ASTM E2262-03(2020)

Standard Practice for Estimating Thurstonian Discriminal Distances

SIGNIFICANCE AND USE
5.1 Under the assumptions of the model, the Thurstonian model approach to measuring the perceived difference between two samples (whether overall or for a specific attribute) is independent of the sensory method used to collect the data. Converting results obtained from different test methods to d' values permits the assessment of relative differences among samples without requiring that the samples be compared to each other directly or that the same test methods be used for all pairs of samples.  
5.2 Thurstonian scaling has been applied to:  
5.2.1 Creating a historical database to track differences between production and reference samples over periods in which different test methods were used to measure the difference,  
5.2.2 Comparing the relative sensitivities of different user groups and consumer segments,  
5.2.3 Comparing trained panels that use different measuring techniques,  
5.2.4 Comparing the relative sensitivities of consumers versus trained panels,  
5.2.5 Comparing different methods of consumer testing (for example, CLT versus HUT, preference versus hedonic scales, etc.), and  
5.2.6 Comparing different discrimination test methods.
SCOPE
1.1 This practice describes procedures to estimate Thurstonian discriminal distances (that is, d' values) from data obtained on two samples. Procedures are presented for four forced-choice methods (that is, the triangle, the Duo-Trio, the 3-alternative-forced-choice (or 3-AFC) and the 2-AFC (also called the directional difference test)), the A/Not-A method, the Same-Different method, and for data obtained from ordered category scales. Procedures for estimating the variance of d' are also presented. Thus, confidence intervals and statistical tests can be calculated for d'.  
1.2 The procedures in this practice pertain only to the unidimensional, equal-variance model. Other, more complicated Thurstonian models, involving multiple dimensions and unequal variances exist but are not addressed in this practice. The procedure for forced-choice methods is limited to dichotomous responses. The procedure for the A/Not-A method assumes equal sample sizes for the two samples. The procedure for the Same-Different method assumes equal sample sizes for the matched and unmatched pairs of samples. For all methods, only unreplicated tests are considered. (Tests in which each assessor performs multiple (that is, replicated) evaluations require different analyses.)  
1.3 Thurstonian scaling is a method for measuring the perceptual difference between two samples based on a probabilistic model for categorical choice decision making. The magnitude of the perceived difference, δ, can be estimated from the assessors' categorical choices using the methods described in this practice. (See Appendix X3 for a more detailed description of Thurstonian scaling.)  
1.4 In theory, the Thurstonian δ does not depend on the method used to measure the difference between two samples. As such, δ provides a common scale of measure for comparing samples measured under a variety of test conditions. For example, Thurstonian scaling can be used to compare products measured under different test conditions, to compare panels (trained, consumer or both) that have evaluated the same samples (using the same or different test methods) and to compare test methods on their ability to discriminate samples that exhibit a fixed sensory difference.  
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.6 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations is...

General Information

• Status: Published
• Publication Date: 31-Jul-2020
• ICS: 67.240 - Sensory analysis
• Technical Committee: E18 - Sensory Evaluation
• Drafting Committee: E18.03 - Sensory Theory and Statistics

Relations

• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E253-19 - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 15-Oct-2019
• Refers: ASTM E253-18a - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 01-Oct-2018
• Refers: ASTM E253-18 - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 15-Jun-2018
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E253-17 - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 01-May-2017
• Refers: ASTM E253-16 - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 01-Jun-2016
• Refers: ASTM E253-15b - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 01-Dec-2015
• Refers: ASTM E253-15a - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 01-Jun-2015
• Refers: ASTM E253-15 - Standard Terminology Relating to Sensory Evaluation of Materials and Products
• Effective Date: 15-Jan-2015
• Refers: ASTM E456-13a - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013

Overview

ASTM E2262-03(2020), "Standard Practice for Estimating Thurstonian Discriminal Distances," provides a method for estimating perceptual differences between two samples based on sensory evaluation data. Developed by ASTM, this international standard guides users in calculating the statistical difference (d' values) using various sensory test methods. Its procedures allow for consistent, reliable comparison of sensory differences across samples, regardless of the data collection technique.

This standard is widely recognized in the sensory sciences, food and beverage industries, and consumer research. Its approach enables the integration of data from multiple methods, allowing companies and researchers to maintain historical records, compare panels, and assess consumer versus trained panel sensitivities.

Key Topics

• Thurstonian Scaling (d', δ Values): A probabilistic approach to quantifying sensory difference, independent of the sensory test method used.
• Applicable Sensory Methods: Procedures are provided for four forced-choice methods (triangle, Duo-Trio, 3-AFC, 2-AFC), A/Not-A test, Same-Different test, and ordered category scales.
• Unidimensional, Equal-Variance Assumption: The method applies only to a single sensory attribute (dimension) and assumes equal variance between samples.
• Conversion Across Test Methods: Results from different sensory discrimination tests can be converted to a common d' metric, facilitating direct comparison.
• Variance and Confidence Intervals: Guidance is given for estimating the variance of d', enabling calculation of confidence intervals and statistical tests.

Applications

ASTM E2262-03(2020) is practically valuable in a variety of sensory and product testing scenarios:

• Historical Comparisons: Enables the creation of historical databases tracking sensory differences between production and reference samples, even when different test methods have been used.
• Panel and Group Sensitivity Analysis: Facilitates the comparison of sensitivity among different user groups (e.g., trained panels vs. general consumers) and between panels using different measurement procedures.
• Quality Control: Supports consistent product evaluation in quality assurance processes by standardizing the measurement of perceptual differences.
• Method Evaluation: Allows comparison of different sensory test methods (e.g., CLT vs. HUT, preference vs. hedonic scaling) for their effectiveness in identifying sensory differences.
• Consumer Studies: Useful for optimizing consumer testing by evaluating sensitivity and discrimination across various demographic segments and product prototypes.

Related Standards

• ASTM E253: Terminology Relating to Sensory Evaluation of Materials and Products.
• ASTM E456: Terminology Relating to Quality and Statistics.
• ISO 5495: Sensory Analysis-Methodology-Paired Comparison.
• ASTM Manual 26: Sensory Testing Methods.

These related standards provide further context, terminology, and methodologies that complement the application and understanding of ASTM E2262-03(2020).

