ASTM E1970-23

Standard Practice for Statistical Treatment of Thermoanalytical Data

SIGNIFICANCE AND USE
5.1 The standard deviation, or one of its derivatives, such as relative standard deviation or pooled standard deviation, derived from this practice, provides an estimate of precision in a measured value. Such results are ordinarily expressed as the mean value ± the standard deviation, that is, X ± s.  
5.2 If the measured values are, in the statistical sense, “normally” distributed about their mean, then the meaning of the standard deviation is that there is a 67 % chance, that is 2 in 3, that a given value will lie within the range of ± one standard deviation of the mean value. Similarly, there is a 95 % chance, that is 19 in 20, that a given value will lie within the range of ± two standard deviations of 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 in the determination of kinetic parameters, vapor pressure or enthalpy of vaporization. This practice describes how to obtain these and other statistically derived values associated with measurements by thermal analysis.
SCOPE
1.1 This practice details the statistical data treatment used in some thermal analysis methods.  
1.2 The method describes the commonly encountered statistical tools of the mean, standard derivation, relative standard deviation, pooled standard deviation, pooled relative standard deviation, the best fit to a (linear regression of a) straight line (or plane), and propagation of uncertainties for all calculations encountered in thermal analysis methods (see Practice E2586).  
1.3 Some thermal analysis methods derive the analytical value from the slope or intercept of a linear regression straight line (or plane) assigned to three or more sets of data pairs. Such methods may require an estimation of the precision in the determined slope or intercept. The determination of this precision is not a common statistical tool. This practice details the process for obtaining such information about precision.  
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.5 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 31-Dec-2022
• ICS: 03.120.30 - Application of statistical methods
• ICS: 17.200.10 - Heat. Calorimetry
• Technical Committee: E37 - Thermal Measurements
• Drafting Committee: E37.10 - Fundamental, Statistical and Mechanical Properties

Relations

• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E2586-19e1 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Apr-2019
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E2161-15 - Standard Terminology Relating to Performance Validation in Thermal Analysis and Rheology
• Effective Date: 01-Sep-2015
• Refers: ASTM E2586-14 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Jun-2014
• Refers: ASTM E177-14 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2014
• Refers: ASTM E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13a - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E2586-13 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Oct-2013
• Refers: ASTM E456-13 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Aug-2013
• Refers: ASTM E177-13 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2013
• Refers: ASTM E691-13 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-May-2013

Overview

ASTM E1970-23 is the international standard practice for the statistical treatment of thermoanalytical data, issued by ASTM International. This standard provides guidance on applying statistical methods to data produced from thermal analysis techniques, with a focus on evaluating the precision and reliability of measured values. Techniques such as mean, standard deviation, relative standard deviation, linear regression, and the propagation of uncertainties are central to this standard. By following ASTM E1970-23, laboratories and researchers can consistently report, interpret, and compare thermoanalytical results, ensuring robust data quality in scientific, academic, and industrial contexts.

Key Topics

• Statistical Tools for Thermoanalytical Data:

  • Calculation and use of mean, standard deviation, and relative standard deviation
  • Assessment and determination of pooled standard deviations
  • Linear regression for obtaining the best fit through data points, including calculation of slope and intercept
  • Application of the law of propagation of uncertainties to quantify and report measurement uncertainties

• Data Precision and Reliability:

  • Estimation of precision for measured values, typically expressed as mean ± standard deviation (X ± s)
  • Determining confidence intervals, with 67% of normal values within one standard deviation and 95% within two
  • Identification of outlying measurements using standard deviation ranges

• Special Considerations in Thermal Analysis:

  • Procedures for handling analytical values derived from linear regression slopes or intercepts from data sets
  • Techniques for calculating the precision of such derived parameters, which are crucial in determining kinetic parameters, vapor pressures, or enthalpy of vaporization

Applications

ASTM E1970-23 is widely employed across research, quality control, and materials science laboratories that utilize thermal analysis methods. Its practical applications include:

• Thermal Analysis Laboratories: Ensures standardization in the reporting and interpretation of results from techniques such as thermogravimetric analysis (TGA), differential scanning calorimetry (DSC), and dynamic mechanical analysis (DMA).
• Product Development: Assists in the quality assurance and validation of materials for industries including polymers, pharmaceuticals, ceramics, and metals by applying reliable statistical data evaluation methods.
• Kinetics and Thermodynamic Studies: Supports researchers in accurately determining kinetic parameters, vapor pressures, and enthalpy of vaporization through precise regression analysis of experimental data.
• Inter-laboratory Studies and Comparisons: Offers a robust statistical framework to compare results across different laboratories or instruments by using pooled deviation and repeatability measures.
• Regulatory Compliance: Provides documentation protocols that align with international trade and quality standards, in accordance with WTO Technical Barriers to Trade (TBT) Committee guidelines.

