# ISO 11843-6:2019

(Amendment)## Capability of detection — Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson distributed measurements by normal approximations

## Capability of detection — Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson distributed measurements by normal approximations

This document presents methods for determining the critical value of the response variable and the minimum detectable value in Poisson distribution measurements. It is applicable when variations in both the background noise and the signal are describable by the Poisson distribution. The conventional approximation is used to approximate the Poisson distribution by the normal distribution consistent with ISO 11843‑3 and ISO 11843‑4. The accuracy of the normal approximation as compared to the exact Poisson distribution is discussed in Annex C.

## Capacité de détection — Partie 6: Méthodologie pour la détermination de la valeur critique et de la valeur minimale détectable pour les mesures distribuées selon la loi de Poisson approximée par la loi Normale

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### Standards Content (sample)

INTERNATIONAL ISO

STANDARD 11843-6

Second edition

2019-02

Capability of detection —

Part 6:

Methodology for the determination

of the critical value and the

minimum detectable value in Poisson

distributed measurements by normal

approximations

Capacité de détection —

Partie 6: Méthodologie pour la détermination de la valeur critique et

de la valeur minimale détectable pour les mesures distribuées selon la

loi de Poisson approximée par la loi Normale

Reference number

ISO 11843-6:2019(E)

ISO 2019

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ISO 11843-6:2019(E)

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© ISO 2019

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

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ii © ISO 2019 – All rights reserved

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ISO 11843-6:2019(E)

Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Measurement system and data handling .................................................................................................................................... 2

5 Computation by approximation............................................................................................................................................................ 2

5.1 The critical value based on the normal distribution .............................................................................................. 2

5.2 Determination of the critical value of the response variable .......................................................................... 4

5.3 Sufficient capability of the detection criterion ............................................................................................................ 5

5.4 Confirmation of the sufficient capability of detection criterion ................................................................... 5

6 Reporting the results from an assessment of the capability of detection .................................................6

7 Reporting the results from an application of the method ......................................................................................... 7

Annex A (informative) Symbols used in this document .................................................................................................................... 8

Annex B (informative) Estimating the mean value and variance when the Poissondistribution is approximated by the normal distribution .....................................................................................10

Annex C (informative) An accuracy of approximations .................................................................................................................11

Annex D (informative) Selecting the number of channels for the detector ..............................................................17

Annex E (informative) Examples of calculations ...................................................................................................................................18

Bibliography .............................................................................................................................................................................................................................23

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ISO 11843-6:2019(E)

Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/patents).Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso

.org/iso/foreword .html.This document was prepared by Technical Committee ISO/TC 69, Application of statistical methods,

Subcommittee SC 6, Measurement methods and results.This second edition cancels and replaces the first edition (ISO 11843-6:2013, corrected version issued

in 2014), of which it constitutes a minor revision. The changes compared to the previous corrected

version are as follows:— the following updates have been made to add clarity or to correct typographic and obvious errors:

— in Formula (2) and the related Note, “±” is replaced with “+”;— in 5.4, 2 paragraph, “100(1-α/2)%” is replaced with “100(1-α)%”, and “described below by the

general theory of estimation” is inserted at the end,— in Clause 6 e), Figure 1, and Annex C 3 paragraph, the average values ( y , y and y ) are used;

b d grd st

— in Annex C, 3 paragraph, 1 sentence, “independent” is inserted before “variables”;

— in the line below Formula (C.1), I • is consistently replaced with I • ;() ()

k y

— in E.2, 2 paragraph, the text has been slightly reworded for clarity;

— thorough the text, minor editorial modifications have been made in line with the 2018 edition of the

ISO/IEC Directives, Part 2.A list of all parts in the ISO 11843 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/members .html.iv © ISO 2019 – All rights reserved

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ISO 11843-6:2019(E)

Introduction

Many types of instruments use the pulse-counting method for detecting signals. X-ray, electron and

ion-spectroscopy detectors, such as X-ray diffractometers (XRD), X-ray fluorescence spectrometers

(XRF), X-ray photoelectron spectrometers (XPS), Auger electron spectrometers (AES), secondary ion

mass spectrometers (SIMS) and gas chromatograph mass spectrometers (GCMS) are of this type. These

signals consist of a series of pulses produced at random and irregular intervals. They can be understood

statistically using a Poisson distribution and the methodology for determining the minimum detectable

value can be deduced from statistical principles.Determining the minimum detectable value of signals is sometimes important in practical work. The

value provides a criterion for deciding when “the signal is certainly not detected”, or when “the signal is

[1]-[8]significantly different from the background noise level” . For example, it is valuable when measuring

the presence of hazardous substances or surface contamination of semi-conductor materials. RoHS

(Restrictions on Hazardous Substances) sets limits on the use of six hazardous materials (hexavalent

chromium, lead, mercury, cadmium and the flame retardant agents, perbromobiphenyl, PBB, and

perbromodiphenyl ether, PBDE) in the manufacturing of electronic components and related goods

sold in the EU. For that application, XRF and GCMS are the testing instruments used. XRD is used to

measure the level of hazardous asbestos and crystalline silica present in the environment or in building

materials.The methods used to set the minimum detectable value have for some time been in widespread use

in the field of chemical analysis, although not where pulse-counting measurements are concerned.