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Keywords: ASTM E2262-03, Thurstonian scaling, sensory discrimination, d' value, sensory difference, forced-choice methods, quality control, sensory evaluation, product testing, consumer research, panel comparison, confidence intervals, sensory standards.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E2262 − 03 (Reapproved 2020)
Standard Practice for
Estimating Thurstonian Discriminal Distances
This standard is issued under the fixed designation E2262; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope (trained, consumer or both) that have evaluated the same
samples (using the same or different test methods) and to
1.1 This practice describes procedures to estimate Thursto-
compare test methods on their ability to discriminate samples
nian discriminal distances (that is, d’ values) from data
that exhibit a fixed sensory difference.
obtained on two samples. Procedures are presented for four
1.5 This standard does not purport to address all of the
forced-choice methods (that is, the triangle, the Duo-Trio, the
safety concerns, if any, associated with its use. It is the
3-alternative-forced-choice (or 3-AFC) and the 2-AFC (also
responsibility of the user of this standard to establish appro-
calledthedirectionaldifferencetest)),theA/Not-Amethod,the
priate safety, health, and environmental practices and deter-
Same-Different method, and for data obtained from ordered
mine the applicability of regulatory limitations prior to use.
category scales. Procedures for estimating the variance of d’
1.6 This international standard was developed in accor-
are also presented. Thus, confidence intervals and statistical
dance with internationally recognized principles on standard-
tests can be calculated for d’.
ization established in the Decision on Principles for the
1.2 The procedures in this practice pertain only to the
Development of International Standards, Guides and Recom-
unidimensional, equal-variance model. Other, more compli-
mendations issued by the World Trade Organization Technical
cated Thurstonian models, involving multiple dimensions and
Barriers to Trade (TBT) Committee.
unequal variances exist but are not addressed in this practice.
Theprocedureforforced-choicemethodsislimitedtodichoto-
2. Referenced Documents
mous responses. The procedure for the A/Not-A method
2.1 ASTM Standards:
assumesequalsamplesizesforthetwosamples.Theprocedure
E253Terminology Relating to Sensory Evaluation of Mate-
for the Same-Different method assumes equal sample sizes for
rials and Products
the matched and unmatched pairs of samples. For all methods,
E456Terminology Relating to Quality and Statistics
only unreplicated tests are considered. (Tests in which each
assessor performs multiple (that is, replicated) evaluations
2.2 ASTM Manual:
require different analyses.)
Manual 26Sensory Testing Methods, 2nd Edition
1.3 Thurstonian scaling is a method for measuring the 2.3 ISO Standard:
perceptual difference between two samples based on a proba- ISO 5495Sensory Analysis—Methodology—Paired Com-
bilistic model for categorical choice decision making. The parison
magnitude of the perceived difference, δ, can be estimated
3. Terminology
from the assessors’ categorical choices using the methods
described in this practice. (See Appendix X3 for a more
3.1 Definitions:
detailed description of Thurstonian scaling.)
3.1.1 For definitions of terms relating to sensory analysis,
see Terminology E253. For terms relating to statistics, see
1.4 In theory, the Thurstonian δ does not depend on the
Terminology E456.
method used to measure the difference between two samples.
3.2 Definitions of Terms Specific to This Standard:
Assuch, δprovidesacommonscaleofmeasureforcomparing
3.2.1 δ—theThurstoniandiscriminaldistanceisthedistance
samples measured under a variety of test conditions. For
between the means of the distributions of sensory magnitudes
example,Thurstonian scaling can be used to compare products
of the two samples in the test (see Appendix X3).
measured under different test conditions, to compare panels
1 2
This practice is under the jurisdiction of ASTM Committee E18 on Sensory For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Evaluation and is the direct responsibility of Subcommittee E18.03 on Sensory contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
Theory and Statistics. Standards volume information, refer to the standard’s Document Summary page on
Current edition approved Aug. 1, 2020. Published September 2020. Originally the ASTM website.
approved in 2003. Last previous edition approved in 2014 as E2262–03 (2014). Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
DOI: 10.1520/E2262-03R20. 4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2262 − 03 (2020)
3.2.2 d’—the statistic used to estimate δ based on the data “A” sample and the other sample is selected to be the “Not-A”
obtained from the test. sample. Choice proportions are tallied for each sample and the
values of d’ and its variance, S (d’), are obtained from Tables
3.2.3 choice proportion (P )—the expected proportion of
c
X1.9 and X1.10, respectively, by the same techniques used in
responses from a forced-choice method. (For example, if there
the A/Not-A method.
is no perceptible difference between the samples in a triangle
test, P = 1/3. If there is a perceptible difference, P > 1/3.)
c c
5. Significance and Use
3.2.4 observed choice proportion (p )—the statistic used to
c
5.1 Under the assumptions of the model, the Thurstonian
estimate choice proportion, P , where p = x/n, where x is the
c c
modelapproachtomeasuringtheperceiveddifferencebetween
observednumberofcorrectresponsesand nisthesamplesize.
two samples (whether overall or for a specific attribute) is
independent of the sensory method used to collect the data.
4. Summary of Practice
Converting results obtained from different test methods to d’
4.1 Determinethetypeofdatacollectedonthetwosamples:
values permits the assessment of relative differences among
data from a forced-choice test, an A/Not-A test, a Same-
samples without requiring that the samples be compared to
Different test or an ordered category scale.
eachotherdirectlyorthatthesametestmethodsbeusedforall
pairs of samples.
