third componrize the data. W, summary staese aspects rers to dispersio

, median, or ma large numbepopulation me

he results of thents. In mathe

f the group, ans it as the numf the calculatio

parameter.

tte liquids com a beaker as asked to rer that the stu

e errors beings a large numbs on the horiz

e, as shown ions will be ute r consideration

other coursesbe used numes advised to setics depict the es about the alone is not is paired with nent of descripWith interval datistics focus oelates to meason (in other w

mode) that giveer of ean and is give

he various ematical notati

nd N is the mber of ons are really a

orrectly. She and weigh it oepeat this 10

udent withdra

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n, the

, this erous eek

a ptive data on sures words,

es an

en

on,

a

is on a

0 aws


20

Attempt Weight (g)

1 1.05

2 0.98

3 0.95

4 1.00

5 1.02

6 1.00

7 1.10

8 1.03

9 0.96

10 0.98

If Xi = 9.99 and N = 10, so 9.99/10 = 0.999. Given the number of significant figures, the average would be reported as 1.00 g, which equals 1 mL since the density of water is 1 g/mL.

The median is the middle value of a range of values when they are arranged in rank order (e.g., from lowest to highest). So, the median value of the list [1, 2, 3, 4, 5] is the number 3. In this case, the mean is also 3. So, which value is a better indicator of the central tendency of the data? The answer in this case is neitherboth indicate central tendency equally well. However, the value of the median as a summary statistic P.9 becomes more obvious when the data set is skewed (in other words, when there are outliers or data points with values that are quite different from most of the others in the data set). For example, in the data set [1, 2, 2, 3, 10] the mean would be 3.6 but the median would be 2. In this case, the median is a better summary statistic than the mean because it gives a better representation of central tendency of the data set. Sometimes the median is referred to as a more robust statistic since it gives a reasonable outcome even with outlier results in the data set. Example 1-7 As you have seen, calculating the median of a data set with an odd number of results is straightforward. But, what do you do when a data set has an even number of members? For example, in the data set [1, 2, 2, 3, 4, 10] you have 6 members to the data set. To calculate


21

the mthem

Althopropas a tendThe pharresucommequaexammodindicVarIn orconvthe vthe mthe pSincfeatupharmeameasourwordbioloThe rougand The scattanalyaboudispethe avaluealgebdevia

in whon edisreYoudgreaFurthestim

median you m. So, the me

ough it is humper to do so un

summary statency of the remode is the v

rmaceutical sclts that tend tomonly occurrinal to 5. Howevmple, the dataes (one mode

cation of the biriability: rder to fully unvey a sense ofvariation in themean deviatiopharmaceuticae much of thisures will be disrmacy have difsurements aresurement erro

rce of variationds, biological vogical processrange is the dh idea of the dminimum valuaverage distater on the targyses is the meut the mean eqersion. The marithmetic meae Xi and the abraic signs, anation of a sam

hich |Xi-[X wither side of th

egarded. den6 discoura

ater precision thermore, the mmates, and acc

need to find edian would

an nature to wnder most circtistic allows yo

esults. value in the daciences but it ho center arounng values). Fo

ver, we someti set [1, 2, 4, 4

e is 5 and the oimodal behaviMeasure

nderstand the f the dispersioe data set cann, or thestandal sciences is s will be a reviscussed. The fferent charace often not peors or may be n is viewed sligvariations that.

difference betwdispersion. It sues are not in ance of all the get. The averaean deviation quals zero; he

mean deviationan of the samprithmetic meand dividing the

mple is express

ith bar above]he term in the

ages the use othan actually emean deviatiocordingly, d is

the two middbe 2.5.

