Designation E2262 − 03 (Reapproved 2014) Standard Practice for Estimating Thurstonian Discriminal Distances1 This standard is issued under the fixed designation E2262; the number immediately following[.]
Designation: E2262 − 03 (Reapproved 2014) Standard Practice for Estimating Thurstonian Discriminal Distances1 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 A number in parentheses indicates the year of last reapproval A superscript epsilon (´) indicates an editorial change since the last revision or reapproval (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 may involve hazardous materials, operations and equipment 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 and health practices and determine the applicability of regulatory limitations prior to use 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’ Referenced Documents 1.2 The procedures in this document 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 standard 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.) 2.1 ASTM Standards:2 E253 Terminology Relating to Sensory Evaluation of Materials and Products E456 Terminology Relating to Quality and Statistics E460 Practice for Determining Effect of Packaging on Food and Beverage Products During Storage E679 Practice for Determination of Odor and Taste Thresholds By a Forced-Choice Ascending Concentration Series Method of Limits E1432 Practice for Defining and Calculating Individual and Group Sensory Thresholds from Forced-Choice Data Sets of Intermediate Size E1593 Guide for Assessing the Efficacy of Air Care Products in Reducing the Perception of Indoor Malodor E1627 Practice for Sensory Evaluation of Edible Oils and Fats E1697 Test Method for Unipolar Magnitude Estimation of Sensory Attributes E1810 Practice for Evaluating Effects of Contaminants on Odor and Taste of Exposed Fish E1879 Guide for Sensory Evaluation of Beverages Containing Alcohol E1885 Test Method for Sensory Analysis—Triangle Test E1958 Guide for Sensory Claim Substantiation E2049 Guide for Quantitative Attribute Evaluation of Fragrance/Odors for Shampoos and Hair Conditioners by Trained Assessors 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 This practice is under the jurisdiction of ASTM Committee E18 on Sensory Evaluation and is the direct responsibility of Subcommittee E18.03 on Sensory Theory and Statistics Current edition approved Sept 1, 2014 Published September 2014 Originally approved in 2003 Last previous edition approved in 2009 as E2262 – 03 (2009) DOI: 10.1520/E2262-03R14 For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org For Annual Book of ASTM Standards volume information, refer to the standard’s Document Summary page on the ASTM website Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959 United States E2262 − 03 (2014) proportion of the “A” responses for the A sample (pa) The same method is used to estimate the variance of d’, S2(d’), using Table X1.10 E2164 Test Method for Directional Difference Test 2.2 ASTM Publication: Manual 26 Sensory Testing Methods, 2nd Edition2 2.3 ISO Standard: ISO 5495 Sensory Analysis—Methodology—Paired Comparison3 4.4 For the Same-Different method, tally the proportion of “same” responses for the matched pairs of samples (that is, A/A or B/B) and the proportion of “same” responses for the unmatched pairs of samples (that is, A/B or B/A) Read the value of d’ from Table X1.11 in the column that corresponds to the observed proportion of “same” responses for the unmatched pairs (ps/u) and the row that corresponds to the observed proportion of the “same” responses for the matched pairs (ps/m) The same method is used to estimate the variance of d’, S2(d’), using Table X1.12 Terminology 3.1 For definitions of terms relating to sensory analysis, see Terminology E253, and for terms relating to statistics, see Terminology E456 3.2 Definitions of Terms Specific to This Standard: 3.2.1 δ—the Thurstonian discriminal distance is the distance between the means of the distributions of sensory magnitudes of the two samples in the test (see Appendix X3) 3.2.2 d’—the statistic used to estimate δ based on the data obtained from the test 3.2.3 choice proportion (Pc)—the expected proportion of responses from a forced-choice method (for example, if there is no perceptible difference between the samples in a triangle test, Pc = 1/3 If there is a perceptible difference, Pc > 1/3) 3.2.4 observed choice proportion (pc) —the statistic used to estimate choice proportion, Pc, where pc = x/n, where x is the observed number of correct responses and n is the sample size 4.5 For ordered category scales, a rapid, table-look-up approach is used For each sample, the category scale data are collapsed into two categories One sample is selected to be the “A” sample and the other sample is selected to be the “Not-A” sample Choice proportions are tallied for each sample and the values of d’ and its variance, S2(d’), are obtained from Tables X1.9 and X1.10, respectively, by the same techniques used in the A/Not A method 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 Summary of Practice 4.1 Determine the type of data collected on the two samples: data from a forced-choice test, an A/Not A test, a samedifferent test or an ordered category scale 4.2 For forced-choice tests, reference the table