4.2 For forced-choice tests, reference the table that corre-
spondstothetestmethod(thatis,triangletest(TablesX1.1and
5.2 Thurstonian scaling has been applied to:
X1.2), Duo-Trio test (Tables X1.3 and X1.4), 3-AFC test
5.2.1 Creating a historical database to track differences
(Tables X1.5 and X1.6), or 2-AFC test (Tables X1.7 and
between production and reference samples over periods in
X1.8)). Identify the entry in the table closest to the observed
which different test methods were used to measure the
choice proportion (p ) from the test. Read the estimated value
c
difference,
of δ (that is, d’) from the corresponding row and column
5.2.2 Comparing the relative sensitivities of different user
headings of the table. Estimate the variance of d’ by referenc-
groups and consumer segments,
ing the appropriate table for the test method. Find the value of
5.2.3 Comparingtrainedpanelsthatusedifferentmeasuring
B that corresponds to the value of d’ obtained in the first step
techniques,
(see Note 1). The estimated variance of d’is S (d’) = B/n,
5.2.4 Comparing the relative sensitivities of consumers
where n is the sample size. Use the estimates d’ and S (d’) to
versus trained panels,
constructconfidenceintervalsandtestsofhypothesesrelatedto
5.2.5 Comparing different methods of consumer testing (for
the objectives of the research.
example, CLT versus HUT, preference versus hedonic scales,
etc.), and
NOTE 1—The variance of d’ is a complicated function of the true value
5.2.6 Comparing different discrimination test methods.
ofδandthedecisionrulewhenassociatedwiththetestmethodbeingused
(see Appendix X3). However, regardless of the test method, the variance
of d’ can always be expressed as S (d’)= B/n, where the parameter B
6. Procedure
captures all of the information concerning the test method, and n is the
6.1 Forced-choice Methods—The relationship between δ
sample size. The values of B have been tabulated to make the calculation
of the variance of d’ a simple task.
andtheexpectedchoiceproportion, P ,isdifferentfordifferent
c
forced-choice methods because the decision rule used by the
4.3 For the A/Not-A method, tally the observed choice
assessors varies from one method to another (see Appendix
proportions of “A” responses for the A sample and the “A”
X3). As a result, different tables are required to estimate δ
responses for the Not-A sample. Read the value of d’ from
depending on the method used. Tables for the four most
Table X1.9 in the column that corresponds to the observed
commonlyusedmethodsarepresented.Theestimatedvalueof
choice proportion of the “A” responses for the Not-A sample
δ (that is, d’) is obtained as follows:
(p ) and the row that corresponds to the observed choice
na
6.1.1 Compute the observed choice proportion as p = x/n,
proportion of the “A” responses for the A sample (p ). The c
a
where x is the observed number of correct responses and n is
same method is used to estimate the variance of d’, S (d’),
the sample size.
using Table X1.10.
6.1.2 Obtain d’ by entering the table in Appendix X1 that
4.4 For the Same-Different method, tally the proportion of
correspondstothetestmethodused:triangletest(TableX1.1),
“same” responses for the matched pairs of samples (that is,
Duo-Trio (Table X1.3), 3-AFC (Table X1.5), or 2-AFC (Table
A/A or B/B) and the proportion of “same” responses for the
X1.7).Findtheentryinthetablethatisclosesttotheobserved
unmatched pairs of samples (that is, A/B or B/A). Read the
value of p . The value of d’, accurate to one decimal place, is
c
valueof d’fromTableX1.11inthecolumnthatcorrespondsto
the row-label of the table corresponding to the selected entry.
the observed proportion of “same” responses for the un-
Theseconddecimalplaceof d’isthecolumn-labelofthetable
matched pairs (p / ) and the row that corresponds to the
s u
corresponding to the selected entry.
observed proportion of the “same” responses for the matched
6.1.3 Obtain the estimated variance of d’ as follows. Enter
pairs (p / ). The same method is used to estimate the variance
s m
the appropriate table in Appendix X1: triangle test (Table
of d’, S (d’), using Table X1.12.
X1.2), Duo-Trio (Table X1.4), 3-AFC (Table X1.6), or 2-AFC
4.5 For ordered category scales, a rapid, table-look-up (Table X1.8). Find the value of B in the row and column that
approach is used. For each sample, the category scale data are correspond to the value of d’ obtained in 6.1.2. Compute the
collapsed into two categories. One sample is selected to be the estimated variance of d’as S (d’) = B/n, where n is the sample
E2262 − 03 (2020)
size. Use the estimates d’ and S (d’) to construct confidence each sample separately and record the totals in the two-by-two
intervalsandtestsofhypothesesrelatedtotheobjectivesofthe table under “High” (that is, the x and x tallies, below).
na a
research.
Sample Low High
Not-A y x
na na
6.2 A/Not-A Method—Compute the choice proportions of
A y x
a a
the two samples, p = x /n and p = x /n, where x is the
a a na na a
6.4.3 Compute the choice proportions of the two samples,
number of times the “A” sample is chosen as being “A,” x is
na
p = x /n and p = x /n, where x and x are obtained from
a a na na a na
the number of times the “Not-A” sample is chosen as being
the table above and n is the sample size, common to both
“A” and n is the sample size.
samples.
NOTE 2—This practice only considers the case where the number of
6.4.4 ApplythesametechniqueusedintheA/Not-Amethod
“A” samples equals the number of “Not-A” samples, n = n = n .
a na
(see 6.2). Read the value of d’ from Table X1.9 in Appendix
X1 in the column that corresponds to the observed choice
6.2.1 Readthevalueof d’fromTableX1.9inAppendixX1
proportion of the Not-A sample (p ) and the row that
in the column that corresponds to the observed choice propor-
na
corresponds to the observed choice proportion of theAsample
tion of the “Not-A” sample (p ) and the row that corresponds
na
(p ).
to the observed choice proportion of the “A” sample (p ).
a
a
6.2.2 To obtain an estimate of the variance of d’, read the 6.4.5 To obtain an estimate of the variance of d’, read the
value of B from Table X1.10 in Appendix X1 using the same
value of B from Table X1.10 in Appendix X1 using the same
2 2