want to throw umstance or aou to use all o

ta set that occhas particular nd more than oor example, inmes see a da

4, 5, 5, 5, 6, 9, other is 11). Tior. Neither wo

es of Dispproperties of t

on or scatter abe calculated

dard deviation.the coefficientew for many oresults obtain

cteristics. In thrfectly reprodudue to errors ghtly differentlwe typically o

ween the largesometimes lealine with the rehits from the bge spread abo of the popul

ence, the algebd for a sampl

ple, is obtainen [X with bar ae sum by the nsed as

| is the sum ofnumerator ind

of the mean deexists when a n of small sub not particular

dle members

w out an outlyiat least withouof the results in

curs most oftevalue in descone value (e.g the data set [

ata set that has 10, 11, 11, 11

Taking the arithould the mediapersionthe data set thround the cen

d. This variabil. Another useft of variation (of the studentsed in the physe physical sciucible. In othein observationly since membobserve are in

est and the smads to ambiguest of the databull's eye wouout the arithmlation. The subraic signs arele, that is, the

ed by taking thabove], addingnumber of valu

f the absolutedicate that the

eviation becausmall number

bsets may be wrly efficient as

s (in this case

ng piece of daut rigorous stan a data set an

en. It is not as ribing the mosg., a bimodal d[1, 2, 4, 4, 5, 5s two clusters1, 11, 13, 14] hmetic mean oan.

hat you are anntral value. Thlity is usually eful measure ofCV), which is s using this tesical, chemicaences, for exa

er words, varians. In the biolobers of a poputrinsic to the i

mallest value inous results, h

a. The range wuld serve as a

metic mean of am of the posite disregardeddeviation of ae difference bg the differencues to obtain t

deviations froe algebraic sig

use it gives a br of values arewidely scattera measure of

e, 2 and 3) th

ata from a datatistical analysind still get an

commonly usest common occdistribution tha5, 5, 6, 9, 10] ts of results rais bimodal andof the data set

alyzing, it is nis is done so texpressed as f dispersion coa dimensionlext, only the mo

al, and biologicample, instrumability may resogical scienceulation differ grndividual, orga

n a group of dowever, when

will not be conconvenient m

a large series ive and negatto obtain a m

an individual obbetween each ces without regthe average. T

om the mean. n of the devia

biased estimate used in the ced around thef precision.

hen average

a set, it is not is. Using medidea of the ce

ed in the ccurrences of at has two the mode valuather than oned thus has twot would not giv

necessary to that an estimathe range, ommonly usedess parameterost pertinent

cal aspects of ment sult from randoes, however, threatly. In otheanism, or

ata and givesn the maximumnsidered furthemeasure of the

of weighings tive deviations

measure of thebservation froindividual gard to the The mean

The vertical liation should be

ate that suggescomputation. e average of th

e

ian entral

ue is e. For o ve an

ate of

d in r.

om he

er

a m er.

or s

om

nes e

sts a

he


22

The squanumof itestandAs pthe pof a P.10 samp

For a

The standThe meapopupopuvariaof frevalueof a less of (NnegliModthat loweHowdeviaforwaA sastandExaA phdividremmethfirst the scoluas 0perc

standard deviares of the devber of measur

ems or measudard deviation

previously notepopulation; thesubset and su

0

ple standard d

a small sampl

term (N - 1) isdard deviationreason for intrsurement or oulation. This siulation. When ability is obtaineedom for esties provide thrsample for obthan N, or (N

N - 1) to estimaigible. ern statistical the estimate o

er than the popwever, for manyation in highlyard.

ample calculatidard deviation

ample 1-8 harmacist recded powdersoves the conhod and thencolumn of Tsign, are give

umn. Based o0.98 0.046 centage devi

iation (the Gviations. This rements, for erements appro

n. Population sed, any finite ge statistic or chupply an estim

deviation and

e, the equatio

s known as then s, which on troducing (N - observation, heingle observata second mea

ned. The statisimating variatiee degrees of

btaining an est- 1), as shown

ate the popula

methods handof the standardpulation standy students stu

y technical term

ion involving tn follows.

ceives a press, each of whntents from en weighs theable 1-2; theen in columnon the use ofg. The variaation by divid