that corresponds to the test method (that is, triangle test—Tables X1.1 and X1.2; duo-trio test—Tables X1.3 and X1.4; 3-AFC test— Tables X1.5 and X1.6; or 2-AFC test—Tables X1.7 and X1.8) Identify the entry in the table closest to the observed choice proportion (pc) from the test Read the estimated value of δ (that is, d’) from the corresponding row and column headings of the table Estimate the variance of d’ by referencing the appropriate table for the test method Find the value of B that corresponds to the value of d’ obtained in the first step.4 The estimated variance of d’ is S2(d’) = B/n, where n is the sample size Use the estimates d’ and S2(d’) to construct confidence intervals and tests of hypotheses related to the objectives of the research 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 4.3 For the A/Not A method, tally the observed choice proportions of “A” responses for the A sample and the “A” responses for the Not-A sample Read the value of d’ from Table X1.9 in the column that corresponds to the observed choice proportion of the “A” responses for the Not-A sample (pna) and the row that corresponds to the observed choice Procedure 6.1 Forced-Choice Methods—The relationship between δ and the expected choice proportion, Pc, is different for different forced-choice methods because the decision rule used by the assessors varies from one method to another (see Appendix X3) As a result, different tables are required to estimate δ depending on the method used Tables for the four most commonly used methods are presented The estimated value of δ (that is, d’) is obtained as follows: Available from American National Standards Institute (ANSI), 25 W 43rd St., 4th Floor, New York, NY 10036 The variance of d’ is a complicated function of the true value of δ and the decision rule when associated with the test method being used (see Appendix X3) However, regardless of the test method, the variance of d’ can always be expressed as S2(d’) = B/n, where the parameter B captures all of the information concerning the test method, and n is the sample size The values of B have been tabulated to make the calculation of the variance of d’ a simple task E2262 − 03 (2014) on the physical scale used to collect the data It is recognized that information detail is lost by collapsing the data into two categories However, the estimates of d’ and its variance, S2(d’), obtained from the technique are accurate The computational ease offsets the small loss of accuracy incurred 6.4.1 Tally the frequency distributions of category scale ratings for the two samples Select the sample with the lower median rating to be the Not-A sample Select the sample with the higher median rating to be the A sample 6.4.2 Collapse the frequency data for each sample into two categories as follows Identify the category in which the median of the Not-A sample occurs Pool the number of responses from that category and all lower categories for each sample separately and record the totals in the 2-by-2 table under “Low” (that is, the yna and ya tallies, below) Pool the number of responses for the remaining, higher categories for each sample separately and record the totals in the 2-by-2 table under “High” (that is, the xna and xa tallies, below) 6.1.1 Compute the observed choice proportion as pc = x/n, where x is the observed number of correct responses and n is the sample size 6.1.2 Obtain d’ by entering the table in Appendix X1 that corresponds to the test method used: triangle test (Table X1.1), duo-trio (Table X1.3), 3-AFC (Table X1.5) or 2-AFC (Table X1.7) Find the entry in the table that is closest to the observed value of pc The value of d’, accurate to one decimal place, is the row-label of the table corresponding to the selected entry The second decimal place of d’ is the column-label of the table corresponding to the selected entry 6.1.3 Obtain the estimated variance of d’ as follows Enter the appropriate table in Appendix X1: triangle test (Table X1.2), duo-trio (Table X1.4), 3-AFC (Table X1.6) or 2-AFC (Table X1.8) Find the value of B in the row and column that correspond to the value of d’ obtained in 6.1.2 Compute the estimated variance of d’ as S2(d’) = B/n, where n is the sample size Use the estimates d’ and S2(d’) to construct confidence intervals and tests of hypotheses related to the objectives of the research Sample Not-A A 6.2 A/Not A Method—Compute the choice proportions of the two samples, pa = xa/n and pna = xna/n, where xa is the number of times the “A” sample is chosen as being “A,”, xna is the number of times the “Not-A” sample is chosen as being “A” and n is the sample size Low yna ya High xna xa 6.4.3 Compute the choice proportions of the two samples, pa = xa/n and pna = xna/n, where xa and xna are obtained from the table above and n is the sample size, common to both samples 6.4.4 Apply the same technique used in the A/Not A method (see 6.2) Read the value of d’ from Table X1.9 in Appendix X1 in the column that corresponds to the observed choice proportion of the Not-A sample (pna) and the row that corresponds to the observed choice proportion of the A sample (pa) 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 technique as in 6.4.4 The variance estimate is S2(d’) = B/n, where n is the sample size NOTE 1—This practice only