1、Designation: E 1970 06Standard 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 numb
2、er 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 encountere
3、d 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 fr
4、om 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 forobt
5、aining such information about precision.1.4 SI units are the standard.1.5 There are no ISO methods equivalent to this practice.2. Referenced Documents2.1 ASTM Standards:2E 177 Practice for Use of the Terms Precision and Bias inASTM Test MethodsE 456 Terminology Relating to Quality and StatisticsE 69
6、1 Practice for Conducting an Interlaboratory Study toDetermine the Precision of a Test MethodF 1469 Guide for Conducting a Repeatability and Repro-ducibility Study on Test Equipment for NondestructiveTesting3. Terminology3.1 DefinitionsThe technical terms used in this practiceare defined in Practice
7、 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 = m
8、ean 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 coefficientR = gage reproducibility
9、 and repeatability (seeF 1469) an estimation of the combined variationof repeatability and reproducibility4sr= within laboratory repeatability standard devia-tion (see E 691)sR= between laboratory repeatability standard devia-tion (see E 691)4. Summary of Practice4.1 The result of a series of replic
10、ate 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
11、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
12、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 are1This practice is under the jurisdiction of ASTM Committee E37 on ThermalMeasurements and i
13、s the direct responsibility of Subcommittee E37.01 on TestMethods and Recommended Practices.Current edition approved March 1, 2006. Published April 2006. Originallyapproved in 1998. Last previous edition approved in 2001 as E 1970 01.2For referenced ASTM standards, visit the ASTM website, www.astm.o
14、rg, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Taylor, J.K., Handbook for SRM Users, Publication 260-100, National Instituteof Standards and Technology, Gaithersburg, MD, 19
15、93.4Measurement System Analysis, third edition, Automotive Industry ActionGroup, Southfield, MI, 2003, pp. 55, 177184.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.scattered about such a straight line. The least squares method isan
16、 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 approxim
17、ate,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, forevery corresponding xi(controlled) value, there is a corre-sponding yi(response) value.4.2.3 The best fit approach assumes
18、that all xivalues areexact and the yivalues (only) are subject to uncertainty.NOTE 1In 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 se
19、ts 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.5,44.2.4 The best fit approach seeks a
20、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
21、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 wi
22、thin 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 calculat
23、ion 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 withmeasuremen
24、ts by thermal analysis.6. Calculation6.1 Commonly encountered statistical results in thermalanalysis are obtained in the following manner.NOTE 2In the calculation of intermediate or final results, all availablefigures shall be retained with any rounding to take place only at theexpression of the fin
25、al 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 5 FSxi2 X!2n 2 1!G1/2(3)6.1.3 The Relative Standard Deviation (RSD) is given by:RSD 5
26、 s 100 %! / X (4)6.1.4 The Pooled Standard Deviation (sp) is given by:sp5 F$n12 1% s12! 1 $n22 1% s22! 1 1S$ni2 1% si2!n12 1! 1 n22 1! 1 1 ni2 1!G1/2(5)5FS$ni2 1% si!Sni2 1!G1/2(6)NOTE 3For the calculation of pooled relative standard deviation, thevalues of siare replaced by RSDi.6.1.5 The Gage Repe
