Sensory analysis - Methodology - Duo-trio test (ISO 10399:2017)

This document specifies a procedure for determining whether a perceptible sensory difference or
similarity exists between samples of two products. The method is a forced-choice procedure. The
method is applicable whether a difference exists in a single sensory attribute or in several attributes.
The method is statistically less efficient than the triangle test (described in ISO 4120) but is easier to
perform by the assessors.
The method is applicable even when the nature of the difference is unknown (i.e. it determines neither
the size nor the direction of difference between samples, nor is there any indication of the attribute(s)
responsible for the difference). The method is applicable only if the products are fairly homogeneous.
The method is effective for
a) determining that
1) either a perceptible difference results (duo-trio testing for difference), or
2) a perceptible difference does not result (duo-trio testing for similarity) when, for example, a
change is made in ingredients, processing, packaging, handling or storage, and
b) for selecting, training and monitoring assessors.
Two forms of the method are described:
— the constant-reference technique, used when one product is familiar to the assessors (e.g. a sample
from regular production);
— the balanced-reference technique, used when one product is not more familiar than the other.

Sensorische Analyse - Prüfverfahren - Duo-Trio-Prüfung (ISO 10399:2017)

Analyse sensorielle - Méthodologie - Essai duo-trio (ISO 10399:2017)

ISO 10399:2017 spécifie un mode opératoire permettant de déterminer s'il existe une différence sensorielle perceptible ou une similitude entre les échantillons de deux produits. La méthode est une procédure à choix forcé. Elle s'applique s'il existe une différence pour une seule propriété sensorielle ou pour plusieurs.
La méthode est statistiquement moins efficace que l'essai triangulaire (décrit dans l'ISO 4120), mais elle est plus facile à réaliser par les sujets.
La méthode est applicable même lorsque la nature de la différence est inconnue (ce qui signifie qu'elle ne détermine ni la taille ni le sens de la différence entre des échantillons et qu'elle ne donne pas non plus d'indications sur l'attribut/les attributs à l'origine de cette différence). La présente méthode n'est applicable que si les produits sont relativement homogènes.
La méthode est efficace dans les cas suivants:
a)    déterminer
     qu'il existe une différence perceptible (essai duo-trio de différence), ou
     qu'il n'existe pas de différence perceptible (essai duo-trio de similitude), quand, par exemple, des modifications sont apportées aux ingrédients, à la transformation, à l'emballage, aux opérations de manutention ou de stockage, et
b)    pour sélectionner, entraîner et contrôler les sujets.
Deux formes de cette méthode sont décrites:
-      la technique de la référence constante, utilisée lorsque les sujets connaissent bien un produit (un échantillon de fabrication courante par exemple), et
-      la technique de la référence équilibrée, utilisée quand aucun des produits n'est plus connu que l'autre.

Senzorična analiza - Metodologija - Preskus "duo-trio" (ISO 10399:2017)

Ta dokument določa postopek za določanje tega, ali med vzorcema dveh izdelkov obstaja zaznavna senzorična razlika. Pri metodi gre za postopek vsiljene izbire. Metoda je uporabna ne glede na to, ali razlika obstaja v enem samem senzoričnem atributu ali v več atributih.
Metoda je statistično manj učinkovita kot trikotni preskus (opisan v standardu ISO 4120), vendar jo ocenjevalci lažje izvajajo.
Metoda je uporabna celo, kadar je narava razlike neznana (ne določa niti velikosti niti smeri razlike med vzorci, poleg tega pa ni nikakršne indikacije glede atributov, odgovornih za razliko). Metoda je uporabna le, če so izdelki dokaj homogeni.
Metoda je učinkovita pri:
a) določanju,
1) ali pride do zaznavne razlike (duo-trio preskus za razliko), ali
2) do zaznavne razlike ne pride (duo-trio preskus za podobnost), kadar se na primer
izvede sprememba pri sestavinah, obdelavi, pakiranju, ravnanju ali shranjevanju; in
b) izbiranju, usposabljanju in nadziranju ocenjevalcev.
Opisani sta dve obliki metode:
– tehnika konstantne reference, ki se uporablja, ko je en izdelek znan ocenjevalcem (npr. vzorec iz redne proizvodnje);
– tehnika uravnotežene reference, ki se uporablja, ko en izdelek ni bolj znan od drugega.