considers the case where the number of “A” samples equals the number of “Not-A” samples, n = na = nna 6.2.1 Read the value of d’ from Table X1.9 in Appendix X1 in the column that corresponds to the observed choice proportion of the “Not-A” sample (pna) and the row that corresponds to the observed choice proportion of the “A” sample (pa) 6.2.2 To obtain an estimate of the variance of d’, read the value of B from Table X1.10 in Appendix X1 using the same technique as in 6.2.1 The variance estimate is S2(d’) = B/n, where n is the sample size 6.5 Statistical Tests and Confidence Intervals—Often the objective of a sensory discrimination test is to determine if the samples in the test are perceptibly different In other instances it is of interest to obtain an estimate of the size of the perceptible difference (and to measure the precision of the estimated difference) Because testing for a difference and estimating the size of a difference address different goals, it is not surprising that different statistical methods apply to each For the purpose of testing if a perceptible difference exists, the binomial and chi-square tests traditionally associated with the test methods discussed in this standard are appropriate For the purposes of estimating the size of the difference and assessing the precision of that estimate, confidence intervals are appropriate Because δ is the difference between the means of two normal distributions and d’ is an estimate of δ, it can be assumed that d’ is approximately normally distributed Based on this assumption, statistical confidence intervals concerning δ can be constructed using traditional techniques 6.5.1 A 100(1- α)% two-sided confidence interval on δ is calculated as: d’ Zα/2S(d’), where d’ is the estimated value of δ, Zα/2 is the upper-α/2 percentage point of the standard normal distribution (for example, for a 90 % confidence interval Zα/2 = 1.65; for a 95 % confidence interval Zα/2 = 1.96; etc.), and S(d’) 6.3 Same-Different Method—Compute the choice proportions for the matched (m) and unmatched (u) pairs of samples, ps/m = xs/m/n and ps/u = xs/u/n, where xs/m is the number of “same” responses for the matched pairs (A/A or B/B) evaluated, xs/u is the number of “same” responses for the unmatched pair and n is the number of matched or unmatched pairs evaluated NOTE 2—This practice only considers the case where the number of matched pairs equals the number of unmatched pairs, n = nm = nu 6.3.1 Read the value of d’ from Table X1.11 in Appendix X1 in the column that corresponds to the observed proportion of “same” responses for unmatched pair (ps/u) and the row that corresponds to the observed proportion of “same” responses for the matched pair (ps/m) 6.3.2 To obtain an estimate of the variance of d’, read the value of B from Table X1.12 in Appendix X1 using the same technique as in 6.3.1 The variance estimate is S2(d’) = B/n, where n is the sample size 6.4 Ordered Category Scales—A rapid, table-look-up method is described The method collapses the category-scale data into two categories, regardless of the number of categories E2262 − 03 (2014) is the standard deviation of d’, that is, the square root of, S2(d’) = B/n Similarly, 100(1 - α)% one-sided confidence intervals on δ are calculated as: d’ + ZαS(d’) for a one-sided upper confidence interval and d’ − ZαS(d’) for a one-sided lower confidence interval, where Zα is the upper-α percentage point of the standard normal distribution (for example, for a 90 % confidence interval Zα = 1.28; for a 95 % confidence interval Zα = 1.65; etc.) and d’ and S(d’) are as defined above 6.5.2 To test if δ is greater than zero, that is, that the two samples in the test are perceptibly different, use the binomial or chi-square test that is traditionally associated with the discrimination method used 6.5.3 To test if it is reasonable to believe two δ’s have the same value, that is, to test the hypotheses: H0: δ1 = δ2 versus Ha: δ1 ≠ δ2 form the ratio: T5 ? d’2d’ ? =S 21 1S 22 where d1’ and d2’ are the estimated values of δ1 and δ2, respectively, and S12 and S22 are the variances of d1’ and d2’, respectively If T > Zα/2, then conclude the two δ values are unequal at the α-level of significance APPENDIXES (Nonmandatory Information) X1 STATISTICAL TABLES TABLE X1.1 Observed Choice Proportions, pc, (x104) as a Function of d’ for the Triangle TestA 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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 3333 3343 3370 3415 3478 3558 3654 3766 3891 4030 4180 4342 4512 4690 4875 5065 5258 5455 5653 5851 6048 6244 6437 6627 6812 6993 7169 7340 7504 7662 7814 7960 8098 8231 8356 8475 8588 8694 8794 8888 8977 9059 9137 9209 9276 3333 3344 3374 3421 3486 3567 3665 3778 3905 4045 4196 4358 4529 4708 4893 5084 5278 5474 5672 5870 6068 6263 6456 6645 6831 7011 7187 7356 7520 7678 7829 7974 8112 8243 8368 8487 8599 8704 8804 8897 8985 9067 9144 9216 9282 3334 3347 3378 3427 3493 3576 3676 3790 3918 4059 4212 4375 4547 4726 4912 5103 5297 5494 5692 5890 6087 6283 6475 6664 6849 7029 7204 7373 7536 7693 7844 7988 8125 8256 8381 8498 8610 8715 8813 8906 8994 9075 9151 9223 9289 3334 3349 3382 3432 3501 3586 3686 3802 3932 4074 4228 4392 4564 4745 4931 5122 5317 5514 5712 5910 6107 6302 6494 6683 6867 7047 7221 7390 7552 7709 7859 