Related Standards

Organizations and laboratories using ASTM E1970-23 may also find the following ASTM standards relevant:

• ASTM E177 - Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• ASTM E456 - Terminology Relating to Quality and Statistics
• ASTM E691 - Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• ASTM E2161 - Terminology Relating to Performance Validation in Thermal Analysis and Rheology
• ASTM E2586 - Practice for Calculating and Using Basic Statistics
• ASTM F1469 (Withdrawn) - Guide for Conducting a Repeatability and Reproducibility Study on Test Equipment for Nondestructive Testing

Summary

ASTM E1970-23 streamlines the statistical treatment of thermoanalytical data, promoting high standards of precision, repeatability, and transparent reporting. Use of this practice ensures that thermal analysis results are both scientifically valid and internationally recognized, fostering confidence in data-driven decision-making across industries. For laboratories aiming to enhance thermal analysis quality and comparability, adherence to ASTM E1970-23 is of significant practical value.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E1970 − 23
Standard Practice for
Statistical Treatment of Thermoanalytical Data
This standard is issued under the fixed designation E1970; 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 (´) indicates an editorial change since the last revision or reapproval.
1. Scope* E691 Practice for Conducting an Interlaboratory Study to
Determine the Precision of a Test Method
1.1 This practice details the statistical data treatment used in
E2161 Terminology Relating to Performance Validation in
some thermal analysis methods.
Thermal Analysis and Rheology
1.2 The method describes the commonly encountered sta-
E2586 Practice for Calculating and Using Basic Statistics
tistical tools of the mean, standard derivation, relative standard
F1469 Guide for Conducting a Repeatability and Reproduc-
deviation, pooled standard deviation, pooled relative standard
ibility Study on Test Equipment for Nondestructive Test-
deviation, the best fit to a (linear regression of a) straight line
ing (Withdrawn 2018)
(or plane), and propagation of uncertainties for all calculations
encountered in thermal analysis methods (see Practice E2586).
3. Terminology
1.3 Some thermal analysis methods derive the analytical
3.1 Definitions—The technical terms used in this practice
value from the slope or intercept of a linear regression straight
are defined in Practice E177 and Terminologies E456 and
line (or plane) assigned to three or more sets of data pairs. Such
E2161 including precision, relative standard deviation,
methods may require an estimation of the precision in the
repeatability, reproducibility, slope, standard deviation,
determined slope or intercept. The determination of this
thermoanalytical, and variance.
precision is not a common statistical tool. This practice details 4
3.2 Symbols (1):
the process for obtaining such information about precision.
a, c, m = slope
1.4 This standard does not purport to address all of the
b, d = intercept
safety concerns, if any, associated with its use. It is the
n = number of data sets (that is, x , y )
i i
responsibility of the user of this standard to establish appro-
r = correlation coefficient
priate safety, health, and environmental practices and deter-
R = gage reproducibility and repeatability (see Guide
mine the applicability of regulatory limitations prior to use.
F1469) an estimation of the combined variation of
1.5 This international standard was developed in accor-
repeatability and reproducibility (2)
dance with internationally recognized principles on standard-
R = coefficient of determination
ization established in the Decision on Principles for the
RSD = relative standard deviation
Development of International Standards, Guides and Recom-
s = standard deviation
mendations issued by the World Trade Organization Technical
s = standard deviation of the line intercept
b
Barriers to Trade (TBT) Committee.
s = standard deviation of the “ith” measurement
i
s = standard deviation of the slope of a line
m
2. Referenced Documents
s = pooled standard deviation
pooled
s = within laboratory repeatability standard deviation
2.1 ASTM Standards:
r
(see Practice E691)
E177 Practice for Use of the Terms Precision and Bias in
s = between laboratory repeatability standard devia-
ASTM Test Methods R
tion (see Practice E691)
E456 Terminology Relating to Quality and Statistics
s = standard deviation of Y values
y
X = mean x value
1 x = an individual independent variable observation
This practice is under the jurisdiction of ASTM Committee E37 on Thermal
i
Measurements and is the direct responsibility of Subcommittee E37.10 on Y = mean y value
Fundamental, Statistical and Mechanical Properties.
y = an individual dependent variable observation
i
Current edition approved Jan. 1, 2023. Published February 2023. Originally
approved in 1998. Last previous edition approved in 2021 as E1970 – 16(2021).
DOI: 10.1520/E1970-23.
2 3
For referenced ASTM standards, visit the ASTM website, www.astm.org, or The last approved version of this historical standard is referenced on www.ast-
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM m.org.
Standards volume information, refer to the standard’s Document Summary page on The boldface numbers in parentheses refer to a list of references at the end of
the ASTM website. this standard.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1970 − 23
4.3.1 Variance is the square of the standard deviation(s).
Z = mean z value