technique as in 6.2.1. The variance estimate is S (d’) = B/n, technique as in 6.4.4. The variance estimate is S (d’) = B/n,
where n is the sample size.
where n is the sample size.
6.3 Same-Different Method—Compute the choice propor- 6.5 Statistical Tests and Confidence Intervals—Often the
tions for the matched (m) and unmatched (u) pairs of samples, objective of a sensory discrimination test is to determine if the
p = x /n and p = x /n, where x is the number of samples in the test are perceptibly different. In other instances
s/m s/m s/u s/u s/m
“same” responses for the matched pairs (A/A or B/B)
it is of interest to obtain an estimate of the size of the
evaluated, x is the number of “same” responses for the perceptible difference (and to measure the precision of the
s/u
unmatched pair and n is the number of matched or unmatched
estimated difference). Because testing for a difference and
pairs evaluated. estimating the size of a difference address different goals, it is
not surprising that different statistical methods apply to each.
NOTE 3—This practice only considers the case where the number of
For the purpose of testing if a perceptible difference exists, the
matched pairs equals the number of unmatched pairs, n = n = n .
m u
binomial and chi-square tests traditionally associated with the
6.3.1 Read the value of d’ from Table X1.11 in Appendix
test methods discussed in this practice are appropriate. For the
X1 in the column that corresponds to the observed proportion
purposes of estimating the size of the difference and assessing
of “same” responses for unmatched pair (p ) and the row that
s/u
the precision of that estimate, confidence intervals are appro-
corresponds to the observed proportion of “same” responses
priate. Because δ is the difference between the means of two
for the matched pair (p ).
s/m
normal distributions and d’ is an estimate of δ, it can be
6.3.2 To obtain an estimate of the variance of d’, read the
assumed that d’ is approximately normally distributed. Based
value of B from Table X1.12 in Appendix X1 using the same
on this assumption, statistical confidence intervals concerning
technique as in 6.3.1. The variance estimate is S (d’) = B/n,
δ can be constructed using traditional techniques.
where n is the sample size.
6.5.1 A 100(1 - α) % two-sided confidence interval on δ is
calculatedas: d’ 6 Z S(d’),where d’istheestimatedvalueof
6.4 Ordered Category Scales—A rapid, table-look-up α/2
δ, Z istheupper-α/2percentagepointofthestandardnormal
method is described. The method collapses the category-scale
α/2
distribution(forexample,fora90%confidenceinterval Z =
dataintotwocategories,regardlessofthenumberofcategories α/2
1.65;fora95%confidenceintervalZ =1.96;etc.),andS(d’)
on the physical scale used to collect the data. It is recognized
α/2
isthestandarddeviationof d’,thatis,thesquarerootof, S (d’)
that information detail is lost by collapsing the data into two
= B/n. Similarly, 100(1 - α) % one-sided confidence intervals
categories. However, the estimates of d’ and its variance,
on δ are calculated as: d’+ Z S(d’) for a one-sided upper
S (d’), obtained from the technique are accurate. The compu- α
confidence interval and d’− Z S(d’) for a one-sided lower
tational ease offsets the small loss of accuracy incurred.
α
confidence interval, where Zα is the upper-α percentage point
6.4.1 Tally the frequency distributions of category scale
of the standard normal distribution (for example, for a 90%
ratings for the two samples. Select the sample with the lower
confidenceintervalZ =1.28;fora95%confidenceintervalZ
median rating to be the Not-Asample. Select the sample with α α
= 1.65; etc.) and d’ and S(d’) are as defined above.
the higher median rating to be the A sample.
6.5.2 To test if δ is greater than zero, that is, that the two
6.4.2 Collapse the frequency data for each sample into two
samplesinthetestareperceptiblydifferent,usethebinomialor
categories as follows. Identify the category in which the
chi-squaretestthatistraditionallyassociatedwiththediscrimi-
median of the Not-A sample occurs. Pool the number of
nation method used.
responses from that category and all lower categories for each
sampleseparatelyandrecordthetotalsinthetwo-by-twotable 6.5.3 To test if it is reasonable to believe two δ’s have the
samevalue,thatis,totestthehypothesesH :δ =δ versusH :
under “Low” (that is, the y and y tallies, below). Pool the
na a 0 1 2 a
number of responses for the remaining, higher categories for δ ≠δ form the ratio:
1 2
E2262 − 03 (2020)
d’2d’
? 2 ? d ’ = estimated value of δ ,
2 2
T 5
2 2
S = variances of d ’, and
=S 1S 1 1
1 2
S = variances of d ’.
2 2
where:
If T> Z ,thenconcludethetwo δvaluesareunequalatthe
α/2
d ’ = estimated value of δ ,
1 1
α-level of significance.
APPENDIXES
(Nonmandatory Information)
X1. STATISTICAL TABLES
4 A
TABLE X1.1 Observed Choice Proportions, p , (×10 ) as Function of d’ for Triangle Test
c
NOTE 1—Find the entry in the table closest to the choice proportion observed in the test. Read the estimated value of d’ from the corresponding row
and column headings.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 3333 3333 3334 3334 3335 3336 3337 3338 3339 3341
0.1 3343 3344 3347 3349 3351 3354 3357 3360 3363 3366
0.2 3370 3374 3378 3382 3386 3390 3395 3400 3405 3410
0.3 3415 3421 3427 3432 3439 3445 3451 3458 3464 3471
0.4 3478 3486 3493 3501 3508 3516 3524 3533 3541 3550
0.5 3558 3567 3576 3586 3595 3604 3614 3624 3634 3644
0.6 3654 3665 3676 3686 3697 3708 3719 3731 3742 3754
0.7 3766 3778 3790 3802 3814 3827 3839 3852 3865 3878
0.8 3891 3905 3918 3932 3945 3959 3973 3987 4001 4016
0.9 4030 4045 4059 4074 4089 4104 4119 4134 4149 4165
1.0 4180 4196 4212 4228 4244 4260 4276 4292 4309 4325
1.1 4342 4358 4375 4392 4409 4426 4443 4460 4477 4494
1.2 4512 4529 4547 4564 4582 4600 4618 4636 4654 4672
1.3 4690 4708 4726 4745 4763 4782 4800 4819 4837 4856
1.4 4875 4893 4912 4931 4950 4969 4988 5007 5026 5045
1.5 5065 5084 5103 5122 5142 5161 5180 5200 5219 5239
1.6 5258 5278 5297 5317 5337 5356 5376 5396 5415 5435
1.7 5455 5474 5494 5514 5534 5554 5573 5593 5613 5633
1.8 5653 5672 5692 5712 5732 5752 5771 5791 5811 5831
1.9 5851 5870 5890 5910 5930 5950 5969 5989 6009 6028
2.0 6048 6068 6087 6107 6127 6146 6166 6185 6205 6224
2.1 6244 6263 6283 6302 6321 6341 6360 6379 6398 6418