The need to establish a methodology for determining the minimum detectable value in that area is

[9]recognized .

In this document the Poisson distribution is approximated by the normal distribution, ensuring

consistency with the IUPAC approach laid out in the ISO 11843 series. The conventional approximation

is used to generate the variance, the critical value of the response variable, the capability of detection

[10]criteria and the minimum detectability level .

In this document:

— α is the probability of erroneously detecting that a system is not in the basic state, when really it is

in that state;— β is the probability of erroneously not detecting that a system is not in the basic state when the

value of the state variable is equal to the minimum detectable value (x ).This document is fully compliant with ISO 11843-1, ISO 11843-3 and ISO 11843-4.

© ISO 2019 – All rights reserved v

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INTERNATIONAL STANDARD ISO 11843-6:2019(E)

Capability of detection —

Part 6:

Methodology for the determination of the critical value

and the minimum detectable value in Poisson distributed

measurements by normal approximations

1 Scope

This document presents methods for determining the critical value of the response variable and the

minimum detectable value in Poisson distribution measurements. It is applicable when variations in

both the background noise and the signal are describable by the Poisson distribution. The conventional

approximation is used to approximate the Poisson distribution by the normal distribution consistent

with ISO 11843-3 and ISO 11843-4.The accuracy of the normal approximation as compared to the exact Poisson distribution is discussed

in Annex C.2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO Guide 30, Reference materials — Selected terms and definitionsISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in

probabilityISO 11843-1, Capability of detection — Part 1: Terms and definitions

ISO 11843-2, Capability of detection — Part 2: Methodology in the linear calibration case

ISO 11843-3, Capability of detection — Part 3: Methodology for determination of the critical value for the

response variable when no calibration data are usedISO 11843-4, Capability of detection — Part 4: Methodology for comparing the minimum detectable value

with a given value3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 11843-1,

ISO 11843-2, ISO 11843-3, ISO 11843-4, and ISO Guide 30 apply.ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https: //www .iso .org/obp— IEC Electropedia: available at http: //www .electropedia .org/

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ISO 11843-6:2019(E)

4 Measurement system and data handling

The conditions under which Poisson counts are made are usually specified by the experimental set-up.

The number of pulses that are detected increases with both the time and with the width of the region

over which the spectrum is observed. These two parameters should be noted and not changed during

the course of the measurement.The following restrictions should be observed if the minimum detectable value is to be determined

reliably:a) Both the signal and the background noise should follow the Poisson distributions. The signal is the

mean value of the gross count.b) The raw data should not receive any processing or treatment, such as smoothing.

c) Time interval: Measurement over a long period of time is preferable to several shorter

measurements. A single measurement taken for over one second is better than 10 measurements

over 100 ms each. The approximation of the Poisson distribution by the normal distribution is more

reliable with higher mean values.d) The number of measurements: Since only mean values are used in the approximations presented

here, repeated measurements are needed to determine them. The power of test increases with the

number of measurements.e) Number of channels used by the detector: There should be no overlap of neighbouring peaks. The

number of channels that are used to measure the background noise and the sample spectra should

be identical (Annex D, Figure D.1).f) Peak width: The full width at half maximum (FWHM) is the recommended coverage for monitoring

a single peak. It is preferable to measurements based on the top and/or the bottom of a noisy

peak. The appropriate FWHM should be assessed beforehand by measuring a standard sample.

An identical value of the FWHM should be used for both the background noise and the sample

measurements.Additional factors are: the instrument should work correctly; the detector should be operating within

its linear counting range; both the ordinate and the abscissa axes should be calibrated; there should

be no signal that cannot be clearly identified as not being noise; degradation of the specimen during

measurement should be negligibly small; at least one signal or peak belonging to the element under

consideration should be observable.5 Computation by approximation

5.1 The critical value based on the normal distribution

The decision on whether a measured signal is significant or not can be made by comparing the

arithmetic mean y of the actual measured values with a suitably chosen value y . The value y , which

c cis referred to as the critical value, satisfies the requirement

Py()>=yx 0 ≤α (1)

where the probability is computed under the condition that the system is in the basic state (x = 0) and