Conversely the standard deviation is the positive square root of
z = an individual independent variable observation
i
the variance.
Σ = mathematical operation which means “the sum of
all” for the term(s) following the operator 4.3.2 The sensitivity coefficient is the partial derivative of
δy = variance in y parameter the function with respect to the individual variable.
i
5. Significance and Use
4. Summary of Practice
5.1 The standard deviation, or one of its derivatives, such as
4.1 The result of a series of replicate measurements of a
relative standard deviation or pooled standard deviation, de-
value are typically reported as the mean value plus some
rived from this practice, provides an estimate of precision in a
estimation of the precision in the mean value. The standard
measured value. Such results are ordinarily expressed as the
deviation is the most commonly encountered tool for estimat-
mean value 6 the standard deviation, that is, X 6 s.
ing precision, but other tools, such as relative standard devia-
5.2 If the measured values are, in the statistical sense,
tion or pooled standard deviation, also may be encountered in
“normally” distributed about their mean, then the meaning of
specific thermoanalytical test methods. This practice describes
the standard deviation is that there is a 67 % chance, that is 2
the mathematical process of achieving mean value, standard
in 3, that a given value will lie within the range of 6 one
deviation, relative standard deviation and pooled standard
standard deviation of the mean value. Similarly, there is a 95 %
deviation and other terms relating to the statistical treatment of
chance, that is 19 in 20, that a given value will lie within the
thermoanalytical data.
range of 6 two standard deviations of the mean. The two
4.2 In some thermal analysis experiments, a linear or a
standard deviation range is sometimes used as a test for
straight line, response is assumed and desired values are
outlying measurements.
obtained from the slope or intercept of the straight line through
5.3 The calculation of precision in the slope and intercept of
the experimental data. In any practical experiment, however,
i
a line, derived from experimental data, commonly is required
there will be some uncertainty in the data so that results are
in the determination of kinetic parameters, vapor pressure or
scattered about such a straight line. The linear regression (also
enthalpy of vaporization. This practice describes how to obtain
known as “least squares”) method is an objective tool for
these and other statistically derived values associated with
determining the “best fit” straight line drawn through a set of
measurements by thermal analysis.
experimental results and for obtaining information concerning
the precision of determined values.
6. Calculation
4.2.1 For the purposes of this practice, it is assumed that the
6.1 Commonly encountered statistical results in thermal
physical behavior, which the experimental results approximate,
analysis are obtained in the following manner.
are linear with respect to the controlled value, and may be
represented by the algebraic function in Eq 1:
NOTE 2—In the calculation of intermediate or final results, all available
figures shall be retained with any rounding to take place only at the
y 5 mx1b (1)
expression of the final results according to specific instructions or to be
consistent with the precision and bias statement.
4.2.2 Should the physical behavior be linear with respect to
6.1.1 The mean value (X) is given by:
two control variables, then the relationship is a plane and may
be represented by the algebraic function in Eq 16.
x 1x 1x 1. . . .1x Σx
1 2 3 i i
X 5 5 (2)
4.2.3 Experimental results are gathered in pairs (or sets),
n n
A similar relation exists for Y and Z.
that is, for every corresponding x (and z ) (controlled) value(s),
i i
there is a corresponding y (response) value.
i
6.1.2 The standard deviation (s) is given by:
4.2.4 The best fit (linear regression) approach assumes that
2 1/2
Σ~x 2 X!
i
all x (and z ) values are exact and the y values (only) are
i i i s 5 (3)
F G
n 2 1
~ !
subject to uncertainty.
6.1.3 The relative standard deviation (RSD) is given by:
NOTE 1—In experimental practice, both x and y values are subject to
uncertainty. If the uncertainty in x and y are of the same relative order of RSD 5 ~s·100 %!/X (4)
i i
magnitude, other more elaborate fitting methods should be considered. For
6.1.4 The pooled standard deviation (s ) is given by:
many sets of data, however, the results obtained by use of the assumption p
of exact values for the x data constitute such a close approximation to
1/2
i
Σ~$n 2 1%·s !
i i
those obtained by the more elaborate methods that the extra work and
s 5 (5)
F G
p
Σ~n 2 1!
i
additional complexity of the latter is hardly justified (2 and 3).
NOTE 3—For the calculation of pooled relative standard deviation, the
values of s are replaced by RSD .
4.2.5 The best fit approach seeks a straight line, which
i i
minimizes the uncertainty in the y value.
i 6.1.5 The gage repeatability and reproducibility (R) is given
by:
4.3 The law of propagation of uncertainties is a tool for
2 2 1/2
estimating the precision in a determined value from the sum of R 5 s 1s (6)
@ #
R r
NOTE 4—For the calculation of relative gage repeatability and
the variance of the respective measurements from which that
reproducibility, the values of s and s are replaced with RSD and RSD .
r R r R
value is derived weighted by the square of their respective
sensitivity coefficients. 6.2 Linear Regression (Best) Fit Straight Line:
--------
...

