1、Designation: E1970 11E1970 16Standard 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.
2、A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope Scope*1.1 This practice details the statistical data treatment used in some thermal analysis methods.1.2 The method describes the common
3、ly encountered statistical tools of the mean, standard derivation, relative standarddeviation, pooled standard deviation, pooled relative standard deviation and deviation, the best fit to a straight line, (linearregression of a) straight line, and propagation of uncertainties for all calculations en
4、countered in thermal analysis methods.methods(see Practice E2586).1.3 Some thermal analysis methods derive the analytical value from the slope or intercept of a best fitlinear regression straightline assigned to three or more sets of data pairs. Such methods may require an estimation of the precisio
5、n in the determined slopeor intercept. The determination of this precision is not a common statistical tool. This practice details the process for obtaining suchinformation about precision.1.4 There are no ISO methods equivalent to this practice.2. Referenced Documents2.1 ASTM Standards:2E177 Practi
6、ce for Use of the Terms Precision and Bias in ASTM Test MethodsE456 Terminology Relating to Quality and StatisticsE691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test MethodE2161 Terminology Relating to Performance Validation in Thermal Analysis and RheologyE258
7、6 Practice for Calculating and Using Basic StatisticsF1469 Guide for Conducting a Repeatability and Reproducibility Study on Test Equipment for Nondestructive Testing3. Terminology3.1 DefinitionsThe technical terms used in this practice are defined in Practice E177 and Terminologies E456 and E2161in
8、cluding precision, relative standard deviation, repeatability, reproducibility, slope, standard deviation, thermoanalytical, andvariance.3.2 Symbols: Symbols (1): 3m = slopeb = interceptn = number of data sets (that is, xi, yi)xi = an individual independent variable observationyi = an individual dep
9、endent variable observation = mathematical operation which means “the sum of all” for the term(s) following the operatorX = mean value1 This practice is under the jurisdiction of ASTM Committee E37 on Thermal Measurements and is the direct responsibility of Subcommittee E37.10 on Fundamental,Statist
10、ical and Mechanical Properties.Current edition approved Aug. 1, 2011April 1, 2016. Published August 2011April 2016. Originally approved in 1998. Last previous edition approved in 20062011 asE1970 06.E1970 11. DOI: 10.1520/E1970-11.10.1520/E1970-16.2 For referencedASTM standards, visit theASTM websit
11、e, www.astm.org, or contactASTM Customer Service at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.3 Taylor, J.K., Handbook for SRM Users, Publication 260-100, National Institute of Standards and Technology, Gait
12、hersburg, MD, 1993.4 Measurement System Analysis, third edition, Automotive Industry Action Group, Southfield, MI, 2003, pp. 55, 177184.3 Mandel, J., The Statistical Analysis of Experimental Data, Dover Publications, New York, NY, 1964. The boldface numbers in parentheses refer to a list of referenc
13、esat the end of this standard.This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommend
14、s that users consult prior editions as appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered the official document.*A Summary of Changes section appears at the end of this standardCopyright ASTM International, 100 Barr Harbor Drive, PO Box C700, W
15、est Conshohocken, PA 19428-2959. United States1s = 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 deviationyi = variance in y parameterr = corre
16、lation coefficientR = gage reproducibility and repeatability (see Guide F1469) an estimation of the combined variation of repeatability andreproducibility (2)sr = within laboratory repeatability standard deviation (see Practice E691)sR = between laboratory repeatability standard deviation (see Pract
17、ice E691)si = standard deviation of the “ith” measurement4. Summary of Practice4.1 The result of a series of replicate measurements of a value are typically reported as the mean value plus some estimationof the precision in the mean value. The standard deviation is the most commonly encountered tool
18、 for estimating precision, butother tools, such as relative standard deviation or pooled standard deviation, also may be encountered in specific thermoanalyticaltest methods. This practice describes the mathematical process of achieving mean value, standard deviation, relative standarddeviation and
