ASTM E 1970-01 热分析数据统计处理的标准操作方法.pdf
Designation:E 1970 01Standard Practice forStatistical Treatment of Thermoanalytical Data1This standard is issued under the fixed designation E 1970;the number immediately following the designation indicates the year oforiginal adoption or,in the case of revision,the year of last revision.A number in parentheses indicates the year of last reapproval.Asuperscript epsilon(e)indicates an editorial change since the last revision or reapproval.1.Scope1.1 This practice details the statistical data treatment used insome thermal analysis methods.1.2 The method describes the commonly encountered sta-tistical tools of the mean,standard derivation,relative standarddeviation,pooled standard deviation,pooled relative standarddeviation and the best fit to a straight line,all calculationsencountered in thermal analysis methods.1.3 Some thermal analysis methods derive the analyticalvalue from the slope or intercept of a best fit straight lineassigned to three or more sets of data pairs.Such methods mayrequire an estimation of the precision in the determined slopeor intercept.The determination of this precision is not acommon statistical tool.This practice details the process forobtaining such information about precision.1.4 Computer or electronic-based instruments,techniquesor data treatment equivalent to this practice may also be used.NOTE1Users of this practice are expressly advised that some suchinstruments or techniques may not be equivalent.It is the responsibility ofthe user of this standard to determine the necessary equivalency prior touse.1.5 SI units are the standard.1.6 There are no ISO methods equivalent to this practice.2.Referenced Documents2.1 ASTM Standards:E 177 Practice for Use of the Terms Precision and Bias inASTM Test Methods2E 456 Terminology Relating to Quality and Statistics23.Terminology3.1 DefinitionsThe technical terms used in this practiceare defined in Practice E 177 and Terminology E 456.3.2 Symbols:3m=slopeb=interceptn=number of data sets(that is,xi,yi)xi=an individual independent variable observationyi=an individual dependent variable observationS=mathematical operation which means“the sumof all”for the term(s)following the operatorX=mean values=standard deviationspooled=pooled standard deviationsb=standard deviation of the line interceptsm=standard deviation of the slope of a linesy=standard deviation of Y valuesRSD=relative standard deviationdyi=variance in y parameterr=correlation coefficient4.Summary of Practice4.1 The result of a series of replicate measurements of avalue are typically reported as the mean value plus someestimation of the precision in the mean value.The standarddeviation is the most commonly encountered tool for estimat-ing precision,but other tools,such as relative standard devia-tion or pooled standard deviation,also may be encountered inspecific thermoanalytical test methods.This practice describesthe mathematical process of achieving mean value,standarddeviation,relative standard deviation and pooled standarddeviation.4.2 In some thermal analysis experiments,a linear or astraight line,response is assumed and desired values areobtained from the slope or intercept of the straight line throughthe experimental data.In any practical experiment,however,there will be some uncertainty in the data so that results arescattered about such a straight line.The least squares method isan objective tool for determining the“best fit”straight linedrawn through a set of experimental results and for obtaininginformation concerning the precision of determined values.4.2.1 For the purposes of this practice,it is assumed that thephysical behavior,which the experimental results approximate,are linear with respect to the controlled value,and may berepresented by the algebraic function:y 5 mx 1 b(1)4.2.2 Experimental results are gathered in pairs,that is,for1This practice is under the jurisdiction of ASTM Committee E37 on ThermalMeasurements and is the direct responsibility of Subcommittee E37.01 on TestMethods and Recommended Practices.Current edition approved Aug.10,2001.Published November 2001.Originallypublished as E 1970 98.Last previous edition E 1970 00.2Annual Book of ASTM Standards,Vol 14.02.3Taylor,J.K.,Handbook for SRM Users,Publication 260-100,National Instituteof Standards and Technology,Gaithersburg,MD,1993.1Copyright ASTM