8002 8139 8269 8393 8510 8620 8725 8823 8915 9002 9083 9159 9229 9295 3335 3351 3386 3439 3508 3595 3697 3814 3945 4089 4244 4409 4582 4763 4950 5142 5337 5534 5732 5930 6127 6321 6513 6701 6885 7064 7238 7406 7568 7724 7873 8016 8152 8282 8405 8521 8631 8735 8833 8924 9010 9091 9166 9236 9301 3336 3354 3390 3445 3516 3604 3708 3827 3959 4104 4260 4426 4600 4782 4969 5161 5356 5554 5752 5950 6146 6341 6532 6720 6903 7082 7255 7423 7584 7739 7888 8030 8165 8294 8417 8532 8642 8745 8842 8933 9019 9099 9173 9243 9308 3337 3357 3395 3451 3524 3614 3719 3839 3973 4119 4276 4443 4618 4800 4988 5180 5376 5573 5771 5969 6166 6360 6551 6739 6922 7100 7272 7439 7600 7754 7902 8044 8179 8307 8428 8544 8652 8755 8851 8942 9027 9106 9180 9250 9314 3338 3360 3400 3458 3533 3624 3731 3852 3987 4134 4292 4460 4636 4819 5007 5200 5396 5593 5791 5989 6185 6379 6570 6757 6940 7117 7289 7455 7616 7769 7917 8057 8192 8319 8440 8555 8663 8765 8861 8951 9035 9114 9188 9256 9320 3339 3363 3405 3464 3541 3634 3742 3865 4001 4149 4309 4477 4654 4837 5026 5219 5415 5613 5811 6009 6205 6398 6589 6776 6958 7135 7306 7472 7631 7784 7931 8071 8205 8332 8452 8566 8673 8775 8870 8959 9043 9122 9195 9263 9326 3341 3366 3410 3471 3550 3644 3754 3878 4016 4165 4325 4494 4672 4856 5045 5239 5435 5633 5831 6028 6224 6418 6608 6794 6976 7152 7323 7488 7647 7799 7945 8085 8218 8344 8464 8577 8684 8784 8879 8968 9051 9129 9202 9269 9332 E2262 − 03 (2014) TABLE X1.1 A Continued d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 9338 9396 9450 9500 9546 9588 9627 9662 9695 9725 9753 9778 9800 9821 9840 9857 9344 9402 9455 9504 9550 9592 9630 9666 9698 9728 9755 9780 9803 9823 9842 9859 9350 9407 9460 9509 9554 9596 9634 9669 9701 9731 9758 9782 9805 9825 9843 9860 9356 9413 9465 9514 9559 9600 9638 9673 9704 9734 9760 9785 9807 9827 9845 9862 9362 9418 9470 9518 9563 9604 9641 9676 9707 9736 9763 9787 9809 9829 9847 9863 9368 9424 9475 9523 9567 9608 9645 9679 9710 9739 9765 9789 9811 9831 9849 9865 9374 9429 9480 9528 9571 9612 9648 9682 9713 9742 9768 9792 9813 9833 9850 9866 9379 9434 9485 9532 9575 9615 9652 9686 9716 9745 9770 9794 9815 9834 9852 9868 9385 9440 9490 9537 9580 9619 9655 9689 9719 9747 9773 9796 9817 9836 9854 9869 9391 9445 9495 9541 9584 9623 9659 9692 9722 9750 9775 9798 9819 9838 9855 9871 Adapted from Ennis, D M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp 353-370 TABLE X1.2 The B Values for Estimating the Variance of d’ Obtained from a Triangle TestA 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 S2(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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 ******* 661.08 167.61 76.236 44.273 29.496 21.489 16.681 13.582 11.479 9.998 8.926 8.136 7.547 7.106 6.778 6.538 6.369 6.258 6.197 6.178 6.198 6.251 6.336 6.450 6.594 6.765 6.963 7.190 7.445 7.729 8.044 8.390 8.770 9.185 9.638 10.131 10.668 11.252 11.887 12.578 13.330 14.148 15.039 16.009 17.068 18.225 19.488 20.871 22.385 65800 546.88 152.31 71.597 42.294 28.477 20.898 16.310 13.335 11.308 9.875 8.836 8.069 7.497 7.068 6.750 6.518 6.355 6.250 6.193 6.179 6.201 6.258 6.346 6.463 6.609 6.783 6.985 7.214 7.472 7.760 8.077 8.427 8.810 9.228 9.685 10.183 10.724 11.313 11.954 12.651 13.409 14.233 15.132 16.111 17.180 18.346 19.621 21.016 22.544 16452 460.03 139.06 67.387 40.455 27.517 20.336 15.954 13.097 11.142 9.756 8.749 8.004 7.448 7.032 6.723 6.499 6.342 6.242 6.190 6.179 6.206 6.265 6.356 6.477 6.626 6.802 7.007 7.239 7.500 7.790 8.111 8.463 8.850 9.272 9.733 10.235 10.781 11.375 12.021 12.723 13.488 14.320 15.226 16.214 17.292 18.468 19.755 21.163 22.705 7314 392.44 127.50 63.554 38.742 26.611 19.801 15.614 12.868 10.982 9.641 8.664 7.941 7.401 6.997 6.697 6.480 6.330 6.235 6.187 6.180 6.210 6.273 6.367 6.490 6.642 6.821 7.029 7.264 7.527 7.821 8.144 8.500 8.890 9.317 9.781 10.287 10.838 11.437 12.089 12.797 13.568 14.407 15.321 16.318 17.405 18.592 19.890 21.310 22.867 4115 338.81 117.35 60.054 37.146 25.755 19.291 15.287 12.647 10.827 9.529 8.582 7.880 7.355 6.962 6.672 6.462 6.318 6.228 6.185 6.182 6.215 6.281 6.378 6.504 6.659 6.841 7.051 7.289 7.555 7.852 8.179 8.538 8.931 9.361 9.830 10.340 10.896 11.500 12.157 12.871 13.649 14.495 15.417 16.422 17.519 18.717 20.026 21.460 23.031 2635 295.54 108.40 56.850 35.655 24.945 18.805 14.973 12.435 10.677 9.421 8.502 7.820 7.310 6.929 6.648 6.445 6.307 6.222 6.183 6.184 6.220 6.289 6.389 6.518 6.676 6.861 7.073 7.314 7.584 7.883 8.213 8.576 8.973 9.406 9.879 10.394 10.954 11.563 12.225 12.946 13.730 14.584 15.514 16.527 17.634 18.842 20.164 21.610 23.196 1831 260.13 100.45 53.910 34.261 24.179 18.341 14.672 12.230 10.532 9.316 8.424 7.762 7.267 6.897 6.624 6.429 6.296 6.216 6.181 6.186 6.226 6.298 6.401 6.533 6.693 6.881 7.096 7.340 7.612 7.914 8.248 8.614 9.014 9.452 9.929 10.448 11.012 11.627 12.295 13.022 13.812 14.673 15.611 16.634 17.750 18.969 20.303 21.763 23.363 1346 230.78 93.38 51.205 32.954 23.452 17.897 14.383 12.032 10.392 9.214 8.349 7.706 7.225 6.866 6.601 6.413 6.286 6.211 6.180 6.188 6.231 6.307 6.413 6.548 6.710 6.901 7.119 7.366 7.641 7.946 8.283 8.652 9.056 9.498 9.979 10.502 11.072 11.691 12.365 13.098 13.895 14.673 15.709 16.741 17.867 19.097 20.443 21.916 23.531 1031 206.19 87.05 48.711 31.729 22.764 17.474 14.106 11.841 10.256 9.115 8.276 7.651 