2.2 6437 6456 6475 6494 6513 6532 6551 6570 6589 6608
2.3 6627 6645 6664 6683 6701 6720 6739 6757 6776 6794
2.4 6812 6831 6849 6867 6885 6903 6922 6940 6958 6976
2.5 6993 7011 7029 7047 7064 7082 7100 7117 7135 7152
2.6 7169 7187 7204 7221 7238 7255 7272 7289 7306 7323
2.7 7340 7356 7373 7390 7406 7423 7439 7455 7472 7488
2.8 7504 7520 7536 7552 7568 7584 7600 7616 7631 7647
2.9 7662 7678 7693 7709 7724 7739 7754 7769 7784 7799
3.0 7814 7829 7844 7859 7873 7888 7902 7917 7931 7945
3.1 7960 7974 7988 8002 8016 8030 8044 8057 8071 8085
3.2 8098 8112 8125 8139 8152 8165 8179 8192 8205 8218
3.3 8231 8243 8256 8269 8282 8294 8307 8319 8332 8344
3.4 8356 8368 8381 8393 8405 8417 8428 8440 8452 8464
3.5 8475 8487 8498 8510 8521 8532 8544 8555 8566 8577
3.6 8588 8599 8610 8620 8631 8642 8652 8663 8673 8684
3.7 8694 8704 8715 8725 8735 8745 8755 8765 8775 8784
3.8 8794 8804 8813 8823 8833 8842 8851 8861 8870 8879
3.9 8888 8897 8906 8915 8924 8933 8942 8951 8959 8968
4.0 8977 8985 8994 9002 9010 9019 9027 9035 9043 9051
4.1 9059 9067 9075 9083 9091 9099 9106 9114 9122 9129
4.2 9137 9144 9151 9159 9166 9173 9180 9188 9195 9202
4.3 9209 9216 9223 9229 9236 9243 9250 9256 9263 9269
4.4 9276 9282 9289 9295 9301 9308 9314 9320 9326 9332
4.5 9338 9344 9350 9356 9362 9368 9374 9379 9385 9391
4.6 9396 9402 9407 9413 9418 9424 9429 9434 9440 9445
4.7 9450 9455 9460 9465 9470 9475 9480 9485 9490 9495
4.8 9500 9504 9509 9514 9518 9523 9528 9532 9537 9541
4.9 9546 9550 9554 9559 9563 9567 9571 9575 9580 9584
5.0 9588 9592 9596 9600 9604 9608 9612 9615 9619 9623
5.1 9627 9630 9634 9638 9641 9645 9648 9652 9655 9659
5.2 9662 9666 9669 9673 9676 9679 9682 9686 9689 9692
5.3 9695 9698 9701 9704 9707 9710 9713 9716 9719 9722
E2262 − 03 (2020)
TABLE X1.1 Continued
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
5.4 9725 9728 9731 9734 9736 9739 9742 9745 9747 9750
5.5 9753 9755 9758 9760 9763 9765 9768 9770 9773 9775
5.6 9778 9780 9782 9785 9787 9789 9792 9794 9796 9798
5.7 9800 9803 9805 9807 9809 9811 9813 9815 9817 9819
5.8 9821 9823 9825 9827 9829 9831 9833 9834 9836 9838
5.9 9840 9842 9843 9845 9847 9849 9850 9852 9854 9855
6.0 9857 9859 9860 9862 9863 9865 9866 9868 9869 9871
A
Adapted from Ennis, D. M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp. 353–370.
A
TABLE X1.2 The B Values for Estimating Variance of d’ Obtained From Triangle Test
NOTE 1—Enter the table in the row and column corresponding to the value of d’ obtained from Table X1.1. The variance of d’ is S (d’)= B/n, where
n is the sample size.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 ******* 65800. 16452. 7314. 4115. 2635. 1831. 1346. 1031. 815.
0.1 661.08 546.88 460.03 392.44 338.81 295.54 260.13 230.78 206.19 185.38
0.2 167.61 152.31 139.06 127.50 117.35 108.40 100.45 93.38 87.05 81.36
0.3 76.236 71.597 67.387 63.554 60.054 56.850 53.910 51.205 48.711 46.406
0.4 44.273 42.294 40.455 38.742 37.146 35.655 34.261 32.954 31.729 30.578
0.5 29.496 28.477 27.517 26.611 25.755 24.945 24.179 23.452 22.764 22.110
0.6 21.489 20.898 20.336 19.801 19.291 18.805 18.341 17.897 17.474 17.069
0.7 16.681 16.310 15.954 15.614 15.287 14.973 14.672 14.383 14.106 13.839
0.8 13.582 13.335 13.097 12.868 12.647 12.435 12.230 12.032 11.841 11.657
0.9 11.479 11.308 11.142 10.982 10.827 10.677 10.532 10.392 10.256 10.125
1.0 9.998 9.875 9.756 9.641 9.529 9.421 9.316 9.214 9.115 9.019
1.1 8.926 8.836 8.749 8.664 8.582 8.502 8.424 8.349 8.276 8.205
1.2 8.136 8.069 8.004 7.941 7.880 7.820 7.762 7.706 7.651 7.598
1.3 7.547 7.497 7.448 7.401 7.355 7.310 7.267 7.225 7.184 7.144
1.4 7.106 7.068 7.032 6.997 6.962 6.929 6.897 6.866 6.835 6.806
1.5 6.778 6.750 6.723 6.697 6.672 6.648 6.624 6.601 6.579 6.558
1.6 6.538 6.518 6.499 6.480 6.462 6.445 6.429 6.413 6.398 6.383
1.7 6.369 6.355 6.342 6.330 6.318 6.307 6.296 6.286 6.276 6.267
1.8 6.258 6.250 6.242 6.235 6.228 6.222 6.216 6.211 6.206 6.201
1.9 6.197 6.193 6.190 6.187 6.185 6.183 6.181 6.180 6.179 6.179
2.0 6.178 6.179 6.179 6.180 6.182 6.184 6.186 6.188 6.191 6.194
2.1 6.198 6.201 6.206 6.210 6.215 6.220 6.226 6.231 6.238 6.244
2.2 6.251 6.258 6.265 6.273 6.281 6.289 6.298 6.307 6.316 6.326
2.3 6.336 6.346 6.356 6.367 6.378 6.389 6.401 6.413 6.425 6.438
2.4 6.450 6.463 6.477 6.490 6.504 6.518 6.533 6.548 6.563 6.578
2.5 6.594 6.609 6.626 6.642 6.659 6.676 6.693 6.710 6.728 6.746
2.6 6.765 6.783 6.802 6.821 6.841 6.861 6.881 6.901 6.921 6.942
2.7 6.963 6.985 7.007 7.029 7.051 7.073 7.096 7.119 7.143 7.166
2.8 7.190 7.214 7.239 7.264 7.289 7.314 7.340 7.366 7.392 7.418
2.9 7.445 7.472 7.500 7.527 7.555 7.584 7.612 7.641 7.670 7.700
3.0 7.729 7.760 7.790 7.821 7.852 7.883 7.914 7.946 7.979 8.011
3.1 8.044 8.077 8.111 8.144 8.179 8.213 8.248 8.283 8.318 8.354
3.2 8.390 8.427 8.463 8.500 8.538 8.576 8.614 8.652 8.691 8.730
3.3 8.770 8.810 8.850 8.890 8.931 8.973 9.014 9.056 9.099 9.142
3.4 9.185 9.228 9.272 9.317 9.361 9.406 9.452 9.498 9.544 9.591
3.5 9.638 9.685 9.733 9.781 9.830 9.879 9.929 9.979 10.029 10.080
3.6 10.131 10.183 10.235 10.287 10.340 10.394 10.448 10.502 10.557 10.612
3.7 10.668 10.724 10.781 10.838 10.896 10.954 11.012 11.072 11.131 11.191
3.8 11.252 11.313 11.375 11.437 11.500 11.563 11.627 11.691 11.756 11.821
3.9 11.887 11.954 12.021 12.089 12.157 12.225 12.295 12.365 12.435 12.506
4.0 12.578 12.651 12.723 12.797 12.871 12.946 13.022 13.098 13.174 13.252
4.1 13.330 13.409 13.488 13.568 13.649 13.730 13.812 13.895 13.979 14.063
4.2 14.148 14.233 14.320 14.407 14.495 14.584 14.673 14.673 14.854 14.946
4.3 15.039 15.132 15.226 15.321 15.417 15.514 15.611 15.709 15.809 15.909
4.4 16.009 16.111 16.214 16.318 16.422 16.527 16.634 16.741 16.849 16.958
4.5 17.068 17.180 17.292 17.405 17.519 17.634 17.750 17.867 17.985 18.104
4.6 18.225 18.346 18.468 18.592 18.717 18.842 18.969 19.097 19.226 19.357
4.7 19.488 19.621 19.755 19.890 20.026 20.164 20.303 20.443 20.584 20.727
4.8 20.871 21.016 21.163 21.310 21.460 21.610 21.763 21.916 22.071 22.227
4.9 22.385 22.544 22.705 22.867 23.031 23.196 23.363 23.531 23.701 23.872
5.0 24.046 24.220 24.397 24.575 24.754 24.936 25.119 25.304 25.491 25.679
5.1 25.869 26.061 26.255 26.451 26.648 26.847 27.049 27.252 27.457 27.665
A
Adapted from Bi, J., Ennis, D. M., and O’Mahony, M., “How to Estimate and Use the Variance of d’ from Difference Tests,” Journal of Sensory Studies, 12, 1997,