International,100 Barr Harbor Drive,PO Box C700,West Conshohocken,PA 19428-2959,United States.every corresponding xi(controlled)value,there is a corre-sponding yi(response)value.4.2.3 The best fit approach assumes that all xivalues areexact and the yivalues(only)are subject to uncertainty.NOTE2In experimental practice,both x and y values are subject touncertainty.If the uncertainty in xiand yiare of the same relative order ofmagnitude,other more elaborate fitting methods should be considered.Formany sets of data,however,the results obtained by use of the assumptionof exact values for the xidata constitute such a close approximation tothose obtained by the more elaborate methods that the extra work andadditional complexity of the latter is hardly justified.44.2.4 The best fit approach seeks a straight line,whichminimizes the uncertainty in the yivalue.5.Significance and Use5.1 The standard deviation,or one of its derivatives,such asrelative standard deviation or pooled standard deviation,de-rived from this practice,provides an estimate of precision in ameasured value.Such results are ordinarily expressed as themean value 6 the standard deviation,that is,X 6 s.5.2 If the measured values are,in the statistical sense,“normally”distributed about their mean,then the meaning ofthe standard deviation is that there is a 67%chance,that is 2in 3,that a given value will lie within the range of 6 onestandard deviation of the mean value.Similarly,there is a 95%chance,that is 19 in 20,that a given value will lie within therange of 6 two standard deviations of the mean.The twostandard deviation range is sometimes used as a test foroutlying measurements.5.3 The calculation of precision in the slope and intercept ofa line,derived from experimental data,commonly is requiredin the determination of kinetic parameters,vapor pressure orenthalpy of vaporization.This practice describes how to obtainthese and other statistically derived values associated withmeasurements by thermal analysis.6.Calculation6.1 Commonly encountered statistical results in thermalanalysis are obtained in the following manner.NOTE3In the calculation of intermediate or final results,all availablefigures shall be retained with any rounding to take place only at theexpression of the final results according to specific instructions or to beconsistent with the precision and bias statement.6.1.1 The mean value(X)is given by:X 5x11 x21 x31.1 xin5Sxin(2)6.1.2 The standard deviation(s)is given by:s 5FSxi2 X!2n 2 1!G1/2(3)6.1.3 The Relative Standard Deviation(RSD)is given by:RSD 5s 100%!/X(4)6.1.4 The Pooled Standard Deviation(sp)is given by:sp5F$n12 1%s12!1$n22 1%s22!1.1 S$ni2 1%si2!n12 1!1n22 1!1.1ni2 1!G1/2(5)5FS$ni2 1%si!Sni2 1!G1/2(6)NOTE4For the calculation of pooled relative standard deviation,thevalues of siare replaced by RSDi.6.2 Best Fit to a Straight Line:6.2.1 The best fit slope(m)is given by:m 5nSxiyi!2Sxi!Syi!nSxi22Sxi!2(7)6.2.2 The best fit intercept(b)is given by:b 5Sxi2!Syi!2Sxi!Sxiyi!nSxi22Sxi!2(8)6.2.3 The individual dependent parameter variance(dyi)ofthe dependent variable(yi)is given by:dyi5 yi2mxi1 b!(9)6.2.4 The standard deviation syof the set of y values is givenby:sy5FSdyi!2n 2 2G1/2(10)6.2.5 The standard deviation(sm)of the slope is given by:sm5 syFnnSxi22Sxi!2G1/2(11)6.2.6 The standard deviation(sb)of the intercept(b)is givenby:sb5 syFSxi2nSxi2Sxi!2G1/2(12)6.2.7 The denominators in Eqs 6,7,9,and 10 are the same.It is convenient to obtain the denominator(D)as a separatefunction for use in manual calculation of each of theseequations.D 5 nSxi22Sxi!2(13)6.2.8 The linear correlation coefficient(r),a measure of themutual dependence between paired x and y values,is given by:r 5nSxy 2Sxi!Syi!