7.184 6.835 6.579 6.398 6.276 6.206 6.179 6.191 6.238 6.316 6.425 6.563 6.728 6.921 7.143 7.392 7.670 7.979 8.318 8.691 9.099 9.544 10.029 10.557 11.131 11.756 12.435 13.174 13.979 14.854 15.809 16.849 17.985 19.226 20.584 22.071 23.701 815 185.38 81.36 46.406 30.578 22.110 17.069 13.839 11.657 10.125 9.019 8.205 7.598 7.144 6.806 6.558 6.383 6.267 6.201 6.179 6.194 6.244 6.326 6.438 6.578 6.746 6.942 7.166 7.418 7.700 8.011 8.354 8.730 9.142 9.591 10.080 10.612 11.191 11.821 12.506 13.252 14.063 14.946 15.909 16.958 18.104 19.357 20.727 22.227 23.872 E2262 − 03 (2014) TABLE X1.2 Continued d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 5.0 5.1 24.046 25.869 24.220 26.061 24.397 26.255 24.575 26.451 24.754 26.648 24.936 26.847 25.119 27.049 25.304 27.252 25.491 27.457 25.679 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 TABLE X1.3 Observed Choice Proportions, pc, (x104) as a Function of d’ for the Duo Trio TestA 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.) A d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.6 5.7 5.8 5.9 6.0 6.1 6.2 5000 5009 5037 5082 5144 5223 5318 5427 5548 5682 5825 5976 6135 6298 6465 6635 6805 6974 7142 7307 7468 7625 7777 7923 8062 8196 8323 8443 8557 8664 8765 8859 8947 9030 9107 9178 9245 9307 9364 9417 9467 9512 9554 9593 9629 9662 9693 9721 9747 9770 9792 9812 9830 9847 9862 9876 9888 9900 9910 9920 9928 5000 5011 5040 5087 5152 5232 5328 5438 5561 5695 5840 5992 6151 6315 6482 6652 6822 6991 7159 7323 7484 7640 7791 7937 8076 8209 8335 8455 8568 8674 8774 8868 8956 9038 9114 9185 9251 9313 9370 9423 9471 9517 9558 9597 9633 9665 9696 9723 9749 9772 9794 9814 9832 9848 9863 9877 9890 9901 9911 9921 9929 5000 5013 5044 5093 5159 5241 5339 5450 5574 5709 5854 6007 6167 6331 6499 6669 6839 7008 7175 7340 7500 7656 7806 7951 8090 8222 8347 8466 8579 8685 8784 8877 8964 9046 9121 9192 9258 9319 9375 9428 9476 9521 9562 9601 9636 9669 9698 9726 9751 9775 9796 9816 9833 9850 9865 9878 9891 9902 9912 9922 9930 5001 5015 5048 5099 5166 5250 5349 5462 5587 5724 5869 6023 6183 6348 6516 6686 6856 7025 7192 7356 7516 7671 7821 7965 8103 8235 8360 8478 8590 8695 8794 8886 8973 9053 9129 9199 9264 9324 9381 9433 9481 9525 9566 9604 9639 9672 9701 9729 9754 9777 9798 9817 9835 9851 9866 9880 9892 9903 9913 9922 9931 5001 5018 5053 5105 5174 5259 5360 5474 5600 5738 5884 6039 6200 6365 6533 6703 6873 7042 7208 7372 7531 7686 7836 7979 8117 8247 8372 8489 8600 8705 8803 8895 8981 9061 9136 9206 9270 9330 9386 9438 9485 9529 9570 9608 9643 9675 9704 9731 9756 9779 9800 9819 9837 9853 9868 9881 9893 9904 9914 9923 9932 5002 5021 5057 5111 5182 5269 5371 5486 5614 5752 5900 6055 6216 6381 6550 6720 6890 7058 7225 7388 7547 7701 7850 7993 8130 8260 8384 8501 8611 8715 8813 8904 8989 9069 9143 9212 9276 9336 9391 9443 9490 9534 9574 9612 9646 9678 9707 9734 9759 9781 9802 9821 9839 9854 9869 9882 9894 9905 9915 9924 9932 5003 5023 5062 5117 5190 5278 5382 5498 5627 5766 5915 6071 6232 6398 6567 6737 6907 7075 7241 7404 7563 7717 7865 8007 8143 8273 8396 8512 8622 8725 8822 8913 8997 9077 9150 9219 9283 9342 9397 9447 9494 9538 9578 9615 9649 9681 9710 9736 9761 9783 9804 9823 9840 9856 9870 9884 9895 9906 9916 9925 9933 5005 5026 5066 5124 5198 5288 5393 5511 5641 5781 5930 6087 6249 6415 6584 6754 6923 7092 7258 7420 7578 7732 7879 8021 8156 8285 8408 8523 8632 8735 8831 8921 9006 9084 9157 9225 9289 9347 9402 9452 9499 9542 9582 9619 9653 9684 9713 9739 9763 9786 9806 9825 9842 9857 9872 9885 9897 9907 9917 9926 9934 5006 5030 5071 5131 5206 5298 5404 5523 5654 5795 5945 6102 6265 6432 6601 6771 6940 7109 7274 7436 7594 7747 7894 8035 8170 8298 8420 8535 8643 8745 8841 8930 9014 9092 9164 9232 9295 9353 9407 9457 9503 9546 9586 9622 9656 9687 9715 9742 9766 9788 9808 9827 9843 9859 9873 9886 9898 9908 9918 9927 9935 5007 5033 5077 5137 5215 5308 5415 5536 5668 5810 5961 6119 6282 6448 6618 6788 6957 7125 7291 7452 7610 7762 7908 8049 8183 8310 8431 8546 8653 8755 8850 8939 9022 9099 9171 9239 9301 9359 9412 9462 9508 9550 9589 9626 9659 9690 9718 9744 9768 9790 9810 9828 9845 9860 9874 9887 9899 9909 9919 9928 9935 Adapted from Ennis, D M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp 353-370 E2262 − 03 (2014) TABLE X1.4 The B Values for Estimating the Variance of d’ Obtained from a Duo-Trio TestA 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 S2(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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 ***** 743.52 188.36 85.576 49.624 33.010 24.012 18.618 15.148 12.803 11.160 9.983 9.127 8.502 8.051 7.734 7.524 7.404 7.360 7.382 7.467 7.608 7.804 8.055 8.359 8.717 9.132 9.604 10.136 10.731 11.391 12.121 12.923 13.802 14.762 15.807 16.942 18.173 19.505 20.946 22.503 24.184 26.000 27.962 30.082 32.376 34.859 37.553 40.478 43.660 47.127 50.911 74025 615.05 171.16 80.358 47.398 31.864 23.349 18.202 14.872 12.612 11.025 9.885 9.055 8.450 8.014 7.709 7.509 7.396 7.359 7.388 7.478 7.625 7.827 8.083 8.392 8.756 9.176 9.654 10.192 10.794 11.461 12.198 13.007 13.894 14.862 15.916 17.061 18.301 19.644 21.096 22.665 24.360 26.189 28.166 30.303 32.615 35.119 37.835 40.784 43.993 47.490 51.308 18508 517.34 156.25 75.622 45.330 30.785 22.719 17.803 14.606 12.427 10.893 9.789 8.986 9.400 7.978 7.684 7.494 7.389 7.359 7.395 7.490 7.643 