pp. 87–104.
E2262 − 03 (2020)
4 A
TABLE X1.3 Observed Choice Proportions, p , (×10 ) as Function of d’ for Duo-Trio Test
c
NOTE 1—Find the entry in the table closest to the choice proportion observed in the test. Read the estimated value of d’ from the corresponding row
and column headings.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 5000 5000 5000 5001 5001 5002 5003 5005 5006 5007
0.1 5009 5011 5013 5015 5018 5021 5023 5026 5030 5033
0.2 5037 5040 5044 5048 5053 5057 5062 5066 5071 5077
0.3 5082 5087 5093 5099 5105 5111 5117 5124 5131 5137
0.4 5144 5152 5159 5166 5174 5182 5190 5198 5206 5215
0.5 5223 5232 5241 5250 5259 5269 5278 5288 5298 5308
0.6 5318 5328 5339 5349 5360 5371 5382 5393 5404 5415
0.7 5427 5438 5450 5462 5474 5486 5498 5511 5523 5536
0.8 5548 5561 5574 5587 5600 5614 5627 5641 5654 5668
0.9 5682 5695 5709 5724 5738 5752 5766 5781 5795 5810
1.0 5825 5840 5854 5869 5884 5900 5915 5930 5945 5961
1.1 5976 5992 6007 6023 6039 6055 6071 6087 6102 6119
1.2 6135 6151 6167 6183 6200 6216 6232 6249 6265 6282
1.3 6298 6315 6331 6348 6365 6381 6398 6415 6432 6448
1.4 6465 6482 6499 6516 6533 6550 6567 6584 6601 6618
1.5 6635 6652 6669 6686 6703 6720 6737 6754 6771 6788
1.6 6805 6822 6839 6856 6873 6890 6907 6923 6940 6957
1.7 6974 6991 7008 7025 7042 7058 7075 7092 7109 7125
1.8 7142 7159 7175 7192 7208 7225 7241 7258 7274 7291
1.9 7307 7323 7340 7356 7372 7388 7404 7420 7436 7452
2.0 7468 7484 7500 7516 7531 7547 7563 7578 7594 7610
2.1 7625 7640 7656 7671 7686 7701 7717 7732 7747 7762
2.2 7777 7791 7806 7821 7836 7850 7865 7879 7894 7908
2.3 7923 7937 7951 7965 7979 7993 8007 8021 8035 8049
2.4 8062 8076 8090 8103 8117 8130 8143 8156 8170 8183
2.5 8196 8209 8222 8235 8247 8260 8273 8285 8298 8310
2.6 8323 8335 8347 8360 8372 8384 8396 8408 8420 8431
2.7 8443 8455 8466 8478 8489 8501 8512 8523 8535 8546
2.8 8557 8568 8579 8590 8600 8611 8622 8632 8643 8653
2.9 8664 8674 8685 8695 8705 8715 8725 8735 8745 8755
3.0 8765 8774 8784 8794 8803 8813 8822 8831 8841 8850
3.1 8859 8868 8877 8886 8895 8904 8913 8921 8930 8939
3.2 8947 8956 8964 8973 8981 8989 8997 9006 9014 9022
3.3 9030 9038 9046 9053 9061 9069 9077 9084 9092 9099
3.4 9107 9114 9121 9129 9136 9143 9150 9157 9164 9171
3.5 9178 9185 9192 9199 9206 9212 9219 9225 9232 9239
3.6 9245 9251 9258 9264 9270 9276 9283 9289 9295 9301
3.7 9307 9313 9319 9324 9330 9336 9342 9347 9353 9359
3.8 9364 9370 9375 9381 9386 9391 9397 9402 9407 9412
3.9 9417 9423 9428 9433 9438 9443 9447 9452 9457 9462
4.0 9467 9471 9476 9481 9485 9490 9494 9499 9503 9508
4.1 9512 9517 9521 9525 9529 9534 9538 9542 9546 9550
4.2 9554 9558 9562 9566 9570 9574 9578 9582 9586 9589
4.3 9593 9597 9601 9604 9608 9612 9615 9619 9622 9626
4.4 9629 9633 9636 9639 9643 9646 9649 9653 9656 9659
4.5 9662 9665 9669 9672 9675 9678 9681 9684 9687 9690
4.6 9693 9696 9698 9701 9704 9707 9710 9713 9715 9718
4.7 9721 9723 9726 9729 9731 9734 9736 9739 9742 9744
4.8 9747 9749 9751 9754 9756 9759 9761 9763 9766 9768
4.9 9770 9772 9775 9777 9779 9781 9783 9786 9788 9790
5.0 9792 9794 9796 9798 9800 9802 9804 9806 9808 9810
5.1 9812 9814 9816 9817 9819 9821 9823 9825 9827 9828
5.2 9830 9832 9833 9835 9837 9839 9840 9842 9843 9845
5.3 9847 9848 9850 9851 9853 9854 9856 9857 9859 9860
5.6 9862 9863 9865 9866 9868 9869 9870 9872 9873 9874
5.7 9876 9877 9878 9880 9881 9882 9884 9885 9886 9887
5.8 9888 9890 9891 9892 9893 9894 9895 9897 9898 9899
5.9 9900 9901 9902 9903 9904 9905 9906 9907 9908 9909
6.0 9910 9911 9912 9913 9914 9915 9916 9917 9918 9919
6.1 9920 9921 9922 9922 9923 9924 9925 9926 9927 9928
6.2 9928 9929 9930 9931 9932 9932 9933 9934 9935 9935
A
Adapted from Ennis, D. M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp. 353–370.
E2262 − 03 (2020)
A
TABLE X1.4 B Values for Estimating Variance of d’ Obtained From Duo-Trio Test
NOTE 1—Enter the table in the row and column corresponding to the value of d’ obtained from Table X1.3. The variance of d’ is S (d’)= B/n, where
n is the sample size.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 ***** 74025 18508 8228 4630 2964 2059 1514 1160 917
0.1 743.52 615.05 517.34 441.30 380.96 332.29 292.45 259.43 231.77 208.35
0.2 188.36 171.16 156.25 143.24 131.83 121.75 112.82 104.86 97.74 91.34
0.3 85.576 80.358 75.622 71.310 67.374 63.770 60.463 57.420 54.615 52.024
0.4 49.624 47.398 45.330 43.405 41.610 39.933 38.365 36.897 35.520 34.226
0.5 33.010 31.864 30.785 29.766 28.804 27.895 27.034 26.218 25.444 24.710
0.6 24.012 23.349 22.719 22.118 21.545 21.000 20.479 19.981 19.506 19.052
0.7 18.618 18.202 17.803 17.422 17.056 16.705 16.368 16.044 15.733 15.435
0.8 15.148 14.872 14.606 14.350 14.104 13.867 13.638 13.418 13.205 13.000
0.9 12.803 12.612 12.427 12.249 12.078 11.911 11.751 11.596 11.446 11.301
1.0 11.160 11.025 10.893 10.766 10.643 10.524 10.409 10.297 10.189 10.084
1.1 9.983 9.885 9.789 9.697 9.608 9.521 9.437 9.356 9.277 9.201
1.2 9.127 9.055 8.986 8.918 8.853 8.790 8.729 8.669 8.612 8.556
1.3 8.502 8.450 9.400 8.351 8.304 8.258 8.214 8.171 8.130 8.090
1.4 8.051 8.014 7.978 7.943 7.910 7.878 7.847 7.817 7.788 7.760
1.5 7.734 7.709 7.684 7.661 7.638 7.617 7.597 7.577 7.559 7.541
1.6 7.524 7.509 7.494 7.480 7.466 7.454 7.442 7.432 7.422 7.412
1.7 7.404 7.396 7.389 7.383 7.378 7.373 7.369 7.365 7.363 7.361
1.8 7.360 7.359 7.359 7.360 7.361 7.363 7.366 7.369 7.373 7.377
1.9 7.382 7.388 7.395 7.401 7.409 7.417 7.426 7.435 7.445 7.456
2.0 7.467 7.478 7.490 7.503 7.516 7.530 7.545 7.560 7.575 7.591
2.1 7.608 7.625 7.643 7.661 7.680 7.699 7.719 7.740 7.761 7.782
2.2 7.804 7.827 7.850 7.874 7.898 7.923 7.948 7.974 8.000 8.027
2.3 8.055 8.083 8.111 8.140 8.170 8.200 8.231 8.262 8.294 8.326
2.4 8.359 8.392 8.426 8.461 8.496 8.531 8.567 8.604 8.641 8.679
2.5 8.717 8.756 8.796 8.836 8.876 8.917 8.959 9.001 9.044 9.088
2.6 9.132 9.176 9.221 9.267 9.313 9.360 9.408 9.456 9.505 9.554
2.7 9.604 9.654 9.705 9.757 9.809 9.862 9.916 9.970 10.024 10.080
2.8 10.136 10.192 10.250 10.308 10.366 10.425 10.485 10.545 10.606 10.668
2.9 10.731 10.794 10.857 10.922 10.987 11.052 11.119 11.186 11.254 11.322
3.0 11.391 11.461 11.531 11.603 11.675 11.747 11.820 11.894 11.969 12.045
3.1 12.121 12.198 12.275 12.354 12.433 12.513 12.593 12.675 12.757 12.839
