1、Designation: E1970 11Standard Practice forStatistical Treatment of Thermoanalytical Data1This standard is issued under the fixed designation E1970; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number
2、 in parentheses indicates the year of last reapproval. Asuperscript epsilon () 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 s
3、ta-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
4、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 forobtain
5、ing such information about precision.1.4 There are no ISO methods equivalent to this practice.2. Referenced Documents2.1 ASTM Standards:2E177 Practice for Use of the Terms Precision and Bias inASTM Test MethodsE456 Terminology Relating to Quality and StatisticsE691 Practice for Conducting an Interla
6、boratory Study toDetermine the Precision of a Test MethodE2161 Terminology Relating to Performance Validation inThermal AnalysisF1469 Guide for Conducting a Repeatability and Reproduc-ibility Study on Test Equipment for Nondestructive Testing3. Terminology3.1 DefinitionsThe technical terms used in t
7、his practiceare defined in Practice E177 and Terminologies E456 andE2161 including precision, relative standard deviation, repeat-ability, reproducibility, slope, standard deviation, thermoana-lytical, and variance.3.2 Symbols:3m = slopeb = interceptn = number of data sets (that is, xi,yi)xi= an ind
8、ividual 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= s
9、tandard deviation of the slope of a linesy= standard deviation of Y valuesRSD = relative standard deviationdyi= variance in y parameterr = correlation coefficientR = gage reproducibility and repeatability (see GuideF1469) an estimation of the combined variationof repeatability and reproducibility4sr
10、= within laboratory repeatability standard devia-tion (see Practice E691)sR= between laboratory repeatability standard devia-tion (see Practice E691)4. Summary of Practice4.1 The result of a series of replicate measurements of avalue are typically reported as the mean value plus someestimation of th
11、e 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 mathe
12、matical 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 through1This
13、practice is under the jurisdiction of ASTM Committee E37 on ThermalMeasurements and is the direct responsibility of Subcommittee E37.10 on Funda-mental, Statistical and Mechanical Properties.Current edition approved Aug. 1, 2011. Published August 2011. Originallyapproved in 1998. Last previous editi
14、on approved in 2006 as E1970 06. DOI:10.1520/E1970-11.2For referenced ASTM standards, visit the ASTM website, www.astm.org, 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
15、, J.K., Handbook for SRM Users, Publication 260-100, NationalInstitute of Standards and Technology, Gaithersburg, MD, 1993.4Measurement System Analysis, third edition, Automotive Industry ActionGroup, Southfield, MI, 2003, pp. 55, 177184.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C
16、700, West Conshohocken, PA 19428-2959, United States.the 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 li
17、nedrawn 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 m
18、ay 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 that all xivalues areexact and the yivalues (only) are sub
19、ject 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 sets of data, however, the results obtained by use of the as
20、sumptionof 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 straight line, whichminimizes the uncertainty in the yival
21、ue.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 d
22、eviation, 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 valu
23、e. 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, deri
24、ved 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 encounte
25、red 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 final results according to specific instructions or to becons
26、istent 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 s 100 %! / X (4)6.1.4 The Pooled Standard Deviation (sp)
27、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$ni2 1% si!Sni2 1!G1/2(6)NOTE 3For the calculation of pooled relative standard deviation, thevalues of siare replaced by RSDi.6.1.5 The Gage Repeatability and Reproducibility (R)isgiven by:R 5 sr21 sr2#1
28、/2(7)NOTE 4For the calculation of relative Gage Repeatability and repro-ducibility, the values of srand sRare replaced with RSDrand RSDR.6.2 Best Fit to a Straight Line:6.2.1 The best fit slope (m) is given by:m 5nSxiyi! 2 Sxi! Syi!nSxi22 Sxi!2(8)6.2.2 The best fit intercept (b) is given by:b 5Sxi2!
29、 Syi! 2 Sxi! Sxiyi!nSxi22 Sxi!2(9)6.2.3 The individual dependent parameter variance (dyi)ofthe dependent variable (yi) is given by:dyi5 yi2 mxi1 b! (10)6.2.4 The standard deviation syof the set of y values is givenby:sy5 FS dyi!2n 2 2G1/2(11)6.2.5 The standard deviation (sm) of the slope is given by
30、:sm5 syFnnSxi22 Sxi!2 G1/2(12)6.2.6 The standard deviation (sb) of the intercept (b) is givenby:sb5 sy FSxi2nSxi2 Sxi!2 G1/2(13)6.2.7 The denominators in Eq 8, Eq 9, Eq 12, and Eq 13 arethe same. It is convenient to obtain the denominator (D)asaseparate function for use in manual calculation of each
31、 of theseequations.D 5 nSxi22 Sxi!2(14)6.2.8 The linear correlation coefficient (r), a measure of themutual dependence between paired x and y values, is given by:5Mandel, J., The Statistical Analysis of Experimental Data, Dover Publica-tions, New York, NY, 1964.E1970 112r 5nSxy 2 Sxi! Syi!$nSxi22 Sx
32、i!2#1/2nSyi2! 2 Syi!2#1/2%(15)NOTE 5r may vary from 1 to +1, where values of +1 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:
33、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 x
34、ibeing 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, and D are summed(added) to obtain Sxi= 76.0, Syi= 86.7, S
35、xi2= 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 values n =6Sxiyi= 1753.9, Sxi= 76.0, Syi= 86.7, and D = 3640
36、.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 and6.3.1.4 and Eq 9:b 51540.0 86.7! 2 76.0 1753.9!3464.0513
37、3518.0 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 1.6.3.1.8 From the values in Column F of Table 1, the six
38、values 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)6.3.1.11 The value for sm(expressed to two significantfigure
39、s) 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 = 3464.0, andsy= 0.119, from 6.3.1.3, 6.3.1.4, and 6.3.1.10
40、, 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 example set of data that may beused to examine the manu
41、al 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 76.0!#1/26 1997.57! 2 86.7 86.7!#1/2%(25)5$10523.4 2 658
42、9.2%$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: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
43、.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 13.7 144.0 164.0 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 8
44、12.25_ _ _ _ _ _S 76.0 86.7 1540.0 1753.9 0.050 920 20 1997.57E1970 1137.1.1 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 devia
45、tion;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 the validity of any such patent rights, and the riskof in
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48、iews 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). Permission rights to photocopy the standard may also be secured from the ASTM website (www.astm.org/COPYRIGHT/).E1970 114