7.850 8.111 8.426 8.796 9.221 9.705 10.250 10.857 11.531 12.275 13.093 13.987 14.964 16.026 17.180 18.431 19.784 21.248 22.829 24.536 26.380 28.373 30.526 32.857 35.381 38.119 41.093 44.329 47.857 51.709 8228 441.30 143.24 71.310 43.405 29.766 22.118 17.422 14.350 12.249 10.766 9.697 8.918 8.351 7.943 7.661 7.480 7.383 7.360 7.401 7.503 7.661 7.874 8.140 8.461 8.836 9.267 9.757 10.308 10.922 11.603 12.354 13.179 14.081 15.066 16.137 17.301 18.561 19.926 21.400 22.994 24.714 26.573 28.580 30.751 33.100 35.645 38.405 41.404 44.668 48.227 52.113 4630 380.96 131.83 67.374 41.610 28.804 21.545 17.056 14.104 12.078 10.643 9.608 8.853 8.304 7.910 7.638 7.466 7.378 7.361 7.409 7.516 7.680 7.898 8.170 8.496 8.876 9.313 9.809 10.366 10.987 11.675 12.433 13.265 14.176 15.169 16.250 17.422 18.693 20.068 21.554 23.160 24.894 26.767 28.790 30.978 33.345 35.911 38.694 41.718 45.010 48.600 52.521 2964 332.29 121.75 63.770 39.933 27.895 21.000 16.705 13.867 11.911 10.524 9.521 8.790 8.258 7.878 7.617 7.454 7.373 7.363 7.417 7.530 7.699 7.923 8.200 8.531 8.917 9.360 9.862 10.425 11.052 11.747 12.513 13.353 14.271 15.273 16.363 17.545 18.826 20.212 21.709 23.327 25.075 26.962 29.001 31.206 33.593 36.179 38.985 42.035 45.356 48.976 52.933 2059 292.45 112.82 60.463 38.365 27.034 20.479 16.368 13.638 11.751 10.409 9.437 8.729 8.214 7.847 7.597 7.442 7.369 7.366 7.426 7.545 7.719 7.948 8.231 8.567 8.959 9.408 9.916 10.485 11.119 11.820 12.593 13.441 14.368 15.378 16.477 17.668 18.960 20.356 21.866 23.496 25.257 27.159 29.214 31.436 33.842 36.449 39.279 42.355 45.704 49.357 53.349 1514 259.43 104.86 57.420 36.897 26.218 19.981 16.044 13.418 11.596 10.297 9.356 8.669 8.171 7.817 7.577 7.432 7.365 7.369 7.435 7.560 7.740 7.974 8.262 8.604 9.001 9.456 9.970 10.545 11.186 11.894 12.675 13.530 14.465 15.484 16.591 17.793 19.094 20.502 22.023 23.666 25.441 27.357 29.428 31.668 34.093 36.722 39.575 42.677 46.054 49.740 53.768 1160 231.77 97.74 54.615 35.520 25.444 19.506 15.733 13.205 11.446 10.189 9.277 8.612 8.130 7.788 7.559 7.422 7.363 7.373 7.445 7.575 7.761 8.000 8.294 8.641 9.044 9.505 10.024 10.606 11.254 11.969 12.757 13.620 14.653 15.591 16.707 17.919 19.230 20.649 22.182 23.838 25.626 27.557 29.645 31.902 34.347 36.996 39.873 43.001 46.409 50.127 54.191 917 208.35 91.34 52.024 34.226 24.710 19.052 15.435 13.000 11.301 10.084 9.201 8.556 8.090 7.760 7.541 7.412 7.361 7.377 7.456 7.591 7.782 8.027 8.326 8.679 9.088 9.554 10.080 10.668 11.322 12.045 12.839 13.711 14.662 15.698 16.824 18.045 19.367 20.797 22.342 24.010 25.812 27.759 29.862 32.138 34.602 37.274 40.174 43.329 46.766 50.517 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 TABLE X1.5 Observed Choice Proportions, pc, (x104) as a Function of d’ for the 3-AFC TestA 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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3333 3620 3914 4214 4519 4826 5134 5441 5745 6044 3362 3649 3944 4244 4549 4857 5165 5471 5775 6074 3390 3678 3974 4275 4580 4888 5195 5502 5805 6103 3418 3707 4003 4305 4611 4918 5226 5532 5835 6133 3447 3737 4033 4336 4641 4949 5257 5563 5865 6162 3475 3766 4063 4366 4672 4980 5288 5593 5895 6191 3504 3795 4093 4396 4703 5011 5318 5624 5925 6221 3533 3825 4124 4427 4734 5042 5349 5654 5955 6250 3562 3855 4154 4458 4764 5072 5380 5684 5985 6279 3591 3884 4184 4488 4795 5103 5410 5714 6014 6308 E2262 − 03 (2014) TABLE X1.5 A Continued d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 6337 6622 6898 7163 7417 7658 7886 8100 8300 8486 8658 8815 8959 9090 9208 9314 9408 9492 9566 9631 9688 9737 9780 9816 9848 9874 9897 9915 9931 9944 9955 9964 9971 9977 9982 9986 9989 9991 9993 9995 9996 9997 9998 9998 6366 6650 6925 7189 7442 7682 7908 8121 8319 8504 8674 8830 8973 9102 9219 9324 9417 9500 9573 9637 9693 9742 9784 9820 9850 9877 9899 9917 9932 9945 9956 9965 9972 9978 9982 9986 9989 9992 9993 9995 9996 9997 9998 9998 6395 6678 6952 7215 7466 7705 7930 8141 8339 8522 8690 8845 8986 9114 9230 9333 9426 9508 9580 9643 9698 9746 9788 9823 9853 9879 9901 9919 9934 9946 9957 9965 9972 9978 9983 9986 9989 9992 9994 9995 9996 9997 9998 9998 6423 6706 6979 7241 7491 7728 7952 8162 8357 8539 8706 8860 9000 9127 9241 9343 9434 9515 9587 9649 9703 9751 9791 9826 9856 9881 9903 9920 9935 9948 9958 9966 9973 9979 9983 9987 9990 9992 9994 9995 9996 9997 9998 9998 6452 6734 7005 7266 7515 7751 7973 8182 8376 8556 8722 8874 9013 9138 9252 9353 9443 9523 9593 9655 9709 9755 9795 9829 9859 9884 9904 9922 9937 9949 9959 9967 9974 9979 9984 9987 9990 9992 9994 9995 9996 9997 9998 9998 6481 6761 7032 7292 7539 7774 7995 8202 8395 8574 8738 8889 9026 9150 9262 9362 9451 9530 9600 9661 9713 9759 9799 9833 9861 9886 9906 9924 9938 9950 9960 9968 9974 9980 9984 9987 9990 9992 9994 9995 9996 9997 9998 9998 6509 6789 7059 7317 7563 7796 8016 8222 8413 8591 8754 8903 9039 9162 9273 9372 9460 9538 9606 9666 9718 9764 9802 9836 9864 9888 9908 9925 9939 9951 9961 9968 9975 9980 9984 9988 9990 9993 9994 9996 9997 9997 9998 9999 6538 6816 7085 7342 7587 7819 8037 8242 8432 8608 8769 8917 9052 9174 9283 9381 9468 9545 9613 9672 9723 9768 9806 9839 9867 9890 9910 9927 9940 9952 9961 9969 9975 9981 9985 9988 9991 9993 9994 9996 9997 9997 9998 9999 6566 6844 7111 7367 7611 7842 8058 8261 8450 8624 8785 8931 9065 9185 9293 9390 9476 9552 9619 9677 9728 