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. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E1970 − 16 (Reapproved 2021) E1970 − 23
Standard Practice for
Statistical Treatment of Thermoanalytical Data
This standard is issued under the fixed designation E1970; 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 (´) 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 commonly encountered statistical tools of the mean, standard derivation, relative standard deviation,
pooled standard deviation, pooled relative standard deviation, the best fit to a (linear regression of a) straight line, line (or plane),
and propagation of uncertainties for all calculations encountered in thermal analysis methods (see Practice E2586).
1.3 Some thermal analysis methods derive the analytical value from the slope or intercept of a linear regression straight line (or
plane) assigned to three or more sets of data pairs. Such methods may require an estimation of the precision in the determined slope
or intercept. The determination of this precision is not a common statistical tool. This practice details the process for obtaining such
information about precision.
1.4 There are no ISO methods equivalent to this practice.
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.5 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
E456 Terminology Relating to Quality and Statistics
E691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
E2161 Terminology Relating to Performance Validation in Thermal Analysis and Rheology
E2586 Practice for Calculating and Using Basic Statistics
F1469 Guide for Conducting a Repeatability and Reproducibility Study on Test Equipment for Nondestructive Testing
(Withdrawn 2018)
This practice is under the jurisdiction of ASTM Committee E37 on Thermal Measurements and is the direct responsibility of Subcommittee E37.10 on Fundamental,
Statistical and Mechanical Properties.
Current edition approved Oct. 1, 2021Jan. 1, 2023. Published November 2021February 2023. Originally approved in 1998. Last previous edition approved in 20162021
as E1970 – 16.E1970 – 16(2021). DOI: 10.1520/E1970-16R21.10.1520/E1970-23.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
The last approved version of this historical standard is referenced on www.astm.org.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1970 − 23
3. Terminology
3.1 Definitions—The technical terms used in this practice are defined in Practice E177 and Terminologies E456 and E2161
including precision, relative standard deviation, repeatability, reproducibility, slope, standard deviation, thermoanalytical, and
variance.
3.2 Symbols (1):
a, c, m = slope
b = intercept
b, d = intercept
n = number of data sets (that is, x , y )
i i
x = an individual independent variable observation
i
r = correlation coefficient
y = an individual dependent variable observation
i
R = gage reproducibility and repeatability (see Guide F1469) an estimation of the combined variation of repeatability and
reproducibility (2)
Σ = mathematical operation which means “the sum of all” for the term(s) following the operator
R = coefficient of determination
X = mean value
s = standard deviation
RSD = relative standard deviation
s = pooled standard deviation
pooled
s = standard deviation
s = standard deviation of the line intercept
b
s = standard deviation of the “ith” measurement
i
s = standard deviation of the slope of a line
m
s = standard deviation of Y values
y
s = pooled standard deviation
pooled
RSD = relative standard deviation
δy = variance in y parameter
i
r = correlation coefficient
R = gage reproducibility and repeatability (see Guide F1469) an estimation of the combined variation of repeatability and
reproducibility (2)
s = within laboratory repeatability standard deviation (see Practice E691)
r
s = between laboratory repeatability standard deviation (see Practice E691)
R
s = standard deviation of the “ith” measurement
i
s = standard deviation of Y values
y
X = mean x value
x = an individual independent variable observation
i
Y = mean y value
y = an individual dependent variable observation
i
Z = mean z value
z = an individual independent variable observation
i
Σ = mathematical operation which means “the sum of all” for the term(s) following the operator
δy = variance in y parameter
i
4. Summary of Practice
4.1 The result of a series of replicate measurements of a value are typically reported as the mean value plus some estimation of