Greek lowercasparameter is uxample, the woximates the pstandard deviagroup of experharacteristic o

mate of the sta

is designated

n is written

e number of dthe average is1) is as followe or she obtaition, however,asurement is tstician states tions in the unif freedom, andtimate of the sn in equation (ation standard

dle small samd deviation beard deviation

udying pharmams. So, we wil

he arithmetic

scription for ahich is to weigeach paper ae powders cae deviations on 2, and the sf the mean dbility of a sinding the mea

se letter sigmaused to measuweights of the population anations are shorimental data mof a sample frondard deviatio

by the letter s

degrees of frees lower than thws. When a sta

ns at least a r, can give no htaken, howevethis fact by saiverse. Three d so on. Therestandard devia(1-9). When N deviation bec

mples quite weecomes less reas fewer sam

acy there is noll simply refer

mean, the me

a patient witgh 1.00 g. To

after filling thearefully. The of each valuesquares of th

deviation, thengle powder can deviation

a) is the squarure the dispercontents of sed is, therefo

own in Figure may be considom the universon of the popu

s. The formula

edom. It replache universe staatistician selecrough estimatehint as to the der, a first basisying that two ovalues provide

efore, we do nation of the poN is large, saycause the diffe

ll; however, theproducible anples are used

o compelling reto standard d

ean deviation,

h rheumatoido check his se prescriptionresults of thee from the arhe deviationse weight of thcan also be eby the arithm

re root of the msion or variabeveral million cre, called the 1-1. dered as a subse used to expulation is know

a is

ces N to reducandard deviaticts a sample ae of the mean degree of varis for estimatinobservations se two degreesot have accespulation. InsteN > 100, we c

erence betwee

he investigatornd, on the aveto compute th

eason to vieweviation as S

and the estim

d arthritis calskill in filling tn by the bloce weighings arithmetic meas are shown he powders cexpressed inmetic mean a

mean of the bility of a large capsules. Thispopulation

bset or samplpress the variawn as the

ce the bias of ion.

and makes a sof the parent ability in the

ng the populatsupply one des of freedom, fss to all N valuead, we must can use N insten the two is

r should recogerage, becomehe estimate.

w standard SD from this p

mate of the

lling for sevethe powders

ck-and-divideare given in an, disregardin the last

can be expren terms of and multiplyin

s set

e of ability

the

single

ion egree four ues use 1 tead

gnize es

point

en s, he e the ding

essed

ng


23

by 1from

The dataStatiwithideviaGoldSaunpharor sexpeThe

and expe

00. The resum the papers

Table 1-2 S

Total

Average

standard devi, it is approximsticians have n one standarations, and 99

dstein7 selectenders and Flermaceutical wospread of the ected to fall ouestimate of th

2s is equal toect that roughl

ult is 0.98% and weighin

Statistical A

WeighPowd

Content

1.00

0.98

1.00

1.05

0.81

0.98

1.02

= 6.84

0.98

iation is used mately 25% laestimated tha

rd deviation on9.7% within 3ed 1.73 as anming8 advocaork, it should bdata in small s

utside this ranghe standard de

o 0.156 g. Thy 90% to 95%

4.6%; of cong the powde

Analysis of

t of der ts (g)

Ign

0.

0.

0.

0.

0.

0.

0.

4

0.

more frequentrger than the m

at owing to chan either side o

3 standard devn equitable tolated the use obe consideredsamples. Thege if only chaneviation in Exa

hat is, based u% of the sample

ourse, it incluers in the ana

f Divided P

Deviation nored), |Xi-

bar abov

.02

.00

.02

.07

.17

.00

.04

= 0.32

.046

tly than the mmean deviatioance errors, aof the arithmetviations, as selerance standa

of 3 as approd permissible tn, roughly 5%nce errors occample 1-8 is ca

upon the analye values woul

udes errors dalysis.

owder Com

(Sign -[X with ve]|

S(X

ean deviation on, that is, =bout 68% of atic mean, 95.5een in Figure 1ard for prescrioximate limitsto accept 2s

% to 10% of thecur. alculated as fo

ysis of this expd fall within 0

due to remov

mpounding

Square of thXi-[X with b

0.0004

0.0000

0.0004

0.0049

0.0289

0.0000

0.0016

= 0.0362

in research. F1.25.

all results in a 5% within 2 st1-1. ption productsof error for a

as a measuree individual res

ollows:

periment, the 0.156 g of the

ving the powd

g Technique

he Deviatiobar above]

2

For large sets

large set will fstandard

s, whereas single result.

e of the variabsults will be

pharmacist sh sample mean

ders

e

on, ])2

of

fall

In ility

hould n.