9772 9809 9842 9869 9892 9912 9928 9942 9953 9962 9970 9976 9981 9985 9988 9991 9993 9995 9996 9997 9998 9998 9999 6594 6871 7137 7392 7635 7864 8079 8281 8468 8641 8800 8945 9077 9197 9304 9399 9484 9559 9625 9683 9733 9776 9813 9845 9872 9894 9914 9930 9943 9954 9963 9970 9977 9981 9985 9989 9991 9993 9995 9996 9997 9998 9998 9999 Adapted from Ennis, D M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp 353-370 TABLE X1.6 The B Values for Estimating the Variance of d’ Obtained from a 3-AFC TestA 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 S2(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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.7925 2.7357 2.6921 2.6610 2.6419 2.6344 2.6385 2.6542 2.6815 2.7210 2.7731 2.8386 2.9185 3.0140 3.1265 3.2580 3.4104 3.5866 3.7895 4.0229 4.2911 4.5996 2.7862 2.7307 2.6884 2.6585 2.6406 2.6343 2.6396 2.6564 2.6949 2.7256 2.7790 2.8459 2.9273 3.0244 3.1388 3.2722 3.4269 3.6056 3.8114 4.0480 4.3201 4.6328 2.7801 2.7259 2.6849 2.6562 2.6395 2.6343 2.6407 2.6587 2.6884 2.7304 2.7851 2.8534 2.9363 3.0351 3.1512 3.2867 3.4437 3.6249 3.8336 4.0736 4.3494 4.6666 2.7740 2.7212 2.6814 2.6540 2.6384 2.6345 2.6420 2.6611 2.6921 2.7353 2.7913 2.8610 2.9454 3.0459 3.1639 3.3014 3.4607 3.6445 3.8561 4.0995 4.3792 4.7008 2.7681 2.7167 2.6782 2.6519 2.6375 2.6347 2.6434 2.6637 2.6958 2.7403 2.7976 2.8688 2.9547 3.0569 3.1767 3.3163 3.4779 3.6643 3.8790 4.1257 4.4093 4.7356 2.7624 2.7123 2.6750 2.6499 2.6367 2.6351 2.6449 2.6664 2.6997 2.7454 2.8041 2.8767 2.9642 3.0680 3.1898 3.3314 3.4954 3.6845 3.9021 4.1523 4.4399 4.7708 2.7568 2.7080 2.6719 2.6481 2.6360 2.6355 2.6465 2.6692 2.7037 2.7507 2.8107 2.8847 2.9738 3.0794 3.2030 3.3468 3.5131 3.7049 3.9256 4.1793 4.4710 4.8065 2.7513 2.7038 2.6690 2.6464 2.6355 2.6361 2.6483 2.6721 2.7079 2.7561 2.8175 2.8930 2.9836 3.0909 3.2164 3.3623 3.5311 3.7256 3.9494 4.2067 4.5025 4.8427 2.7460 2.6998 2.6662 2.6448 2.6350 2.6368 2.6501 2.6751 2.7121 2.7616 2.8244 2.9013 2.9936 3.1026 3.2301 3.3781 3.5493 3.7466 3.9735 4.2344 4.5344 4.8794 2.7408 2.6959 2.6635 2.6433 2.6347 2.6376 2.6521 2.6783 2.7165 2.7673 2.8314 2.9098 3.0037 3.1145 3.2439 3.3942 3.5678 3.7679 3.9980 4.2626 4.5667 4.9167 E2262 − 03 (2014) 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 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 4.9545 5.3635 5.8357 6.3819 7.0156 7.7526 8.6125 9.619 10.801 12.195 13.844 15.802 18.138 20.936 24.299 28.362 33.290 39.295 46.642 55.676 66.835 80.676 97.94 119.54 146.75 181.14 224.81 280.57 352.08 444.34 4.9928 5.4077 5.8868 6.4411 7.0843 7.8327 8.7061 9.729 10.930 12.347 14.025 16.018 18.396 21.245 24.672 29.813 33.838 39.964 47.463 56.687 68.085 82.231 99.89 122.00 149.83 185.06 229.85 296.97 360.42 454.81 5.0317 5.4525 5.9386 6.5012 7.1542 7.9141 8.8012 9.840 11.062 12.503 14.209 16.237 18.658 21.559 25.051 29.273 34.397 40.647 48.301 57.722 69.363 83.826 101.87 124.49 153.00 189.07 234.91 293.44 368.70 465.84 5.0711 5.4981 5.9912 6.5622 7.2250 7.9967 8.8979 9.954 11.195 12.660 14.396 16.460 18.925 21.880 25.438 29.741 34.968 41.343 49.157 58.778 70.679 85.456 103.91 127.06 156.23 193.16 240.13 300.11 377.37 477.06 5.1111 5.5442 6.0446 6.6241 7.2970 8.0806 8.9960 10.069 11.331 12.821 14.587 16.687 19.196 22.206 25.832 30.218 35.549 42.055 50.031 59.856 72.011 87.122 106.00 129.67 159.55 197.35 245.52 307.05 386.31 488.24 5.1517 5.5910 6.0988 6.6869 7.3701 8.1658 9.0957 10.186 11.469 12.984 14.781 16.918 19.473 22.539 26.234 30.706 36.142 42.781 50.925 60.959 73.378 88.824 108.13 132.36 162.92 201.65 251.01 314.10 395.30 500.04 5.1929 5.6386 6.1538 6.7507 7.4443 8.2524 9.1971 10.305 11.609 13.150 14.978 17.154 19.755 22.878 26.643 31.203 36.748 43.522 51.835 62.083 74.776 90.568 110.30 135.11 166.39 206.08 256.65 321.37 404.63 512.35 5.2346 5.6868 6.2096 6.8154 7.5196 8.3403 9.3000 10.426 11.752 13.319 15.179 17.393 20.042 23.224 27.061 31.710 37.365 44.278 52.765 63.232 76.203 92.347 112.54 137.91 169.94 210.59 262.39 328.68 414.26 524.87 5.2769 5.7357 6.2662 6.8811 7.5961 8.4296 9.4047 10.549 11.897 13.491 15.383 17.637 20.334 23.575 27.486 32.225 37.996 45.051 53.716 64.406 77.663 94.168 114.82 140.79 173.60 215.24 268.29 336.35 423.99 537.15 5.3199 5.7853 6.3236 6.9479 7.6737 8.5204 9.5110 10.674 12.045 13.666 15.591 17.886 20.632 23.934 27.920 32.752 38.638 45.837 54.686 65.609 79.153 96.029 117.15 143.72 177.32 219.93 274.38 344.18 433.91 550.34 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 TABLE X1.7 Observed Choice Proportions, pc, (x104) as a Function of d’ for the 2-AFC TestA 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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 5000 5282 5562 5840 6114 6382 6643 6897 7142 7377 7602 7817 8019 8210 8389 8556 8711 8853 8985 9104 9214 9312 9401 9481 9552 9615 9670 9719 9761 9798 9831 9858 9882 9902 9919 5028 5310 5590 5868 6141 6408 6669 6922 7166 7400 7624 7837 8039 8229 8406 8572 8725 8867 8997 9116 9224 9321 9409 9488 9558 9620 9675 9723 9765 9802 9833 9861 9884 9904 9921 5056 5338 5618 5895 6168 6434 6695 6947 7190 7423 7646 7858 8058 8247 8423 8588 8740 8881 9009 9127 9234 9331 9418 9495 9565 9626 9680 9728 9769 9805 9836 9863 9886 9906 9922 5085 5366 5646 5923 6195 6461 6720 6971 7214 7446 7668 7879 8078 8265 8440 8603 8755 8894 9022 9138 9244 9340 9426 