$nSxi22Sxi!2#1/2nSyi2!2Syi!2#1/2%(14)NOTE5r may vary from 1 to+1,where values of+or 1 indicateperfect(100%)correlation and 0 indicates no(0%)correlation,that is,random scatter.A positive(+)value indicates a positive slope and anegative(-)indicates a negative slope.6.3 Example Calculations:6.3.1 Table 1 provides an example set of data and interme-diate calculations which may be used to examine the manualcalculation of slope(m)and its standard deviation(sm)and ofthe intercept(b)and its standard deviation(sb).6.3.1.1 The values in Columns A and B are experimentalparameters with xibeing the independent parameter and yithedependent parameter.6.3.1.2 From the individual values of xiand yiin ColumnsAand B in Table 1,the values for xi2and xiyiare calculated andplaced in Columns C and D.4Mandel,J.,The Statistical Analysis of Experimental Data,Dover Publications,New York,NY,1964.E 197026.3.1.3 The values in columns A,B,C,and D are summed(added)to obtain Sxi=76.0,Syi=86.7,Sxi2=1540.0,andSxiyi=1753.9,respectively.6.3.1.4 The denominator(D)is calculated using Eq 13 andthe values Sxi2=1540.0 and Sxi=76.0 from 6.3.1.3.D 56 1540.0!276.0 76.0!5 3464.0(15)6.3.1.5 The value for m is calculated using the values n=6Sxi yi=1753.9,Sxi=76.0,Syi=86.7,and D=3640.0,from6.3.1.3 and 6.3.1.4 and Eq 6:m 5nSxiyi!2 SxiSyiD(16)m 56 1753.9!276.0 86.7!3464.0510523.4 2 6589.23464.05 1.1357(17)6.3.1.6 The value for b is calculated using the values n=6,Sxi yi=1753.9,Sxi=76.0,and Syi=86.7,from 6.3.1.3 and6.3.1.4 and Eq 7:b 51540.0 86.7!276.0 1753.9!3464.05133518.0 2 133296.43464.05 0.064(18)6.3.1.7 Using the values for m=1.1357 and b=0.064 from6.3.1.5 and 6.3.1.6,and the value Sxi=76.0 from Table 1,then=6,values for dyiare calculated values using Eq 8 andrecorded in Column F in Table 1.6.3.1.8 From the values in Column F of Table 1,the sixvalues for(dyi)2are calculated and recorded in Column G.6.3.1.9 The values in Column G of Table 1 are summed toobtain S(dyi)2.6.3.1.10 The value of syis calculated using the value from6.3.1.9 and Eq 10:sy50.050 092 02/4#1/25 0.1119(19)6.3.1.11 The value for sm(expressed to two significantfigures)is calculated using the values of D=3464.0 and sy=0.1119 from 6.3.1.4 and 6.3.1.10,respectively.sm5 0.1119F63464.0G1/25 0.0047(20)6.3.1.12 The value for sb(expressed to two significantfigures)is calculated using the values ofSxi2,D=3464.0,andsy=0.119,from 6.3.1.3,6.3.1.4,and 6.3.1.10,respectively.sb5 0.1119F1540.03464.0G1/25 0.075(21)6.3.1.13 The value of the slope along with its estimation ofprecision is obtained from 6.3.1.5 and 6.3.1.11 and reported asfollows:m 6 sm(22)m 5 1.1357 6 0.0047(23)6.3.2 Table 1 provides an example set of data that may beused to examine the manual calculation of the correlationcoefficient(r).6.3.2.1 The value of r is calculated using the values n=6,Sxi=76.0,Syi=86.7,Sxi2=1540.0,Sxiyi=1753.9,andS(yi)2=1997.57 from Table 1 and Eq 13.r 5$6 1753.9!276.0 86.7!%$6 1540.0!276.0 76.0!#1/26 1997.57!286.7 86.7!#1/2%(24)5$10523.4 2 6589.2%$9240 2 5776#1/211985.42 2 7516.89#1/2%53934.2$3464#1/24468.53#1/2%53934.2$58.856 66.847%5 0.999967.Report7.1 Report the following information:7.1.1 All of the statistical values required to meet the needsof the respective applications method.7.1.2 The specific dated version of this practice that is used.8.Keywords8.1 intercept;mean;precision;relative standard deviation;slope;standard deviationTABLE 1 Example Set of Data and Intermediate CalculationsColumnABCDEFGHExperi-mentxiyixi2xiyim xi+bdyi(dyi)2(yi)211.01.21.01.21.19970.00030.000 000 091.4421.01.31.01.31.19970.10030.010 060 091.69312.013.7144.0164.013.69240.00760.000 057 76187.69412.013.5144.0162.013.6924-0.19240.037 017 76182.25525.028.5625.0712.528.45650.04350.001 892 25812.25625.028.5625.0712.528.45650.04350.001 892 25812.25_S76.086.71540.01753.90.050 920 201997.57NOTE1n=6.E 19703ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentionedin this standard.Users of this standard are expressly advised that determination of the validity of any such patent rights,and the riskof infringement of such rights,are entirely their own responsibility.This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years andif not revised,either reapproved or withdrawn.Your comments are invited either for revision of this standard or for additional standardsand should be addressed to ASTM International Headquarters.Your comments will receive careful consideration at a meeting of theresponsible technical committee,which you may attend.If you feel that your comments have not received a fair hearing you shouldmake your views known to the ASTM Committee on Standards,at the address shown below.This standard is copyrighted by ASTM International,100 Barr Harbor Drive,PO Box C700,West Conshohocken,PA 19428-2959,United States.Individual reprints(single or multiple copies)of this standard may be obtained by contacting ASTM at the aboveaddress or at 610-832-9585(phone),610-832-9555(fax),or serviceastm.org(e-mail);or through the ASTM website(www.astm.org).E 19704