19、pooled standard deviation.4.2 In some thermal analysis experiments, a linear or a straight line, response is assumed and desired values are obtained fromthe slope or intercept of the straight line through the experimental data. In any practical experiment, however, there will be someuncertainty in t
20、he data so that results are scattered about such a straight line. The least squares linear regression (also known as“least squares”) method is an objective tool for determining the “best fit” straight line drawn through a set of experimental resultsand for obtaining information concerning the precis
21、ion of determined values.4.2.1 For the purposes of this practice, it is assumed that the physical behavior, which the experimental results approximate, arelinear with respect to the controlled value, and may be represented by the algebraic function:y 5mx1b (1)4.2.2 Experimental results are gathered
22、in pairs, that is, for every corresponding xi (controlled) value, there is a correspondingyi (response) value.4.2.3 The best fit (linear regression) approach assumes that all xi values are exact and the yi values (only) are subject touncertainty.NOTE 1In experimental practice, both x and y values ar
23、e subject to uncertainty. If the uncertainty in xi and yi are of the same relative order ofmagnitude, other more elaborate fitting methods should be considered. For many sets of data, however, the results obtained by use of the assumption ofexact values for the xi data constitute such a close approx
24、imation to those obtained by the more elaborate methods that the extra work and additionalcomplexity of the latter is hardly justified.justified (,2 and 3).4.2.4 The best fit approach seeks a straight line, which minimizes the uncertainty in the yi value.4.3 The law of propagation of uncertainties i
25、s a tool for estimating the precision in a determined value from the sum of thevariance of the respective measurements from which that value is derived weighted by the square of their respective sensitivitycoefficients.4.3.1 Variance is the square of the standard deviation(s). Conversely the standar
26、d deviation is the positive square root of thevariance.4.3.2 The sensitivity coefficient is the partial derivative of the function with respect to the individual variable.5. Significance and Use5.1 The standard deviation, or one of its derivatives, such as relative standard deviation or pooled stand
27、ard deviation, derivedfrom this practice, provides an estimate of precision in a measured value. Such results are ordinarily expressed as the mean value6 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
28、 meaning of the standarddeviation is that there is a 67 % chance, that is 2 in 3, that a given value will lie within the range of 6 one standard deviation ofthe mean value. Similarly, there is a 95 % chance, that is 19 in 20, that a given value will lie within the range of 6 two standarddeviations o
29、f the mean. The two standard deviation range is sometimes used as a test for outlying measurements.5.3 The calculation of precision in the slope and intercept of a line, derived from experimental data, commonly is required inthe determination of kinetic parameters, vapor pressure or enthalpy of vapo
30、rization. This practice describes how to obtain theseand other statistically derived values associated with measurements by thermal analysis.E1970 1626. Calculation6.1 Commonly encountered statistical results in thermal analysis are obtained in the following manner.NOTE 2In the calculation of interm
31、ediate or final results, all available figures shall be retained with any rounding to take place only at the expressionof the final results according to specific instructions or to be consistent with the precision and bias statement.6.1.1 The mean value (X) is given by:X 5x11x21x311xin 5xin (2)6.1.2
32、 The standard deviation (s) is given by:s 5Fxi 2X!2n 21! G1/2(3)6.1.3 The Relative Standard Deviationrelative standard deviation (RSD) is given by:RSD5s100%!/X (4)6.1.4 The Pooled Standard Deviationpooled standard deviation (sp) is given by:sp 5F$n121%s12!1 $n221%s22!11$ni 21%si2!n121!1n221!11ni 21!
33、 G1/2(5)5F $ni 21%si! ni 21!G1/2 (5)5F$ni 21%si!ni 21!G1/2 (6)NOTE 3For the calculation of pooled relative standard deviation, the values of si are replaced by RSDi.6.1.5 The Gage Repeatabilitygage repeatability and Reproducibilityreproducibility (R) is given by:R 5sr 21sr2#1/2 (6)R 5sR 21sr2#1/2 (6
34、)NOTE 4For the calculation of relative Gage Repeatabilitygage repeatability and reproducibility, the values of sr and sR are replaced with RSDr andRSDR.6.2 Best Fit to aLinear Regression (Best) Fit Straight Line:6.2.1 The best fit slope (m) is given by:m 5nxiyi! 2xi! yi!nxi22xi!2 (7)6.2.2 The best f