24

The In thof drpredP.11 specthe sWhea meerrorThe infinibecoquanThe of anIn Exinvol

in whthesedue systequesof 0.probThe

The percsevein the

The consg witaccuvaluealso due precA stucompdiscoobtadivergreaby ch

smaller the ste filling of cap

rug in each caicting the prob

cific deviation scope of this bereas the averethod, the differ that can oftetrue or absoluitely large setomes progressntity measureddifference bet

n operation; it xample 1-8, thlved in compo

hich the positive results showto accidental eemic error canstionable. This98, the differe

bability that sucmean error in

relative error entage by mu

eral sets of rese case just cit

reader shouldstant error is pth a mean devuracy, howevee by 40%. Conprecise. The to chance is laisely wrong. udy of the indipounding opeordant value, nin a mean of 1rgent result is

ater than the mhance only ab

tandard deviatpsules, precisiopsule and to rbability of occu

in future operabook. The inteage deviation erence between be used as

ute value is ordbecause it issively larger. Td in those casetween the samis known as th

he true value isounding this pr

ve sign signifiws, however, terrors. Hencen be presumeds possibility is ence could be ch a result cou this case is

is obtained byultiplying by 10sults by using ted is

d recognize thapresent. If the cviation of 0.5%er, would havenversely, the fsituation can aarge. Saunder

vidual values rations. Returnamely, 0.81 g1.01 g. The m0.20 g smalle

mean deviationbout once or tw

tion estimate (on is a measureproduce the urrence of a

ations, althougrested reader and the stand

en the arithmea measure of dinarily regards assumed thaThe universe mes in which de

mple arithmetiche mean errors 1.00 g, the arescription is

es that the truthat this differe, the accuracyd. However, oconsidered lastated with asuld occur by c

y dividing the m00 or in parts pthe relative er

at it is possiblecapsule conte

%, the results we been low becfact that the rearise in which rs and Fleming

of a set often rning to the dag. If the arithmean deviation

er than the newn. A deviation wice in 1000 m

(or the mean dure of the abilit

result in subs

gh important ir is referred to dard deviationetic mean andthe accuracy

ded as the uniat the true valmean does noeterminate errc mean and thr. amount reque

ue value is greence is not stay of the operat

on further analyater. If the arithssurance to hachance alone w

mean error byper thousand brror rather tha

e for a result tents in Exampwould have because the aveesult may be athe mean valg8 observed,

throws additioata of Examplemetic mean is without the dw average or, greater than f

measurements

deviation), thety of the pharmsequent opera

n pharmacy, rtreatises on s

n can be used the true or abof the methodverse mean ue is approac

ot, however, corors are inherehe true value g

sted by the ph

eater than the atistically signition in Exampysis it is foundhmetic mean iave statistical would be sma

y the true valueby multiplying n the absolute

to be precise wle 1-8 had yie

een accepted rage weight w

accurate doesue is close to it is better to

onal light on the 1-8 (Table 1recalculated igoubtful result in other word

four times the s; hence, the d

e more precisemacist to put tations. Statistic

require methodstatistical analyas measures

bsolute value ed. that is, the hed as the saoincide with thent in the measgives a measu

hysician. The a

mean value. Aificant but rathle 1-8 is suffic

d that one or snExample 1-8significance bll.

