Sensory analysis - Methodology - Triangle test (ISO 4120:2021)

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 more efficient than the duo-trio test (described in ISO 10399), but has
limited use with products that exhibit strong carryover and/or lingering flavours.
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 homogeneous.
The method is effective for:
a) determining that:
1) either a perceptible difference results (triangle testing for difference);
2) a perceptible difference does not result (triangle testing for similarity),
when, for example, a change is made in ingredients, processing, packaging, handling or storage;
b) selecting, training and monitoring assessors.

Sensorische Analyse - Prüfverfahren - Dreiecksprüfung (ISO 4120:2021)

Dieses Dokument legt ein Verfahren zur Feststellung, ob zwischen den Proben von zwei Prüfmaterialien ein wahrnehmbarer sensorischer Unterschied oder eine Ähnlichkeit vorhanden ist, fest. Bei dem Verfahren wird nach dem „Forced Choice“ Prinzip vorgegangen. Das Verfahren ist anwendbar, wenn ein Unterschied bei einem einzelnen sensorischen Merkmal oder bei mehreren Merkmalen vorliegt.
Das Verfahren ist statistisch aussagekräftiger als die Duo Trio Prüfung (beschrieben in ISO 10399), hat aber eine begrenzte Anwendung bei Prüfmaterialien, die starke überdeckende und/oder nachwirkende olfaktorische/gustatorische Eindrücke hinterlassen.
Das Verfahren ist auch dann anwendbar, wenn die Art des Unterschiedes unbekannt ist (d. h. es bestimmt weder Größe noch Richtung eines Unterschiedes zwischen den Proben noch liefert es Hinweise auf Merkmale, die für den Unterschied verantwortlich sind). Das Verfahren ist nur anwendbar, wenn die Prüfmaterialien homogen sind.
Das Verfahren eignet sich für
a) die Untersuchung daraufhin,
1) ob ein wahrnehmbarer Unterschied resultiert (Dreiecksprüfung auf Unterschied);
2) ob kein wahrnehmbarer Unterschied resultiert (Dreiecksprüfung auf Ähnlichkeit),
wenn zum Beispiel eine Änderung von Zutaten, Verarbeitung, Verpackung, Behandlung oder Lagerung vorgenommen wurde.
b) für die Auswahl, Schulung und Leistungsüberprüfung von Prüfpersonen.

Analyse sensorielle - Méthodologie - Essai triangulaire (ISO 4120:2021)

Le présent document 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 plus efficace que l'essai duo-trio (décrit dans l'ISO 10399), mais est d'une utilisation limitée pour les produits présentant une forte rémanence et/ou des arômes persistants.
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 la ou les propriétés à l'origine de cette différence]. La présente méthode n'est applicable que si les produits sont homogènes.
La méthode est efficace dans les cas suivants:
a)    pour déterminer:
     qu'il existe une différence perceptible (essai triangulaire de différence);
     qu'il n'existe pas de différence perceptible (essai triangulaire de similitude),
quand, par exemple, des modifications sont apportées aux ingrédients, à la transformation, à l'emballage, aux opérations de manutention ou de stockage;
b)    pour sélectionner, entraîner et contrôler les sujets.

Senzorična analiza - Metodologija - Trikotni "triangel" preskus (ISO 4120:2021)

General Information

Status
Published
Public Enquiry End Date
19-May-2020
Publication Date
24-May-2021
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
30-Mar-2021
Due Date
04-Jun-2021
Completion Date
25-May-2021