α is a pre-selected probability value.Formula (1) gives the probability that yy> under the condition that:

yy=+z σ + (2)

cb 1b−α

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ISO 11843-6:2019(E)

where

is the (1 − α)-quantile of the standard normal distribution where 1 − α is the confidence level;

1−αis the standard deviation under actual performance conditions for the response in the basic state;

is the arithmetic mean of the actual measured response in the basic state;is the number of repeat measurements of the blank reference sample. This represents the value

of the basic state variable;is the number of repeat measurements of the test sample. This gives the value of the actual

state variable.NOTE Only the + sign is used in Formula (2). In the pulse counting method, the response variable is a positive

integer and always increases as the state variable increases.The definition of the critical value follows ISO 11843-1 and ISO 11843-3. Its relationship to the measured

values in the active and basic states is illustrated in Figure 1.© ISO 2019 – All rights reserved 3

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ISO 11843-6:2019(E)

Key

X state variable

Y response variable

α the probability that an error of the first kind has occurred

β the probability that an error of the second kind has occurred

Figure 1 — A conceptual diagram showing the relative position of the critical value and the

measured values of the active and basic states5.2 Determination of the critical value of the response variable

If the response variable follows a Poisson distribution with a sufficiently large mean value, the standard

deviation of the repeated measurements of the response variable in the basic state is estimated as

y . This is an estimate of σ . The standard deviation of the repeated measurements of the response

variable in the actual state of the sample is y , giving an estimate of σ (see Annex B).

The critical value, y , of a response variable that follows the Poisson distribution approximated by the

normal distribution generally satisfies:11 11

yy=+z σ +≈ yz++y (3)

cb 1−−ααbb 1b

JK JK

where y is the arithmetic mean of the actual measured response in the basic state.

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ISO 11843-6:2019(E)

5.3 Sufficient capability of the detection criterion

The sufficient capability of detection criterion enables decisions to be made about the detection of a

signal by comparing the critical value probability with a specified value of the confidence levels, 1 − β. If

the criterion is satisfied, it can be concluded that the minimum detectable value, x , is less than or equal

to the value of the state variable, x . The minimum detectable value then defines the smallest value of

the response variable, η , for which an incorrect decision occurs with a probability, β. At this value,

there is no signal, only background noise, and an ‘error of the second kind’ has occurred.

If the standard deviation of the response for a given value x is σ , the criterion for the probability to

g gbe greater than or equal to 1 − β is set by Formula (4), from which Formulae (5) and (6) can be derived:

2 2η ≥+yz σσ+ (4)

gc 1−β b g

If y is replaced by yz=+ησ + , defined in Formulae (2) and (3), then:

cb 1b−α

11 11

2 2

η −≥ησz ++z σσ+ (5)

gb 11−−αβbb g

JK JK

where

α is the probability that an error of the first kind has occurred;

β is the probability that an error of the second kind has occurred;

η is the expected value under the actual performance conditions for the response in the basic state;

η is the expected value under the actual performance conditions for the response in a sample

with the state variable equal to x .With β = α and K = J, the criterion simplifies to:

2 2

η −≥ησz 2 ++σσ (6)

gb 1−α bb g

If σ is replaced with an estimate of y following 5.2 and similarly σ is replaced with an estimate of

b gy (see Annex B), the criterion becomes Formula (7).

η −≥η z 2yy++ y (7)

gb 1−α bb g

NOTE When validating a method, the capability of detection is usually determined for K = J = 1 in accordance

with ISO 11843-4.5.4 Confirmation of the sufficient capability of detection criterion

The standard deviations and expected values of the response are usually unknown, so an assessment

using criterion Formula (6) has to be made from the experimental data. The expression on the left-hand

side of the simplified criterion Formula (6) is unknown, whereas that on the right-hand side is known.

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ISO 11843-6:2019(E)

A confidence interval of η − η is provided by N repeated measurements in the basic state and N

g brepeated measurements of a sample with the state variable equal to x . A 100(1 − α) % confidence

interval for η − η , described below by the general theory of estimation, is:g b

11 11

2 2 2 2

()yy−−z σσ+≤ηη−≤()yy−+z σ + σσ (8)

gb (/12−−αα)(b g gb gb 12/) b g

NN NN

where z is the 100(1 − α/2) quantile of the standard normal distribution.

(1 − α/2)

To confirm the sufficient capability of detection criterion, a one-sided test is used. With β = α,

100(1 − α) % of the one-sided lower confidence bound on η − η is:g b

2 2

η −≥ησyy− −+z σ (9)

gb gb b g

1−α

where

N is the number of replications of measurements of each reference material used to assess the

capability of detection;is the arithmetic mean of the actual measured response in a sample with the state variable

equal to x ;η is the expected value under actual performance conditions for the response in the basic state;

η is the expected value under actual performance conditions for the respo**...**

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