General Information

Status
Published
Public Enquiry End Date
19-Oct-2017
Publication Date
14-Feb-2018
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
02-Feb-2018
Due Date
09-Apr-2018
Completion Date
15-Feb-2018

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SLOVENSKI STANDARD
SIST EN ISO 10399:2018
01-marec-2018
1DGRPHãþD
SIST EN ISO 10399:2010
6HQ]RULþQDDQDOL]D0HWRGRORJLMD3UHVNXVGXRWULR ,62
Sensory analysis - Methodology - Duo-trio test (ISO 10399:2017)
Analyse sensorielle - Méthodologie - Essai duo-trio (ISO 10399:2017)
Ta slovenski standard je istoveten z: EN ISO 10399:2018
ICS:
67.240 6HQ]RULþQDDQDOL]D Sensory analysis
SIST EN ISO 10399:2018 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 10399:2018

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SIST EN ISO 10399:2018


EN ISO 10399
EUROPEAN STANDARD

NORME EUROPÉENNE

January 2018
EUROPÄISCHE NORM
ICS 67.240 Supersedes EN ISO 10399:2010
English Version

Sensory analysis - Methodology - Duo-trio test (ISO
10399:2017)
Analyse sensorielle - Méthodologie - Essai duo-trio
(ISO 10399:2017)
This European Standard was approved by CEN on 10 December 2017.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland,
Turkey and United Kingdom.





EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2018 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 10399:2018 E
worldwide for CEN national Members.

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SIST EN ISO 10399:2018
EN ISO 10399:2018 (E)
Contents Page
European foreword . 3
2

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SIST EN ISO 10399:2018
EN ISO 10399:2018 (E)
European foreword
This document (EN ISO 10399:2018) has been prepared by Technical Committee ISO/TC 34 "Food
products".
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by July 2018, and conflicting national standards shall be
withdrawn at the latest by July 2018.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 10399:2010.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta,
Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland,
Turkey and the United Kingdom.
Endorsement notice
The text of ISO 10399:2017 has been approved by CEN as EN ISO 10399:2018 without any modification.

3

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SIST EN ISO 10399:2018

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SIST EN ISO 10399:2018
INTERNATIONAL ISO
STANDARD 10399
Third edition
2017-12
Sensory analysis — Methodology —
Duo-trio test
Analyse sensorielle — Méthodologie — Essai duo-trio
Reference number
ISO 10399:2017(E)
©
ISO 2017

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SIST EN ISO 10399:2018
ISO 10399:2017(E)

COPYRIGHT PROTECTED DOCUMENT
© ISO 2017, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. +41 22 749 01 11
Fax +41 22 749 09 47
copyright@iso.org
www.iso.org
ii © ISO 2017 – All rights reserved

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SIST EN ISO 10399:2018
ISO 10399:2017(E)

Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Principle . 2
5 General test conditions and requirements . 3
6 Assessors . 3
6.1 Qualification . 3
6.2 Number of assessors . 3
7 Procedure. 4
8 Analysis and interpretation of results . 5
8.1 When testing for a difference . 5
8.2 When testing for similarity. 5
9 Test report . 5
10 Precision and bias . 6
Annex A (normative) Tables . 7
Annex B (informative) Examples .13
Bibliography .21
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ISO 10399:2017(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 on 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 the following
URL: www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 34, Food products, Subcommittee SC 12,
Sensory analysis.
This third edition cancels and replaces the second edition (ISO 10399:2004), of which it constitutes
a minor revision. The references have been updated, the definition for 3.6 has been replaced and an
expression in A.3 has been corrected.
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SIST EN ISO 10399:2018
INTERNATIONAL STANDARD ISO 10399:2017(E)
Sensory analysis — Methodology — Duo-trio test
1 Scope
This document specifies a procedure for determining whether a perceptible sensory difference or
similarity exists between samples of two products. The method is a forced-choice procedure. The
method is applicable whether a difference exists in a single sensory attribute or in several attributes.
The method is statistically less efficient than the triangle test (described in ISO 4120) but is easier to
perform by the assessors.
The method is applicable even when the nature of the difference is unknown (i.e. it determines neither
the size nor the direction of difference between samples, nor is there any indication of the attribute(s)
responsible for the difference). The method is applicable only if the products are fairly homogeneous.
The method is effective for
a) determining that
1) either a perceptible difference results (duo-trio testing for difference), or
2) a perceptible difference does not result (duo-trio testing for similarity) when, for example, a
change is made in ingredients, processing, packaging, handling or storage, and
b) for selecting, training and monitoring assessors.
Two forms of the method are described:
— the constant-reference technique, used when one product is familiar to the assessors (e.g. a sample
from regular production);
— the balanced-reference technique, used when one product is not more familiar than the other.
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 5492, Sensory analysis — Vocabulary
ISO 8589, Sensory analysis — General guidance for the design of test rooms
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 5492 and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at https://www.iso.org/obp
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ISO 10399:2017(E)