3.2 12.923 13.007 13.093 13.179 13.265 13.353 13.441 13.530 13.620 13.711
3.3 13.802 13.894 13.987 14.081 14.176 14.271 14.368 14.465 14.653 14.662
3.4 14.762 14.862 14.964 15.066 15.169 15.273 15.378 15.484 15.591 15.698
3.5 15.807 15.916 16.026 16.137 16.250 16.363 16.477 16.591 16.707 16.824
3.6 16.942 17.061 17.180 17.301 17.422 17.545 17.668 17.793 17.919 18.045
3.7 18.173 18.301 18.431 18.561 18.693 18.826 18.960 19.094 19.230 19.367
3.8 19.505 19.644 19.784 19.926 20.068 20.212 20.356 20.502 20.649 20.797
3.9 20.946 21.096 21.248 21.400 21.554 21.709 21.866 22.023 22.182 22.342
4.0 22.503 22.665 22.829 22.994 23.160 23.327 23.496 23.666 23.838 24.010
4.1 24.184 24.360 24.536 24.714 24.894 25.075 25.257 25.441 25.626 25.812
4.2 26.000 26.189 26.380 26.573 26.767 26.962 27.159 27.357 27.557 27.759
4.3 27.962 28.166 28.373 28.580 28.790 29.001 29.214 29.428 29.645 29.862
4.4 30.082 30.303 30.526 30.751 30.978 31.206 31.436 31.668 31.902 32.138
4.5 32.376 32.615 32.857 33.100 33.345 33.593 33.842 34.093 34.347 34.602
4.6 34.859 35.119 35.381 35.645 35.911 36.179 36.449 36.722 36.996 37.274
4.7 37.553 37.835 38.119 38.405 38.694 38.985 39.279 39.575 39.873 40.174
4.8 40.478 40.784 41.093 41.404 41.718 42.035 42.355 42.677 43.001 43.329
4.9 43.660 43.993 44.329 44.668 45.010 45.356 45.704 46.054 46.409 46.766
5.0 47.127 47.490 47.857 48.227 48.600 48.976 49.357 49.740 50.127 50.517
5.1 50.911 51.308 51.709 52.113 52.521 52.933 53.349 53.768 54.191 54.618
A
Adapted from Bi, J., Ennis, D. M., and O’Mahony, M., “How to Estimate and Use the Variance of d’ from Difference Tests,” Journal of Sensory Studies, 12, 1997,
pp. 87–104.
4 A
TABLE X1.5 Observed Choice Proportions, p , (×10 ) as Function of d’ for 3-AFC Test
c
NOTE 1—Find the entry in the table closest to the choice proportion observed in the test. Read the estimated value of d’ from the corresponding row
and column headings.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 3333 3362 3390 3418 3447 3475 3504 3533 3562 3591
0.1 3620 3649 3678 3707 3737 3766 3795 3825 3855 3884
0.2 3914 3944 3974 4003 4033 4063 4093 4124 4154 4184
0.3 4214 4244 4275 4305 4336 4366 4396 4427 4458 4488
0.4 4519 4549 4580 4611 4641 4672 4703 4734 4764 4795
0.5 4826 4857 4888 4918 4949 4980 5011 5042 5072 5103
0.6 5134 5165 5195 5226 5257 5288 5318 5349 5380 5410
0.7 5441 5471 5502 5532 5563 5593 5624 5654 5684 5714
0.8 5745 5775 5805 5835 5865 5895 5925 5955 5985 6014
0.9 6044 6074 6103 6133 6162 6191 6221 6250 6279 6308
1.0 6337 6366 6395 6423 6452 6481 6509 6538 6566 6594
E2262 − 03 (2020)
TABLE X1.5 Continued
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
1.1 6622 6650 6678 6706 6734 6761 6789 6816 6844 6871
1.2 6898 6925 6952 6979 7005 7032 7059 7085 7111 7137
1.3 7163 7189 7215 7241 7266 7292 7317 7342 7367 7392
1.4 7417 7442 7466 7491 7515 7539 7563 7587 7611 7635
1.5 7658 7682 7705 7728 7751 7774 7796 7819 7842 7864
1.6 7886 7908 7930 7952 7973 7995 8016 8037 8058 8079
1.7 8100 8121 8141 8162 8182 8202 8222 8242 8261 8281
1.8 8300 8319 8339 8357 8376 8395 8413 8432 8450 8468
1.9 8486 8504 8522 8539 8556 8574 8591 8608 8624 8641
2.0 8658 8674 8690 8706 8722 8738 8754 8769 8785 8800
2.1 8815 8830 8845 8860 8874 8889 8903 8917 8931 8945
2.2 8959 8973 8986 9000 9013 9026 9039 9052 9065 9077
2.3 9090 9102 9114 9127 9138 9150 9162 9174 9185 9197
2.4 9208 9219 9230 9241 9252 9262 9273 9283 9293 9304
2.5 9314 9324 9333 9343 9353 9362 9372 9381 9390 9399
2.6 9408 9417 9426 9434 9443 9451 9460 9468 9476 9484
2.7 9492 9500 9508 9515 9523 9530 9538 9545 9552 9559
2.8 9566 9573 9580 9587 9593 9600 9606 9613 9619 9625
2.9 9631 9637 9643 9649 9655 9661 9666 9672 9677 9683
3.0 9688 9693 9698 9703 9709 9713 9718 9723 9728 9733
3.1 9737 9742 9746 9751 9755 9759 9764 9768 9772 9776
3.2 9780 9784 9788 9791 9795 9799 9802 9806 9809 9813
3.3 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845
3.4 9848 9850 9853 9856 9859 9861 9864 9867 9869 9872
3.5 9874 9877 9879 9881 9884 9886 9888 9890 9892 9894
3.6 9897 9899 9901 9903 9904 9906 9908 9910 9912 9914
3.7 9915 9917 9919 9920 9922 9924 9925 9927 9928 9930
3.8 9931 9932 9934 9935 9937 9938 9939 9940 9942 9943
3.9 9944 9945 9946 9948 9949 9950 9951 9952 9953 9954
4.0 9955 9956 9957 9958 9959 9960 9961 9961 9962 9963
4.1 9964 9965 9965 9966 9967 9968 9968 9969 9970 9970
4.2 9971 9972 9972 9973 9974 9974 9975 9975 9976 9977
4.3 9977 9978 9978 9979 9979 9980 9980 9981 9981 9981
4.4 9982 9982 9983 9983 9984 9984 9984 9985 9985 9985
4.5 9986 9986 9986 9987 9987 9987 9988 9988 9988 9989
4.6 9989 9989 9989 9990 9990 9990 9990 9991 9991 9991
4.7 9991 9992 9992 9992 9992 9992 9993 9993 9993 9993
4.8 9993 9993 9994 9994 9994 9994 9994 9994 9995 9995
4.9 9995 9995 9995 9995 9995 9995 9996 9996 9996 9996
5.0 9996 9996 9996 9996 9996 9996 9997 9997 9997 9997
5.1 9997 9997 9997 9997 9997 9997 9997 9997 9998 9998
5.2 9998 9998 9998 9998 9998 9998 9998 9998 9998 9998
5.3 9998 9998 9998 9998 9998 9998 9999 9999 9999 9999
A
Adapted from Ennis, D. M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp. 353–370.
A
TABLE X1.6 The B Values for Estimating Variance of d’ Obtained From 3-AFC Test
NOTE 1—Enter the table in the row and column corresponding to the value of d’ obtained from Table X1.5. The variance of d’ is S (d’)= B/n, where
n is the sample size.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 2.7925 2.7862 2.7801 2.7740 2.7681 2.7624 2.7568 2.7513 2.7460 2.7408
0.1 2.7357 2.7307 2.7259 2.7212 2.7167 2.7123 2.7080 2.7038 2.6998 2.6959
0.2 2.6921 2.6884 2.6849 2.6814 2.6782 2.6750 2.6719 2.6690 2.6662 2.6635
0.3 2.6610 2.6585 2.6562 2.6540 2.6519 2.6499 2.6481 2.6464 2.6448 2.6433
0.4 2.6419 2.6406 2.6395 2.6384 2.6375 2.6367 2.6360 2.6355 2.6350 2.6347
0.5 2.6344 2.6343 2.6343 2.6345 2.6347 2.6351 2.6355 2.6361 2.6368 2.6376
0.6 2.6385 2.6396 2.6407 2.6420 2.6434 2.6449 2.6465 2.6483 2.6501 2.6521
0.7 2.6542 2.6564 2.6587 2.6611 2.6637 2.6664 2.6692 2.6721 2.6751 2.6783
0.8 2.6815 2.6949 2.6884 2.6921 2.6958 2.6997 2.7037 2.7079 2.7121 2.7165
0.9 2.7210 2.7256 2.7304 2.7353 2.7403 2.7454 2.7507 2.7561 2.7616 2.7673
1.0 2.7731 2.7790 2.7851 2.7913 2.7976 2.8041 2.8107 2.8175 2.8244 2.8314
1.1 2.8386 2.8459 2.8534 2.8610 2.8688 2.8767 2.8847 2.8930 2.9013 2.9098