the precision in the mean value. The standard deviation is the most commonly encountered tool for estimating precision, but other
tools, such as relative standard deviation or pooled standard deviation, also may be encountered in specific thermoanalytical test
methods. This practice describes the mathematical process of achieving mean value, standard deviation, relative standard deviation
and pooled standard deviation.deviation and other terms relating to the statistical treatment of thermoanalytical data.
4.2 In some thermal analysis experiments, a linear or a straight line, response is assumed and desired values are obtained from
the slope or intercept of the straight line through the experimental data. In any practical experiment, however, there will be some
i
The boldface numbers in parentheses refer to a list of references at the end of this standard.
E1970 − 23
uncertainty in the data so that results are scattered about such a straight line. The 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 results and for obtaining
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 represented by the algebraic function:function in Eq 1:
y 5 mx1b (1)
4.2.2 Should the physical behavior be linear with respect to two control variables, then the relationship is a plane and may be
represented by the algebraic function in Eq 16.
4.2.3 Experimental results are gathered in pairs, pairs (or sets), that is, for every corresponding x (and z ) (controlled)
i i
value,value(s), there is a corresponding y (response) value.
i
4.2.4 The best fit (linear regression) approach assumes that all x (and z ) values are exact and the y values (only) are subject to
i i i
uncertainty.
NOTE 1—In experimental practice, both x and y values are subject to uncertainty. If the uncertainty in x and y are of the same relative order of magnitude,
i i
other more elaborate fitting methods should be considered. For many sets of data, however, the results obtained by use of the assumption of exact values
for the x data constitute such a close approximation to those obtained by the more elaborate methods that the extra work and additional complexity of
i
the latter is hardly justified (2 and 3).
4.2.5 The best fit approach seeks a straight line, which minimizes the uncertainty in the y value.
i
4.3 The law of propagation of uncertainties is a tool for estimating the precision in a determined value from the sum of the variance
of the respective measurements from which that value is derived weighted by the square of their respective sensitivity coefficients.
4.3.1 Variance is the square of the standard deviation(s). Conversely the standard deviation is the positive square root of the
variance.
4.3.2 The sensitivity coefficient is the partial derivative of the function with respect to the individual variable.
5. Significance and Use
5.1 The standard deviation, or one of its derivatives, such as relative standard deviation or pooled standard deviation, derived from
this practice, provides an estimate of precision in a measured value. Such results are ordinarily expressed as the mean 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 of the standard
deviation 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 of
the 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 standard
deviations of 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 in the
determination of kinetic parameters, vapor pressure or enthalpy of vaporization. This practice describes how to obtain these and
other statistically derived values associated with measurements by thermal analysis.
6. Calculation
6.1 Commonly encountered statistical results in thermal analysis are obtained in the following manner.
NOTE 2—In the calculation of intermediate or final results, all available figures shall be retained with any rounding to take place only at the expression
of 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:
E1970 − 23
x 1x 1x 1. . . .1x Σx
1 2 3 i i
X 5 5 (2)
n n
A similar relation exists for Y and Z.
6.1.2 The standard deviation (s) is given by:
2 1/2
Σ x 2 X
~ !
i
s 5 (3)
F G
n 2 1
~ !
6.1.3 The relative standard deviation (RSD) is given by:
RSD 5 ~s·100 %!/X (4)
6.1.4 The pooled standard deviation (s ) is given by:
p
1/2
Σ n 2 1 ·s
~$ % !
i i
5 (5)
F G
Σ n 2 1
~ !
i
1/2
Σ~$n 2 1%·s !
i i
s 5 (5)
F G
p
Σ n 2 1
~ !
i
NOTE 3—For the calculation of pooled relative standard deviation, the values of s are replaced by
...