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SLOVENSKI STANDARD
SIST EN ISO 4120:2021
01-julij-2021
Nadomešča:
SIST EN ISO 4120:2007
Senzorična analiza - Metodologija - Trikotni "triangel" preskus (ISO 4120:2021)
Sensory analysis - Methodology - Triangle test (ISO 4120:2021)
Sensorische Analyse - Prüfverfahren - Dreiecksprüfung (ISO 4120:2021)
Analyse sensorielle - Méthodologie - Essai triangulaire (ISO 4120:2021)
Ta slovenski standard je istoveten z: EN ISO 4120:2021
ICS:
67.240 Senzorična analiza Sensory analysis
SIST EN ISO 4120:2021 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 4120:2021
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SIST EN ISO 4120:2021
EN ISO 4120
EUROPEAN STANDARD
NORME EUROPÉENNE
March 2021
EUROPÄISCHE NORM
ICS 67.240 Supersedes EN ISO 4120:2007
English Version
Sensory analysis - Methodology - Triangle test (ISO
4120:2021)

Analyse sensorielle - Méthodologie - Essai triangulaire Sensorische Analyse - Prüfverfahren - Dreiecksprüfung

(ISO 4120:2021) (ISO 4120:2021)
This European Standard was approved by CEN on 23 February 2021.

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, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,

Poland, Portugal, Republic of North Macedonia, 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

© 2021 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 4120:2021 E

worldwide for CEN national Members.
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SIST EN ISO 4120:2021
EN ISO 4120:2021 (E)
Contents Page

European foreword ....................................................................................................................................................... 3

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SIST EN ISO 4120:2021
EN ISO 4120:2021 (E)
European foreword

This document (EN ISO 4120:2021) has been prepared by Technical Committee ISO/TC 34 "Food

products" in collaboration with CCMC.

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 September 2021, and conflicting national standards

shall be withdrawn at the latest by September 2021.

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 4120:2007.

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, France, Germany, Greece, Hungary, Iceland,

Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of

North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the

United Kingdom.
Endorsement notice

The text of ISO 4120:2021 has been approved by CEN as EN ISO 4120:2021 without any modification.

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SIST EN ISO 4120:2021
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SIST EN ISO 4120:2021
INTERNATIONAL ISO
STANDARD 4120
Third edition
2021-03
Sensory analysis — Methodology —
Triangle test
Analyse sensorielle — Méthodologie — Essai triangulaire
Reference number
ISO 4120:2021(E)
ISO 2021
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SIST EN ISO 4120:2021
ISO 4120:2021(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2021

All rights reserved. Unless otherwise specified, or required in the context of its implementation, 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
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2021 – All rights reserved
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SIST EN ISO 4120:2021
ISO 4120:2021(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 ............................................................................................................................... 2

6 Assessors ...................................................................................................................................................................................................................... 3

6.1 Qualification .............................................................................................................................................................................................. 3

6.2 Number of assessors .......................................................................................................................................................................... 3

7 Procedure..................................................................................................................................................................................................................... 4

8 Analysis and interpretation of results ........................................................................................................................................... 4

8.1 When testing for a difference ..................................................................................................................................................... 4

8.2 When testing for similarity........................................................................................................................................................... 5

9 Test report ................................................................................................................................................................................................................... 5

10 Precision and bias ............................................................................................................................................................................................... 5

Annex A (normative) Tables .......................................................................................................................................................................................... 7

Annex B (informative) Examples ...........................................................................................................................................................................12

Bibliography .............................................................................................................................................................................................................................17

© ISO 2021 – All rights reserved iii
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SIST EN ISO 4120:2021
ISO 4120:2021(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 ISO/TC 34, Food products, Subcommittee SC 12, Sensory analysis, in

collaboration with the European Committee for Standardization (CEN) Technical Committee CEN/

SS C01, Food Products, in accordance with the Agreement on technical cooperation between ISO and

CEN (Vienna Agreement).

This third edition cancels and replaces the second edition (ISO 4120:2004), which has been technically

revised. The main changes compared with the previous edition are as follows:
— the document has been generalized beyond food and beverage applications;

— guidance on how to use the Thurstonian model in addition to the previously emphasized guessing

model has been added.

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 2021 – All rights reserved
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SIST EN ISO 4120:2021
INTERNATIONAL STANDARD ISO 4120:2021(E)
Sensory analysis — Methodology — Triangle 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 more efficient than the duo-trio test (described in ISO 10399), but has

limited use with products that exhibit strong carryover and/or lingering flavours.