3.1
alpha-risk
α-risk
probability of concluding that a perceptible difference (3.3) exists when one does not
Note 1 to entry: This is also known as Type I error, significance level or false positive rate.
3.2
beta-risk
β-risk
probability of concluding that no perceptible difference (3.3) exists when one does
Note 1 to entry: This is also known as Type II error or false negative rate.
3.3
difference
situation in which samples (3.5) can be distinguished based on their sensory properties
Note 1 to entry: The proportion of assessments in which a perceptible difference is detected between the two
products is given the symbol p .
d
3.4
product
material to be evaluated
3.5
sample
unit of product (3.4) prepared, presented and evaluated in the test
3.6
sensitivity
< statistic> statistical parameters that measure the performance characteristics of the test
Note 1 to entry: In statistical terms, the sensitivity of the test is defined by the values of α, β and p .
d
3.7
similarity
situation in which any perceptible differences (3.3) between the samples (3.5) are so small that the
products (3.4) can be used interchangeably
3.8
triad
three samples (3.5) given to an assessor in the duo-trio test
Note 1 to entry: In the duo-trio test, one sample is labelled as the reference, the other two are marked with
different codes. One of the coded samples is the same product as the reference; the other coded sample is the
other product in the test.
4 Principle
The number of assessors is chosen based on the sensitivity desired for the test (see 6.2 and the
discussion in A.3).
Assessors receive a set of three samples (i.e. a triad), one sample of which is labelled as a reference and
the other two samples have different codes. The assessors are informed that one of the coded samples
is the same as the reference and that one is different. Based on their training and the instructions given
prior to the test, the assessors report either which of the coded samples they believe to be same as the
reference, or which of the coded samples they believe to be different from the reference.
The number of correct responses is counted and the significance is determined by reference to a
statistical table.
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ISO 10399:2017(E)

5 General test conditions and requirements
5.1 Clearly define the test objective in writing.
5.2 Carry out the test under conditions that prevent communication among assessors until all the
evaluations have been completed using facilities and booths that conform with ISO 8589.
5.3 Prepare the samples out of sight of the assessors and in an identical manner (i.e. same apparatus,
same vessels, same quantity of product).
5.4 Assessors shall not be able to identify the samples from the way in which they are presented. For
example, in a taste test, avoid any differences in appearance. Mask any irrelevant colour differences using
light filters and/or subdued illumination.
5.5 Code the vessels containing the samples in a uniform manner, preferably using three-digit numbers,
chosen at random for each test. Each triad is composed of three samples, one labelled as the reference
and two labelled with different codes. Preferably, different codes should be used for each assessor
during a session. However, the same two codes may be used for all assessors within a test, provided that
each code is used only once per assessor during a test session (e.g. if several duo-trio tests on different
products are being conducted in the same session).
5.6 The quantity or volume served shall be identical for the three samples in each triad, just as that
of all the other samples in a series of tests on a given type of product. The quantity or volume to be
evaluated may be imposed. If it is not, the assessors should be told to take quantities or volumes that are
always similar whatever the sample.
5.7 The temperature of the three samples in each triad shall be identical, just as that of all the other
samples in a series of tests on a given type of product. It is preferable to present the samples at the
temperature at which the product is generally consumed.
5.8 The assessors shall be told whether or not they are to swallow the samples or whether they are
free to do as they please. In this latter case, they shall be requested to proceed in the same manner for all
samples.
5.9 During the test sessions, avoid giving information about product identity, expected treatment
effects, or individual performance until all testing is completed.
6 Assessors
6.1 Qualification
All assessors should possess the same level of qualification, this level being chosen on the basis of the
test objective (see ISO 8586 for guidance). Experience and familiarity with the product may improve
the performance of an assessor and, therefore, may increase the likelihood of finding a significant
difference. Monitoring the performance of assessors over time may be useful for increased sensitivity.
All assessors shall be familiar with the mechanics of the duo-trio test (i.e. the format, task and evaluation
procedure).
6.2 Number of assessors
Choose the number of assessors so as to obtain the sensitivity required for the test (see discussion in
A.3). Using large numbers of assessors increases the likelihood of detecting small differences between
the products. However, in practice, the number of assessors often is determined by material conditions
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ISO 10399:2017(E)