1.2 2.9185 2.9273 2.9363 2.9454 2.9547 2.9642 2.9738 2.9836 2.9936 3.0037
1.3 3.0140 3.0244 3.0351 3.0459 3.0569 3.0680 3.0794 3.0909 3.1026 3.1145
1.4 3.1265 3.1388 3.1512 3.1639 3.1767 3.1898 3.2030 3.2164 3.2301 3.2439
1.5 3.2580 3.2722 3.2867 3.3014 3.3163 3.3314 3.3468 3.3623 3.3781 3.3942
1.6 3.4104 3.4269 3.4437 3.4607 3.4779 3.4954 3.5131 3.5311 3.5493 3.5678
1.7 3.5866 3.6056 3.6249 3.6445 3.6643 3.6845 3.7049 3.7256 3.7466 3.7679
1.8 3.7895 3.8114 3.8336 3.8561 3.8790 3.9021 3.9256 3.9494 3.9735 3.9980
1.9 4.0229 4.0480 4.0736 4.0995 4.1257 4.1523 4.1793 4.2067 4.2344 4.2626
2.0 4.2911 4.3201 4.3494 4.3792 4.4093 4.4399 4.4710 4.5025 4.5344 4.5667
2.1 4.5996 4.6328 4.6666 4.7008 4.7356 4.7708 4.8065 4.8427 4.8794 4.9167
2.2 4.9545 4.9928 5.0317 5.0711 5.1111 5.1517 5.1929 5.2346 5.2769 5.3199
2.3 5.3635 5.4077 5.4525 5.4981 5.5442 5.5910 5.6386 5.6868 5.7357 5.7853
E2262 − 03 (2020)
TABLE X1.6 Continued
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
2.4 5.8357 5.8868 5.9386 5.9912 6.0446 6.0988 6.1538 6.2096 6.2662 6.3236
2.5 6.3819 6.4411 6.5012 6.5622 6.6241 6.6869 6.7507 6.8154 6.8811 6.9479
2.6 7.0156 7.0843 7.1542 7.2250 7.2970 7.3701 7.4443 7.5196 7.5961 7.6737
2.7 7.7526 7.8327 7.9141 7.9967 8.0806 8.1658 8.2524 8.3403 8.4296 8.5204
2.8 8.6125 8.7061 8.8012 8.8979 8.9960 9.0957 9.1971 9.3000 9.4047 9.5110
2.9 9.619 9.729 9.840 9.954 10.069 10.186 10.305 10.426 10.549 10.674
3.0 10.801 10.930 11.062 11.195 11.331 11.469 11.609 11.752 11.897 12.045
3.1 12.195 12.347 12.503 12.660 12.821 12.984 13.150 13.319 13.491 13.666
3.2 13.844 14.025 14.209 14.396 14.587 14.781 14.978 15.179 15.383 15.591
3.3 15.802 16.018 16.237 16.460 16.687 16.918 17.154 17.393 17.637 17.886
3.4 18.138 18.396 18.658 18.925 19.196 19.473 19.755 20.042 20.334 20.632
3.5 20.936 21.245 21.559 21.880 22.206 22.539 22.878 23.224 23.575 23.934
3.6 24.299 24.672 25.051 25.438 25.832 26.234 26.643 27.061 27.486 27.920
3.7 28.362 29.813 29.273 29.741 30.218 30.706 31.203 31.710 32.225 32.752
3.8 33.290 33.838 34.397 34.968 35.549 36.142 36.748 37.365 37.996 38.638
3.9 39.295 39.964 40.647 41.343 42.055 42.781 43.522 44.278 45.051 45.837
4.0 46.642 47.463 48.301 49.157 50.031 50.925 51.835 52.765 53.716 54.686
4.1 55.676 56.687 57.722 58.778 59.856 60.959 62.083 63.232 64.406 65.609
4.2 66.835 68.085 69.363 70.679 72.011 73.378 74.776 76.203 77.663 79.153
4.3 80.676 82.231 83.826 85.456 87.122 88.824 90.568 92.347 94.168 96.029
4.4 97.94 99.89 101.87 103.91 106.00 108.13 110.30 112.54 114.82 117.15
4.5 119.54 122.00 124.49 127.06 129.67 132.36 135.11 137.91 140.79 143.72
4.6 146.75 149.83 153.00 156.23 159.55 162.92 166.39 169.94 173.60 177.32
4.7 181.14 185.06 189.07 193.16 197.35 201.65 206.08 210.59 215.24 219.93
4.8 224.81 229.85 234.91 240.13 245.52 251.01 256.65 262.39 268.29 274.38
4.9 280.57 296.97 293.44 300.11 307.05 314.10 321.37 328.68 336.35 344.18
5.0 352.08 360.42 368.70 377.37 386.31 395.30 404.63 414.26 423.99 433.91
5.1 444.34 454.81 465.84 477.06 488.24 500.04 512.35 524.87 537.15 550.34
A
Adapted from Bi, J., Ennis, D. M., and O’Mahony, M., “How to Estimate and Use the Variance of d’ from Difference Tests,” Journal of Sensory Studies, 12, 1997,
pp. 87–104.
4 A
TABLE X1.7 Observed Choice Proportions, p , (×10 ) as Function of d’ for 2-AFC Test
c
NOTE 1—Find the entry in the table closest to the choice proportion observed in the test. Read the estimated value of d’ from the corresponding row
and column headings.
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 5000 5028 5056 5085 5113 5141 5169 5197 5226 5254
0.1 5282 5310 5338 5366 5394 5422 5450 5478 5506 5534
0.2 5562 5590 5618 5646 5674 5702 5729 5757 5785 5812
0.3 5840 5868 5895 5923 5950 5977 6005 6032 6059 6086
0.4 6114 6141 6168 6195 6221 6248 6275 6302 6329 6355
0.5 6382 6408 6434 6461 6487 6513 6539 6565 6591 6617
0.6 6643 6669 6695 6720 6746 6771 6796 6822 6847 6872
0.7 6897 6922 6947 6971 6996 7021 7045 7069 7094 7118
0.8 7142 7166 7190 7214 7237 7261 7284 7308 7331 7354
0.9 7377 7400 7423 7446 7469 7491 7514 7536 7558 7580
1.0 7602 7624 7646 7668 7689 7711 7732 7754 7775 7796
1.1 7817 7837 7858 7879 7899 7919 7940 7960 7980 8000
1.2 8019 8039 8058 8078 8097 8116 8135 8154 8173 8192
1.3 8210 8229 8247 8265 8283 8301 8319 8337 8354 8372
1.4 8389 8406 8423 8440 8457 8474 8491 8507 8523 8540
1.5 8556 8572 8588 8603 8619 8635 8650 8665 8681 8696
1.6 8711 8725 8740 8755 8769 8783 8798 8812 8826 8840
1.7 8853 8867 8881 8894 8907 8920 8933 8946 8959 8972
1.8 8985 8997 9009 9022 9034 9046 9058 9070 9081 9093
1.9 9104 9116 9127 9138 9149 9160 9171 9182 9193 9203
2.0 9214 9224 9234 9244 9254 9264 9274 9284 9293 9303
2.1 9312 9321 9331 9340 9349 9358 9367 9375 9384 9393
2.2 9401 9409 9418 9426 9434 9442 9450 9458 9465 9473
2.3 9481 9488 9495 9503 9510 9517 9524 9531 9538 9545
2.4 9552 9558 9565 9571 9578 9584 9590 9596 9603 9609
2.5 9615 9620 9626 9632 9638 9643 9649 9654 9659 9665
2.6 9670 9675 9680 9685 9690 9695 9700 9705 9710 9714
2.7 9719 9723 9728 9732 9737 9741 9745 9749 9753 9757
2.8 9761 9765 9769 9773 9777 9781 9784 9788 9791 9795
2.9 9798 9802 9805 9809 9812 9815 9818 9821 9824 9828
3.0 9831 9833 9836 9839 9842 9845 9848 9850 9853 9856
3.1 9858 9861 9863 9866 9868 9870 9873 9875 9877 9880
3.2 9882 9884 9886 9888 9890 9892 9894 9896 9898 9900
3.3 9902 9904 9906 9907 9909 9911 9912 9914 9916 9917
3.4 9919 9921 9922 9924 9925 9926 9928 9929 9931 9932
3.5 9933 9935 9936 9937 9938 9940 9941 9942 9943 9944
3.6 9945 9947 9948 9949 9950 9951 9952 9953 9954 9955
3.7 9956 9956 9957 9958 9959 9960 9961 9962 9962 9963
E2262 − 03 (2020)
TABLE X1.7 Continued
d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
3.8 9964 9965 9965 9966 9967 9968 9968 9969 9970 9970
3.9 9971 9972 9972 9973 9973 9974 9974 9975 9976 9976
4.0 9977 9977 9978 9978 9979 9979 9980 9980 9980 9981
4.1 9981 9982 9982 9983 9983 9983 9984 9984 9984 9985
4.2 9985 9985 9986 9986 9986 9987 9987 9987 9988 9988
4.3 9988 9988 9989 9989 9989 9990 9990 9990 9990 9990
4.4 9991 9991 9991 9991 9992 9992 9992 9992 9992 9993
4.5 9993 9993 9993 9993 9993 9994 9994 9994 9994 9994
4.6 9994 9994 9995 9995 9995 9995 9995 9995 9995 9995
4.7 9996 9996 9996 9996 9996 9996 9996 9996 9996 9996
4.8 9997 9997 9997 9997 9997 9997 9997 9997 9997 9997
4.9 9997 9997 9997
...