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 homogeneous.

The method is effective for:
a) determining that:
1) either a perceptible difference results (triangle testing for difference);
2) a perceptible difference does not result (triangle testing for similarity),

when, for example, a change is made in ingredients, processing, packaging, handling or storage;

b) selecting, training and monitoring assessors.
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:

— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
3.1
alpha-risk
α-risk
probability of concluding that a perceptible difference exists when one does not

Note 1 to entry: This is also known as Type I error, significance level or false positive rate.

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SIST EN ISO 4120:2021
ISO 4120:2021(E)
3.2
beta-risk
β-risk
probability of concluding that no meaningful difference 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 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 .
3.4
product
material to be evaluated
3.5
sample
unit of product prepared, presented and evaluated in the test
3.6
sensitivity
statistical parameters that measure the performance characteristics of the test

Note 1 to entry: The sensitivity of the test is defined by the values of α, β and p .

3.7
similarity

situation in which any perceptible differences between the samples are so small that the products can

be used interchangeably
3.8
triad
those three samples given to an assessor in the triangle test

Note 1 to entry: In the triangle test, each sample is marked with a different code. Two of the samples are alike (i.e.

from one product) and one is different (i.e. from the other product).
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) and are informed that two of the samples are the

same and that one is different. The assessors report which sample they believe to be different, even if

the selection is based only on a guess.

The number of correct responses is counted, and the significance is determined by reference to a

statistical table or an applicable computer program or app.
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 to ISO 8589.

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ISO 4120:2021(E)

5.3 Prepare the samples out of sight of the assessors and in an identical manner (e.g. same apparatus,

same vessels, same quantities 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 test samples in a uniform manner, preferably using three-digit numbers, chosen at

random for each test. Each triad is composed of three samples, each with a different code. Preferably,

different codes should be used for each assessor during a session. However, the same three 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 triangle tests on different products are being conducted in the same session).

5.6 It is preferable to present the samples under the conditions at which the product is generally used

(e.g. in a taste test, present the samples at the temperature at which the product is generally consumed).

The serving conditions of the three samples in each triad shall be identical (e.g. in a taste test, the three

samples shall be served at the same temperature), just as that of all the other samples in a series of tests

on a given type of product.

5.7 The size, quantity or volume presented 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 size, quantity or volume to

be evaluated may be imposed. If it is not, the assessors should be told to take sizes, quantities or volumes

that are always similar whatever the sample.

5.8 In a taste test, 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 the samples.

5.9 During the test sessions, avoid giving information about product identity, expected treatment

effects, or individual performance until all testing is completed. The only necessary information for the

assessor is the nature of the product to be tested and the task to be performed.
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 triangle test (i.e. 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 the 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 is often determined by material conditions

(e.g. duration of the experiment, number of available assessors, quantity of product). When testing

for a difference, typical numbers of assessors are between 24 and 30. When testing for no meaningful

difference (i.e. similarity), twice as many assessors (i.e. approximately 60) 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

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SIST EN ISO 4120:2021
ISO 4120:2021(E)

each assessor perform the same number of replicate evaluations. For example, if only 10 assessors are

available, have each assessor evaluate 3 triads to obtain a total of 30 evaluations.

NOTE Treating three evaluations performed by 10 assessors as 30 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

evaluations are performed (see References [8] and [9]). References [4], [5] and [6] on replicated discrimination

tests suggest alternative approaches for analysing replicated evaluations in discrimination tests.

7 Procedure

7.1 Prepare worksheets and scoresheets (see B.1 and B.2) in advance of the test so as to utilize an

equal number of the six possible sequences of two products, A and B:
ABB AAB ABA
BAA BBA BAB

Distribute these at random in groups of six among the assessors (i.e. use each sequence once among the

first group of six assessors; use each sequence once again among the next group of six assessors, etc.).

This will minimize the imbalance that results if the total number of assessors is not a multiple of six.