(e.g. duration of the experiment, number of available assessors, quantity of product). When testing for
a difference, the typical number of assessors is between 32 and 36. When testing for no meaningful
difference (i.e. similarity), twice as many assessors (i.e. approximately 72) are needed for equivalent
sensitivity.
Avoid replicate evaluations by the same assessor whenever possible. However, if replicate evaluations
are needed to produce a sufficient number of total evaluations, every effort should be made to have
each assessor perform the same number of replicate evaluations. For example, if only 12 assessors are
available, have each assessor evaluate 3 triads to obtain a total of 36 evaluations.
NOTE Treating 3 evaluations performed by 12 assessors as 36 independent evaluations is not valid when
testing for similarity using Table A.2. However, the test for difference using Table A.1 is valid even when replicate
[8],[9] [6],[7]
evaluations are performed Recent publications on replicated discrimination tests suggest alternative
approaches for analysing replicated evaluations in discrimination tests.
7 Procedure
7.1 If the product is familiar to the assessors (e.g. a control sample from the production line), use
the constant reference technique. If neither product is more familiar than the other, use the balanced-
reference technique
a) Constant-reference technique: Prepare worksheets and scoresheets (see B.2) in advance of the
test so as to utilize an equal number of the two possible sequences of two products, A and B:
A-REF AB A-REF BA
Distribute these at random in groups of two among the assessors (i.e. use each sequence once among
the first two assessors; use each sequence once again among the next two assessors, etc.) This will
minimize the imbalance that results if the total number of assessors is not an even number.
b) Balanced-reference technique: Prepare worksheets and scoresheets (see B.1) in advance of the
test so as to utilize an equal number of the four possible sequences of two products, A and B:
A-REF AB A-REF BA
B-REF AB B-REF BA
where the first two triads contain product A as the reference (i.e. A-REF) and the last two triads
contain product B as the reference (i.e. B-REF). Distribute these at random in groups of four among
the assessors (i.e. use each sequence once among the first group of four assessors; use each sequence
once again among the next group of four assessors, etc.). This will minimize the imbalance that
results if the total number of assessors is not a multiple of four.
7.2 Present the three samples of each triad simultaneously if possible, following the same spatial
arrangement for each assessor (e.g. on a line to be sampled always from left to right, in a triangular
array). Within the triad, assessors are generally allowed to make repeated evaluations of each sample as
desired (if, of course, the nature of the product allows for repeated evaluations).
7.3 Instruct the assessors to evaluate the reference sample first, then evaluate the two coded samples
in the order in which they were presented. Inform the assessors that one of the coded samples is the
same as the reference and that one is different from the reference. Instruct the assessors to indicate
either which of the two coded samples is the same as the reference, or which of the two coded samples is
different from the reference.
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ISO 10399:2017(E)