e. It can be exby 1000. It is

e mean error. T

without being lded an averaas precise. Th

would have diffnot necessarthe true valuebe roughly ac

he exactitude -2), we note ognoring this mis 0.02 g. It is s, its deviationmean deviatio

discrepancy in

e is the operatthe same amocal techniques

ds that are ouysis. of the precisioexpresses the

mean for an ample size he true value oasurements. ure of the accu

apparent erro

An analysis ofher is most likeciently great thseveral results8 were 0.90 insbecause the

xpressed as a easier to comThe relative e

accurate, thatage weight of 0he degree of fered from therily mean that e, but the scatccurate than

of the one rather measurement,

now seen than is 10 times on will occur pn this case is

ion. ount s for

utside

on of e

of the

uracy

r

f ely hat no s are stead

mpare error

t is, a 0.60

e true it is ter

we at the

purely


25

probusefuHavideteor onwideapprhelp pharThe mea

It is vmultibecaresusets VisScievariaP.12 put tthe drelatmathknowThe scaledata is knThe negaWe wrelatstraigexpr

bably caused bul though not ng uncoveredrminate error.n the weighingely from the meraisals as thesthe pharmaci

rmacy. CV is a dimenn and is defin

valid only wheiplied by 100)

ause the standlts. The CV shwith dissimila

sualizing ntists are not

ables under st2

hem in the fordata plotted inionship more hematical equwn as curve fitmagnitude of e, called the x are plotted on

nown as the xcintersection o

ative or positivwill first go thrionship betweght line when

ressed as

by some definalways reliabl

d the variable w The pharmac

g paper or posean, a serious

se in the collegst become a s

nsionless paraed as

en the mean is. For exampledard deviationhould be usedar units or veryResults:

usually so fortudy. Instead,

rm of a table o a manner so clearly and peation. The protting and is trethe independe

x axis. The depn the graph, acoordinate or tf the x axis an

ve. ough some of

een two variabplotted using

ite error in tece criterion for weight amongcist may find thssibly was losts deficiency inge laboratory wsafe and profic

ameter that is

s nonzero. It is, if SD = 2 andof data must instead of the

y different mea: Graphictunate as to bthe investigato

or graph to betas to form a s

erhaps will alloocedure of obteated in booksent variable ispendent variaband a smooth lthe abscissa; nd the y axis is

f the technical bles, in which t

rectangular co

chnique. Statisfinding discre

g the units, onehat some of tht during tritura the compounwill aid the stucient compoun

quite useful. T

s also commod mean = 3, thalways be un

e standard devans. c Methodbegin each proor must collec

tter observe thsmooth curve ow expressiontaining an emps on statistics as customarily mble is measureline is drawn tthe y value is s referred to a

aspects of linthe variables coordinates. Th

sticians rightly epant results.e can proceedhe powder wastion. If severa

nder's techniquudent in locatinnder before en

The CV relates

nly reported ahen the %CV derstood in thviation to asse

ds, Linesoblem with an ct raw data and

he relationshipoften permits

n of the connepirical equatioand graphic ameasured aloned along the vthrough the poknown as the

as the origin. T

nes and linear contain no exphe straight-line

question this

d to investigates left on the sil of the powdeue would be sung and correctntering the pra

s the standard

as a percentagis 67%. The Ce context of thess the differe

equation at had

ps. Constructinthe investigatction in the forn from a plot onalysis. ng the horizonvertical scale, ooints. The x vae y coordinate The x and yval

relationships.ponents other e or linear rela

rule, but it is a

e the cause ofides of the moer weights devuspected. Suc

cting errors anactice of

d deviation to

ge (%CV is CVCV is useful he mean of thence between

and relating th

ng a graph wittor to observe rm of a of the data is

ntal coordinateor the y axis.

alue of each poor the ordinatlues may be e

. The simplestthan 1, yields

ationship is

a

f the ortar viated ch d will

the

V

e data

he

th the

e The oint te. either

t s a


26

inanvchthslanfo

Whe

The If a isbelow

and The concseenthe t

Tab

C

The subs

n which y isnd a and b a

value of b, thhange in x, he x axis. Thlants upwarn angle of 4ollows:

en b = 0, the lin

constant a is s positive, thew the x axis. W

the line passeresults of the

centrations of n to produce awo-point form

ble 1-3 Refr

oncentratio

10.0

25.0

33.0

50.0

60.0

method involvstituting into th

s the dependare const

Martin's Physical Pharmacy 6th.ed 2011 Dr.murtadha Alshareifi

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