35、it intercept (b) is given by:b 5xi2! yi! 2xi! xiyi!nxi22xi!2 (8)6.2.3 The individual dependent parameter variance (yi) of the dependent variable (yi) is given by:yi 5yi 2mxi1b! (9)6.2.4 The standard deviation sy of the set of y values is given by:sy 5F yi!2n 22 G1/2(10)6.2.5 The standard deviation (
36、sm) of the slope is given by:sm 5syF nnxi22xi!2 G1/2(11)6.2.6 The standard deviation (sb) of the intercept (b) is given by:sb 5sy F xi2nxi22xi!2 G1/2(12)6.2.7 The denominators in Eq 87, Eq 98, Eq 1211, and Eq 1312 are the same. It is convenient to obtain the denominator (D )as a separate function fo
37、r use in manual calculation of each of these equations.D 5nxi2 2xi!2 (13)6.2.8 The linear correlation coefficient (r), a measure of the mutual dependence between paired x and y values, is given by:E1970 163r 5 nxy2xi! yi!$nxi22xi!2#1/2 nyi2!2yi!2#1/2% (14)r 5 nxy2xi! yi!nxi22xi!2#1/2 nyi2!2yi!2#1/2
38、(14)NOTE 5r may vary from 1 to +1, where values of +1 or 1 indicate perfect (100 %) correlation and 0 indicates no (0 %) correlation, that is, randomscatter. A positive (+) value indicates a positive slope and a negative () indicates a negative slope.6.3 Propagation of Uncertainties:6.3.1 The law of
39、 propagation of uncertainties, neglecting the cross terms, is given by:sz25z i!si#2 (15)orsz 5$z i! si#2%2 (16)6.3.2 For example, given the function z = a d /c, then the sensitivity coefficient for a is z/a = d/c, for d is z/d = a/c, andfor c is z/c = ad/c2.6.3.3 Eq 16 becomes:sz 5$z a!sa#2 1z d!sd#
40、2 1z c!sc#2%12 (17)orsz 5$d sa c!2 1a sd b!2 1a d sc c2!2%126.3.4 Dividing both sides of the equation by z = a d/c, yields:szz 5$sa a!2 1sd d!2 1sc c!2%12 (18)6.3.5 The form of Eq 17 has been determined for a number of functions and is presented in Table 1.6.4 Example Calculations:6.4.1 Table 12 pro
41、vides an example set of data and intermediate calculations which may be used to examine the manualcalculation of slope (m) and its standard deviation (sm) and of the intercept (b) and its standard deviation (sb).6.4.1.1 The values in Columns A and B are experimental parameters with xi being the inde
42、pendent parameter and yi thedependent parameter.6.4.1.2 From the individual values of xi and yi in Columns A and B in Table 12, the values for xi2 and xiyi are calculated andplaced in Columns C and D.6.4.1.3 The values in columns A, B, C, and D are summed (added) to obtain xi = 76.0, yi = 86.7, xi2
43、= 1540.0, and xiyi= 1753.9, respectively.6.4.1.4 The denominator (D) is calculated using Eq 1413 and the values xi2 = 1540.0 and xi = 76.0 from 6.3.1.36.4.1.3.D 561540.0!276.076.0!53464.0 (26)6.4.1.5 The value for m is calculated using the values n = 6 xi yi = 1753.9, xi = 76.0, yi = 86.7, and D = 3
44、640.0, from6.3.1.36.4.1.3 and 6.3.1.46.4.1.4 and Eq 87:m 5nxi yi! 2xi yiD (27)TABLE 1 Uncertainties (4)Description Example UncertaintyAddition or Subtraction z = a + d c sz5fssad21ssdd21sscd2g12 (19)Multiplication or Division z = a d /c sz5fssa ad21ssd dd21ssc cd2g12 (20)Exponential z = ax szz5xssa
45、ad (21)Logarithmic z = log10a sz50.434saa (22)z = ln a sz5saa (23)Antilogarithm z = 10a szz52.303sa (24)z = ea szz5sa (25)E1970 164m 561753.9!276.086.7!3464.0 510523.426589.23464.0 (28)51.13576.4.1.6 The value for b is calculated using the values n = 6, xi yi = 1753.9, xi = 76.0, and yi = 86.7, from
46、 6.3.1.36.4.1.3and 6.3.1.46.4.1.4 and Eq 98:b 51540.086.7!276.01753.9!3464.0 5133518.02133296.43464.0 (29)50.0646.4.1.7 Using the values for m = 1.1357 and b = 0.064 from 6.3.1.56.4.1.5 and 6.3.1.66.4.1.6, and the value xi = 76.0 fromTable 12, the n = 6, values for yi are calculated values using Eq
47、109 and recorded in Column F in Table 12.6.4.1.8 From the values in Column F of Table 12, the six values for (yi)2 are calculated and recorded in Column G.6.4.1.9 The values in Column G of Table 12 are summed to obtain (yi) 2.6.4.1.10 The value of sy is calculated using the value from 6.3.1.96.4.1.9
48、 and Eq 1110:sy 50.050 092 02/4#1/250.1119 (30)6.4.1.11 The value for sm (expressed to two significant figures) is calculated using the values of D = 3464.0 and sy = 0.1119 from6.3.1.46.4.1.4 and 6.3.1.106.4.1.10, respectively.sm 50.1119 F 63464.0G1/250.0047 (31)6.4.1.12 The value for sb (expressed
49、to two significant figures) is calculated using the values ofxi2,D = 3464.0, and sy = 0.119,from 6.3.1.36.4.1.3, 6.3.1.46.4.1.4, and 6.3.1.106.4.1.10, respectively.sb 50.1119 F1540.03464.0G1/250.075 (32)6.4.1.13 The value of the slope along with its estimation of precision is obtained from 6.3.1.56.4.1.5 and 6.3.1.116.4.1.11 andreported as follows:m6sm (33)m 51.135760
