ASTM E1970-00

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Designation: E 1970 – 00
Standard Practice for
Statistical Treatment of Thermoanalytical Data
This standard is issued under the fixed designation E 1970; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope
m = slope
1.1 This practice details the statistical data treatment used in
b = intercept
some methods based upon thermal analysis.
n = number of data sets (that is, x,y )
i i
1.2 The method describes the commonly encountered sta-
x = an individual independent variable observation
i
tistical tools of the mean, standard derivation, relative standard
y = an individual dependent variable observation
i
deviation, pooled standard deviation, pooled relative standard
S = mathematical operation which means “the sum
deviation and the best fit to a straight line, all calculations
of all” for the term(s) following the operator
encountered in thermal analysis methods.
X = mean value
1.3 Some thermal analysis methods derive the analytical
s = standard deviation
value from the slope or intercept of a best fit straight line
s = pooled standard deviation
pooled
assigned to three or more sets of data pairs. Such methods may
s = standard deviation of the line intercept
b
require an estimation of the precision in the determined slope
s = standard deviation of the slope of a line
m
or intercept. The determination of this precision is not a
s = standard deviation of Y values
y
common statistical tool. This practice details the process for
RSD = relative standard deviation
obtaining such information about precision. dy = variance in y parameter
i
1.4 Computer or electronic-based instruments, techniques
4. Summary of Practice
or data treatment equivalent to this practice may also be used.
4.1 The result of a series of replicate measurements of a
NOTE 1—Users of this practice are expressly advised that some such
value is reported typically as the mean value plus some
instruments or techniques may not be equivalent. It is the responsibility of
estimation of the precision in the mean value. The standard
the user of this standard to determine the necessary equivalency prior to
deviation is the most commonly encountered tool for estimat-
use.
ing precision, but other tools, such as relative standard devia-
1.5 The values stated in SI units are to be regarded as the
tion or pooled standard deviation, also may be encountered in
standard.
specific thermoanalytical test methods. This practice describes
1.6 There are no ISO methods equivalent to this practice.
the mathematical process of achieving mean value, standard
deviation, relative standard deviation and pooled standard
2. Referenced Documents
deviation.
2.1 ASTM Standards:
4.2 In some thermal analysis experiments, a linear, that is, a
E 177 Practice for Use of the Terms Precision and Bias in
2 straight line, response is assumed and desired values are
ASTM Test Methods
2 obtained from the slope or intercept of the straight line through
E 456 Terminology Relating to Quality and Statistics
the experimental data. In any practical experiment, however,
3. Terminology there will be some uncertainty in the data so that results are
scattered about such a straight line. The least squares method is
3.1 Definitions—The technical terms used in this practice
an objective tool for determining the “best fit” straight line
are defined in Practice E 177 and Terminology E 456.
drawn through a set of experimental results and for obtaining
3.2 Symbols:
information concerning the precision of determined values.
4.2.1 For the purposes of this practice, it is assumed that the
physical behavior, which the experimental results approximate,
are linear with respect to the controlled value, and may be
This practice is under the jurisdiction of ASTM Committee E-37 on Thermal
represented by the algebraic function:
Measurements and is the direct responsibility of Subcommittee E37.01 on Thermal
y 5 mx 1 b (1)
Analysis Methods.
Current edition approved March 10, 2000. Published May 2000. Originally
4.2.2 Experimental results are gathered in pairs, that is, for
published as E 1970 – 98. Last previous edition E 1970 – 98.
every corresponding x (controlled) value, there is a corre-
Annual Book of ASTM Standards, Vol 14.02. i
Taylor, J.K., Handbook for SRM Users, Publication 260-100, National Institute sponding y (response) value.
i
of Standards and Technology, Gaithersburg, MD, 1993.
Copyright © ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959, United States.
E 1970
1/2
4.2.3 The best fit approach assumes that all x values are S ~$n 2 1 % •s !
i i i
F G
exact and the y values (only) are subject to uncertainty. S ~n 2 1!
i
i
NOTE 4—For the calculation of pooled relative standard deviation, the
NOTE 2—In experimental practice, both x and y values are subject to
values of s are replaced by RSD .
uncertainty. If the uncertainty in x and y are of the same relative order of
i i
i i
magnitude, other more elaborate fitting methods should be considered. For
6.2 Best Fit to a Straight Line:
many sets of data, however, the results obtained by use of the assumption
6.2.1 The best fit slope (m) is given by:
of exact values for the x data constitute such a close approximation to
i
those obtained by the more elaborate methods that the extra work and n S~x y ! 2 ~S x ! ~S yi!
i i i
m 5 (6)
2 2
additional complexity of the latter is hardly justified.
n S x 2 ~ S x !
i i
4.2.4 The best fit approach seeks a straight line, which
6.2.2 The best fit intercept (b) is given by:
minimizes the uncertainty in the y value.
i
~S x ! ~S y ! 2 ~S x ! ~S x y !
i i i i i
b 5 (7)
2 2
5. Significance and Use
n S x 2 ~ S x !
i i
5.1 The standard deviation, or one of its derivatives, such as
6.2.3 The individual dependent parameter variance (dy)of
i
relative standard deviation or pooled standard deviation, de-
the dependent variable ( y ) is given by:
i
rived from this practice, provides and estimate of precision in
dy 5 y 2 ~mx 1 b! (8)
i i i
a measured value. Such results are expressed ordinarily as the
6.2.4 The standard deviation s of the set of y values is given
y
mean value 6 the standard deviation, that is, X 6 s.
by:
5.2 If the measured values are, in the statistical sense,
2 1/2
“normally” distributed about their mean, then the meaning of
S ~dy !
i
s 5 (9)
F G
y
the standard deviation is that there is a 67 % chance, that is 2
n 2 2
in 3, t
...