NOTE The worksheets also can be prepared digitally and made available for the evaluations.

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 samples in the order in which they were presented. Inform

the assessors that two of the samples are the same and that one is different. Each assessor shall then

indicate which one of the three samples is different from the other two.

7.4 The triangle 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. The assessor has one chance out of three of giving the correct answer randomly.

7.5 The assessor shall not go back to any samples from previous triads or change the verdict on

any previous test. If an assessor is to carry out more than one test in a session, it is imperative that

the assessor shall not be able to change their response once given. For example, collect the completed

scoresheet and unused samples prior to serving the subsequent triad or do not allow the assessor to

return to an earlier answer screen once a response is confirmed.

7.6 Do not ask questions about preference, acceptance or degree of difference after the initial selection

of the odd sample. 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 for guidance.) A comment section asking why the choice was made may

be included for the assessor’s remarks.
8 Analysis and interpretation of results
8.1 When testing for a difference

Use Table A.1 to analyse the data obtained from a triangle 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

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SIST EN ISO 4120:2021
ISO 4120:2021(E)

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 triangle 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 samples (see B.2). If results are to be compared from one test to another, then the same value of p

should 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 samples that were tested, the results and the conclusions. Include

that the test was conducted in accordance with this document, i.e. ISO 4120:2021, and that the data

were analysed according to the methods given in Clause 8. The following additional information is

recommended:
— the purpose of the test and the nature of the treatment studied;

— 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);

— 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. test facility used, simultaneous or sequential presentation, if the identity

of samples was disclosed after the test and, if so, in what manner);
— the location, date of the test and name of the panel leader.
10 Precision and bias

Because results of sensory discrimination tests are a function 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 their training and with exposure to the product (e.g. the precision of the

test conducted using assessors drawn from the population of employees of the company that makes

the product would likely be greater than the precision of the test conducted using the same number of

assessors drawn from the population of naïve consumers).
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SIST EN ISO 4120:2021
ISO 4120:2021(E)

Since 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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SIST EN ISO 4120:2021
ISO 4120:2021(E)
Annex A
(normative)
Tables

A.1 Values given in Table A.1 are the minimum number of correct responses required for significance

at the stated α-risk 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 on a triangle 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 4 5 5 6 — 27 12 13 14 16 18
7 4 5 5 6 7 28 12 14 15 16 18
8 5 5 6 7 8 29 13 14 15 17 19
9 5 6 6 7 8 30 13 14 15 17 19
10 6 6 7 8 9
31 14 15 16 18 20
11 6 7 7 8 10 32 14 15 16 18 20
12 6 7 8 9 10 33 14 15 17 18 21
13 7 8 8 9 11 34 15 16 17 19 21
14 7 8 9 10 11 35 15 16 17 19 22
15 8 8 9 10 12
36 15 17 18 20 22
16 8 9 9 11 12 42 18 19 20 22 25
17 8 9 10 11 13 48 20 21 22 25 27
18 9 10 10 12 13 54 22 23 25 27 30
19 9 10 11 12 14 60 24 26 27 30 33
20 9 10 11 13 14 66 26 28 29 32 35
21 10 11 12 13 15 72 28 30 32 34 38
22 10 11 12 14 15 78 30 32 34 37 40
23 11 12 12 14 16 84 33 35 36 39 43
24 11 12 13 15 16 90 35 37 38 42 45
25 11 12 13 15 17 96 37 39 41 44 48
26 12 13 14 15 17 102 39 41 43 46 50

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 based on the normal approximation to the binomial distribution as follows. Minimum number of responses

(x) = nearest whole number greater than:
xn=+()/z32n / 9

where z varies with the significance level as follows: 0,84 for α = 0,20; 1,28 for α = 0,10; 1,64 for α = 0,05; 2,33 for α = 0,01; 3,09 for

α = 0,001.

NOTE 1 Values of n < 18 are usually not recommended for a triangle test for a difference.

NOTE 2 Adapted f
...

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