NOTE When deciding whether to instruct the assessors to select the sample that is the same as the reference
or to select the sample that is different from the reference, consideration is given to whether or not the panel
routinely uses other discrimination test methods. Many discrimination test methods like the triangle test, for
example, focus on identifying the “odd” or “different” sample in the test. Instructing the assessors to identify the
“different” sample in one method and to identify the “same” sample in another method can cause confusion and
lead to higher levels of incorrect responses.
7.4 Each scoresheet should provide for a single triad of samples. If an assessor is to carry out more
than one test in a session, collect the completed scoresheet and unused samples prior to serving the
subsequent triad. The assessor shall not go back to any of the previous samples or change the verdict on
any previous test.
7.5 Do not ask questions about preference, acceptance or degree of difference after the assessor has
made a selection. The selection the assessor has just made may bias the reply to any additional questions.
Responses to such questions may be obtained through separate tests for preference, acceptance, degree
of difference, etc., see ISO 6658. A comment section asking why the choice was made may be included for
the assessor’s remarks.
7.6 The duo-trio test is a forced-choice procedure; assessors are not allowed the option of reporting
“no difference”. An assessor who detects no difference between the samples should be instructed to
randomly select one of the samples and to indicate that the selection was only a guess in the comments
section of the scoresheet.
8 Analysis and interpretation of results
8.1 When testing for a difference
Use Table A.1 to analyse the data obtained from a duo-trio test. If the number of correct responses is
greater than or equal to the number given in Table A.1 (corresponding to the number of assessors and
the α-risk level chosen for the test), conclude that a perceptible difference exists between the samples
(see B.1).
If desired, calculate a confidence interval on the proportion of the population that can distinguish the
samples. The method is described in B.3.
8.2 When testing for similarity
NOTE In this document, “similar” does not mean “identical”. Rather, “similar” means that the two products
are sufficiently alike to be used interchangeably. It is not possible to prove that two products are identical.
However, it can be demonstrated that any difference that does exist between two products is so small as to have
no practical significance.
Use Table A.2 to analyse the data obtained from a duo-trio test. If the number of correct responses is
less than or equal to the number given in Table A.2 (corresponding to the number of assessors, the β-risk
level and the value of p chosen for the test), conclude that no meaningful difference exists between the
d
samples (see B.2). If results will be compared from one test to another, then the same value of p should
d
be chosen for all tests.
If desired, calculate a confidence interval on the proportion of the population that can distinguish the
samples. The method is described in B.3.
9 Test report
Report the test objective, the results and the conclusions. The following additional information is
recommended:
— the purpose of the test and the nature of the treatment studied;
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ISO 10399:2017(E)

— full identification of the samples (i.e. origin, method of preparation, quantity, shape, storage prior
to testing, serving size, temperature); sample information should communicate that all storage,
handling and preparation was done in such a way as to yield samples that differ only due to the
variable of interest, if at all;
— the number of assessors, the number of correct responses and the result of the statistical evaluation
(including the values of α, β and p used for the test);
d
— assessors: experience (in sensory testing, with the product, with the samples in the test), age and
gender (see ISO 8586 for guidance);
— any information and any specific recommendations given to the assessors in connection with the test;
— the test environment (i.e. the test facility used, simultaneous or sequential presentation, if the
identity of samples was disclosed after the test, if so, in what manner);
— the location, date of the test and name of the panel leader.
10 Precision and bias
Because the results of sensory discrimination tests are functions of individual sensitivities, a general
statement regarding the reproducibility of results that is applicable to all populations of assessors
cannot be made. Precision regarding a particular population of assessors increases as the size of the
panel increases and also with training and with exposure to the product.
As a forced-choice procedure is used, results obtained by this method are bias-free, provided that the
precautions in Clause 7 are fully observed.
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ISO 10399:2017(E)

Annex A
(normative)

Tables
A.1 The values given in Table A.1 are the minimum number of correct responses required for
significance at the stated α level (i.e. column) for the corresponding number of assessors, n (i.e. row).
Reject the assumption of “no difference” if the number of correct responses is greater than or equal to
the value in Table A.1.
Table A.1 — Minimum number of correct responses needed to conclude that a perceptible
difference exists based a duo-trio test
α α
n n
0,20 0,10 0,05 0,01 0,001 0,20 0,10 0,05 0,01 0,001
6 5 6 6 — — 26 16 17 18 20 22
7 6 6 7 7 — 27 17 18 19 20 22
8 6 7 7 8 — 28 17 18 19 21 23
9 7 7 8 9 — 29 18 19 20 22 24
10 7 8 9 10 10 30 18 20 20 22 24
11 8 9 9 10 11 32 19 21 22 24 26
12 8 9 10 11 12 36 22 23 24 26 28
13 9 10 10 12 13 40 24 25 26 28 31
14 10 10 11 12 13 44 26 27 28 31 33
15 10 11 12 13 14 48 28 29 31 33 36
16 11 12 12 14 15 52 30 32 33 35 38
17 11 12 13 14 16 56 32 34 35 38 40
18 12 13 13 15 16 60 34 36 37 40 43
19 12 13 14 15 17 64 36 38 40 42 45
20 13 14 15 16 18 68 38 40 42 45 48
21 13 14 15 17 18 72 41 42 44 47 50
22 13 14 15 17 19 76 43 45 46 49 52
23 15 16 16 18 20 80 45 47 48 51 55
24 15 16 17 19 20 84 47 49 51 54 57
25 16 17 18 19 21 88 49 51 53 56 59
NOTE 1 Values in the table are exact because they are based on the binomial distribution. For values of n not in the table,
compute approximate values for the missing entries
...

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