ISO 11843-6:2025

International
Standard
ISO 11843-6
Third edition
Capability of detection —
2025-10
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 2025
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Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 2
5 Measurement system and data handling . 2
6 Computation by approximation . 3
6.1 The critical value based on the normal distribution .3
6.2 Determination of the critical value of the response variable .4
6.3 Sufficient capability of the detection criterion .5
6.4 Confirmation of the sufficient capability of detection criterion .6
7 Reporting the results from an assessment of the capability of detection . 7
8 Reporting the results from an application of the method . 7
Annex A (informative) Estimating the mean value and variance when the Poisson distribution
is approximated by the normal distribution . 8
Annex B (informative) Accuracy of approximations . 9
Annex C (informative) Selecting the number of channels for the detector .15
Annex D (informative) Examples of calculations .16
Bibliography .21

iii
Foreword
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This document was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee SC 6, Measurement methods and results.
This third edition cancels and replaces the second edition (ISO 11843-6:2019), which has been technically
revised.
The main changes are as follows:
— the symbols were modified to conform to ISO 11843-1;
— a list of symbols was moved from Annex A to Clause 4;
— in 6.3, explanatory text of how to determine the minimum detectable value was added;
— Clause 8 was revised to provide a description of the appropriate approach for determining whether or
not the target substance has been detected;
— typographic and obvious errors were corrected.
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
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]to[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.
Although the methodology employed to determine the minimum detection values has long been established
in the field of chemical analysis, it has hitherto remained undefined within the domain of pulse count
measurements. The necessity of establishing a methodology for determining the minimum detectable value
[9]
in this field is duly acknowledged .
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 criteria and the minimum
[10]
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 ).
D
v
International Standard ISO 11843-6:2025(en)
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 B.
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 definitions
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 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 used
ISO 11843-4, Capability of detection — Part 4: Methodology for comparing the minimum detectable value with
a given value
3 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 terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/

4 Symbols
X
state variable
Y
response variable
number of replications of measurements on the reference material representing the value of the basic
J
state variable (blank sample)
K
number of replications of measurements on the actual state (test sample)
number of replications of measurements of each reference material in assessment of the capability
N
of detection
x
a value of state variable
y
a value of response variable
y
critical value of the response variable defined by ISO 11843-1 and ISO 11843-3
C
x
given value which is tested to determine whether it is greater than the minimum detectable value
g
x
minimum detectable value of the state variable
D
σ
standard deviation under actual performance conditions for the response in the basic state
b
σ
standard deviation under actual performance conditions for the response in a sample with the state
g
variable equal to x
g
η
expected value under the actual performance conditions for the response in the basic state
b
η
expected value under the actual performance conditions for the response in a sample with the state
g
variable equal to x
g
y
the arithmetic mean of the actual measured response in the basic state
b
y the arithmetic mean of the actual measured response in a sample with the state variable equal to x
g g
y minimum detectable response value with the state variable equal to x
D d
λ
mean value corresponding to the expected number of events in Poisson distribution
α
the probability that an error of the first kind has occurred
β
the probability that an error of the second kind has occurred
1−α
confidence level
1−β
confidence level
z ()1−α -quantile of the standard normal distribution
1−α
z ()1−β -quantile of the standard normal distribution
1−β
T
lower confidence limit
5 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
(milliseconds) each. The approximation of the Poisson distribution by the normal distribution is more
reliable with higher mean values.
NOTE In the case of a measurement system that obtains an average signal of 2 000 counts/s, a single
measurement taken over 2 s yields to a measured value of 4 000 counts. In pulse count measurement, the standard
deviation is given by the square root of the measured value (see Annex A), which in this case is 63,20 counts, and
the coefficient of variation, C , is 0,016. On the other hand, if multiple measurements are taken with a measurement
V
time of 100 ms (milliseconds), an average measured value of 200 counts is obtained, with a standard deviation of
14,1 counts and a C of 0,071. Higher measured values provide greater accuracy than lower measured values.
V
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 (see Figure C.1).
f) Peak width: The full width at half maximum (FWHM) is the recommended coverage for monitoring a
single peak. It is preferable the measurements are based on the top, the bottom or both, 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.
6 Computation by approximation
6.1 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 is referred to
g C C
as the critical value, satisfies the requirement:
Py()>=yx 0 ≤α (1)
gC
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 of false decision.

Formula (1) gives the probability that yy> under the condition that:
gC
yy=+z σ + (2)
Cb 1−α b
JK
where
z is the (1 − α)-quantile of the standard normal distribution where 1 − α is the confidence level;
1−α
σ is the standard deviation under actual performance conditions for the response in the basic state;
b
y is the arithmetic mean of the actual measured response in the basic state;
b
J is the number of repeat measurements of the blank reference sample. This represents the value
of the basic state variable;
K is the number of repeat measurements of the test sample. This gives the value of the actual
state variable.
NOTE "The only + sign is used in Formula (2). In the pulse counting method the response variable is 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.
Key
X state variable
Y response variable
α probability that an error of the first kind has occurred
β probability that an error of the second kind has occurred
Figure 1 — Conceptual diagram showing the relative position of the critical value and the measured
values of the active and basic states
6.2 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 .
b
This is an estimate of σ . The standard deviation of the repeated measurements of the response variable in
b
the actual state of the sample is y , giving an estimate of σ (see Annex A).
g g
The critical value, y , of a response variable that follows the Poisson distribution approximated by the
C
normal distribution generally satisfies Formula (3):
11 11
yy=+z σ +≈ yz++y (3)
Cb 11−−ααbb b
JK JK
where y is the arithmetic mean of the actual measured response in the basic state.
b
6.3 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 level, 1−β . If the
criterion is satisfied, it can be concluded that the minimum detectable value, x , is less than or equal to the
D
value of the state variable, x . The minimum detectable value then defines the smallest value of the response
g
variable, η , for which an incorrect decision occurs with a probability, β . At this value, there is no signal,
g
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 be
g g
greater than or equal to 1−β is set by Formula (4), from which Formulae (5) and (6) can be derived:
η ≥+yz σσ+ (4)
g Cb1−β g
JK
If y is replaced by yz=+ησ + , defined in Formulae (2) and (3), then:
C Cb 1−α b
JK
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;
b
η is the expected value under the actual performance conditions for the response in a sample
g
with the state variable
...