27、atability and Reproducibility (R)isgiven by:R 5 sr21 sr2#1/2(7)NOTE 4For the calculation of relative Gage Repeatability and repro-ducibility, the values of srand sRare replaced with RSDrand RSDR.6.2 Best Fit to a Straight Line:6.2.1 The best fit slope (m) is given by:m 5nSxiyi! 2 Sxi! Syi!nSxi22 Sxi
28、!2(8)6.2.2 The best fit intercept (b) is given by:b 5Sxi2! Syi! 2 Sxi! Sxiyi!nSxi22 Sxi!2(9)6.2.3 The individual dependent parameter variance (dyi)ofthe dependent variable (yi) is given by:dyi5 yi2 mxi1 b! (10)6.2.4 The standard deviation syof the set of y values is givenby:sy5 FS dyi!2n 2 2G1/2(11)
29、6.2.5 The standard deviation (sm) of the slope is given by:sm5 syFnnSxi22 Sxi!2 G1/2(12)6.2.6 The standard deviation (sb) of the intercept (b) is givenby:sb5 sy FSxi2nSxi2 Sxi!2 G1/2(13)6.2.7 The denominators in Eq 8, Eq 9, Eq 12, and Eq 13 arethe same. It is convenient to obtain the denominator (D)
30、asaseparate function for use in manual calculation of each of theseequations.D 5 nSxi22 Sxi!2(14)6.2.8 The linear correlation coefficient (r), a measure of themutual dependence between paired x and y values, is given by:r 5nSxy 2 Sxi! Syi!$nSxi22 Sxi!2#1/2nSyi2! 2 Syi!2#1/2%(15)5Mandel, J., The Stat
31、istical Analysis of Experimental Data, Dover Publications,New York, NY, 1964.E1970062NOTE 5r 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
32、(-) 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 val
33、ues 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 ColumnsAandBinTable 1, the values for xi2and xiyiare calculated andplaced in Columns C and D.6.3.1.3 The values in columns A, B, C, a
34、nd 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 14 andthe values Sxi2= 1540.0 and Sxi= 76.0 from 6.3.1.3.D 5 6 1540.0! 2 76.0 76.0! 5 3464.0 (16)6.3.1.5 The value for m is calculated using the valu
35、es n =6Sxiyi= 1753.9, Sxi= 76.0, Syi= 86.7, and D = 3640.0, from6.3.1.3 and 6.3.1.4 and Eq 8:m 5nSxiyi! 2SxiSyiD(17)m 56 1753.9! 2 76.0 86.7!3464.0510523.4 2 6589.23464.05 1.1357 (18)6.3.1.6 The value for b is calculated using the values n =6,Sxiyi= 1753.9, Sxi= 76.0, and Syi= 86.7, from 6.3.1.3 and
36、6.3.1.4 and Eq 9:b 51540.0 86.7! 2 76.0 1753.9!3464.05133518.0 2 133296.43464.05 0.064 (19)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 10 andrecorded in Column F in Table 1
37、.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 11:sy5 0.050 092 02 / 41/25 0.1119 (20)6.3.
38、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 (21)6.3.1.12 The value for sb(expressed to two significantfigures) is calculated using the values ofSxi2, D = 3
39、464.0, 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 (22)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(23)m 5 1.1357 6 0.0047 (24)6.3.2 Table 1 provides an
40、 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 valuesn=6,Sxi= 76.0, Syi= 86.7, Sxi2= 1540.0, Sxiyi= 1753.9, andS(yi)2= 1997.57 from Table 1 and Eq 15.r 5$6 1753.9! 2 76.0 86.7!%$6 1540.0! 2 76.0
41、76.0!#1/26 1997.57! 2 86.7 86.7!#1/2%(25)5$10523.4 2 6589.2%$9240 2 57761/2 11985.42 2 7516.891/2%53934.2$34641/2 4468.531/2%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 metho
42、d.7.1.2 The specific dated version of this practice that is used.TABLE 1 Example Set of Data and Intermediate Calculations (n =6)Column A B C D E F G HExperi-mentxiyixi2xiyimxi+b dyi(dyi)2(yi)21 1.0 1.2 1.0 1.2 1.1997 0.0003 0.000 000 09 1.442 1.0 1.3 1.0 1.3 1.1997 0.1003 0.010 060 09 1.693 12.0 13
43、.7 144.0 164.0 13.6924 0.0076 0.000 057 76 187.694 12.0 13.5 144.0 162.0 13.6924 -0.1924 0.037 017 76 182.255 25.0 28.5 625.0 712.5 28.4565 0.0435 0.001 892 25 812.256 25.0 28.5 625.0 712.5 28.4565 0.0435 0.001 892 25 812.25_ _ _ _ _ _S 76.0 86.7 1540.0 1753.9 0.050 920 20 1997.57E19700638. Keywords
44、8.1 intercept; mean; precision; relative standard deviation;slope; standard deviationASTM 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 th
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47、 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).E1970064