9503 9571 9632 9685 9732 9773 9809 9839 9866 9888 9907 9924 5113 5394 5674 5950 6221 6487 6746 6996 7237 7469 7689 7899 8097 8283 8457 8619 8769 8907 9034 9149 9254 9349 9434 9510 9578 9638 9690 9737 9777 9812 9842 9868 9890 9909 9925 5141 5422 5702 5977 6248 6513 6771 7021 7261 7491 7711 7919 8116 8301 8474 8635 8783 8920 9046 9160 9264 9358 9442 9517 9584 9643 9695 9741 9781 9815 9845 9870 9892 9911 9926 5169 5450 5729 6005 6275 6539 6796 7045 7284 7514 7732 7940 8135 8319 8491 8650 8798 8933 9058 9171 9274 9367 9450 9524 9590 9649 9700 9745 9784 9818 9848 9873 9894 9912 9928 5197 5478 5757 6032 6302 6565 6822 7069 7308 7536 7754 7960 8154 8337 8507 8665 8812 8946 9070 9182 9284 9375 9458 9531 9596 9654 9705 9749 9788 9821 9850 9875 9896 9914 9929 5226 5506 5785 6059 6329 6591 6847 7094 7331 7558 7775 7980 8173 8354 8523 8681 8826 8959 9081 9193 9293 9384 9465 9538 9603 9659 9710 9753 9791 9824 9853 9877 9898 9916 9931 5254 5534 5812 6086 6355 6617 6872 7118 7354 7580 7796 8000 8192 8372 8540 8696 8840 8972 9093 9203 9303 9393 9473 9545 9609 9665 9714 9757 9795 9828 9856 9880 9900 9917 9932 E2262 − 03 (2014) TABLE X1.7 A Continued d’ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 9933 9945 9956 9964 9971 9977 9981 9985 9988 9991 9993 9994 9996 9997 9997 9998 9998 9935 9947 9956 9965 9972 9977 9982 9985 9988 9991 9993 9994 9996 9997 9997 9998 9998 9936 9948 9957 9965 9972 9978 9982 9986 9989 9991 9993 9995 9996 9997 9997 9998 9999 9937 9949 9958 9966 9973 9978 9983 9986 9989 9991 9993 9995 9996 9997 9998 9998 9999 9938 9950 9959 9967 9973 9979 9983 9986 9989 9992 9993 9995 9996 9997 9998 9998 9999 9940 9951 9960 9968 9974 9979 9983 9987 9990 9992 9994 9995 9996 9997 9998 9998 9999 9941 9952 9961 9968 9974 9980 9984 9987 9990 9992 9994 9995 9996 9997 9998 9998 9999 9942 9953 9962 9969 9975 9980 9984 9987 9990 9992 9994 9995 9996 9997 9998 9998 9999 9943 9954 9962 9970 9976 9980 9984 9988 9990 9992 9994 9995 9996 9997 9998 9998 9999 9944 9955 9963 9970 9976 9981 9985 9988 9990 9993 9994 9995 9996 9997 9998 9998 9999 Adapted from Ennis, D M., “The Power of Sensory Discrimination Methods,” Journal of Sensory Studies, 8, 1993, pp 353-370 TABLE X1.8 The B Values for Estimating the Variance of d’ Obtained from a 2-AFC TestA NOTE 1—Enter the table in the row and column corresponding to the value of d’ obtained from Table X1.7 The variance of d’ is S2(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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 3.1416 3.1473 3.1645 3.1934 3.2345 3.2881 3.3550 3.4361 3.5324 3.6453 3.7763 3.9274 4.1007 4.2989 4.5250 4.7828 5.0766 5.4114 5.7933 6.2295 6.7285 7.3005 7.9577 8.7145 9.588 10.601 11.777 13.147 14.750 16.631 18.846 21.465 24.573 28.277 32.706 38.028 44.447 52.221 61.678 73.231 87.41 104.88 126.52 153.42 187.03 229.20 282.42 349.85 435.55 3.1417 3.1485 3.1669 3.1970 3.2392 3.2942 3.3624 3.4450 3.5429 3.6576 3.7905 3.9437 4.1194 4.3202 4.5493 4.8105 5.1081 5.4473 5.8343 6.2764 6.7823 7.3622 8.0286 8.7963 9.683 10.710 11.904 13.296 14.924 16.836 19.088 21.752 24.915 28.685 33.196 38.617 45.158 53.085 62.731 74.522 88.99 106.84 128.95 156.46 190.84 234.00 288.43 357.46 445.37 3.1418 3.1498 3.1694 3.2007 3.2442 3.3004 3.3700 3.4541 3.5536 3.6700 3.8049 3.9602 4.1383 4.3418 4.5739 4.8386 5.1401 5.4838 5.8759 6.3240 6.8367 7.4247 8.1005 8.8793 9.779 10.822 12.034 13.448 15.102 17.045 19.335 22.044 25.262 29.100 33.694 39.217 45.884 53.966 63.805 75.838 90.62 108.85 131.44 159.55 194.72 238.88 294.62 365.30 455.30 3.1421 3.1513 3.1720 3.2045 3.2492 3.3067 3.3778 3.4633 3.5645 3.6826 3.8195 3.9770 4.1574 4.3637 4.5989 4.8670 5.1724 5.5207 5.9181 6.3722 6.8919 7.4881 8.1735 8.9635 9.876 10.935 12.166 13.601 15.282 17.257 19.586 22.342 25.616 29.523 34.201 39.828 46.623 54.864 64.903 77.181 92.27 110.89 133.97 162.72 198.70 243.91 301.00 373.36 465.61 3.1425 3.1528 3.1747 3.2084 3.2544 3.3132 3.3857 3.4727 3.5755 3.6954 3.8343 3.9939 4.1768 4.3858 4.6241 4.8957 5.2052 5.5581 5.9608 6.4211 6.9479 7.5524 8.2475 9.0489 9.975 11.050 12.299 13.758 15.466 17.473 19.841 22.644 25.977 29.953 34.718 40.450 47.377 55.782 66.021 78.552 93.96 112.98 136.58 165.96 202.76 248.99 307.42 381.61 476.25 3.1430 3.1545 3.1775 3.2124 3.2597 3.3198 3.3937 3.4822 3.5867 3.7084 3.8492 4.0111 4.1965 4.4083 4.6497 4.9249 5.2385 5.5960 6.0041 6.4706 7.0047 7.6176 8.3226 9.1356 10.076 11.166 12.435 13.917 15.652 17.692 20.100 22.952 26.343 30.392 35.244 41.085 48.146 56.716 67.163 79.954 95.68 115.12 139.23 169.28 206.92 254.24 314.06 390.07 486.93 3.1436 3.1562 3.1805 3.2166 3.2651 3.3266 3.4019 3.4920 3.5981 3.7216 3.8645 4.0286 4.2165 4.4310 4.6757 4.9544 5.2722 5.6344 6.0480 6.5208 7.0622 7.6837 8.3988 9.2235 10.178 11.285 12.573 14.078 15.841 17.915 20.364 23.265 26.716 30.838 35.780 41.732 48.929 57.669 68.328 81.384 97.45 117.30 141.95 172.68 211.18 259.65 320.90 398.73 498.06 3.1444 3.1581 3.1835 3.2209 3.2706 3.3335 3.4102 3.5018 3.6096 3.7350 3.8799 4.0462 4.2367 4.4541 4.7019 4.9844 5.3063 5.6734 6.0925 6.5717 7.1206 7.7508 8.4760 9.3128 10.281 11.405 12.713 14.242 16.034 18.142 20.632 23.583 27.096 31.292 36.327 42.392 49.728 58.641 69.516 82.844 99.25 119.53 144.72 176.14 215.54 265.13 327.80 407.54 509.36 3.1452 3.1601 3.1867 3.2253 3.2763 3.3405 3.4187 3.5119 3.6213 3.7486 3.8955 4.0642 4.2571 4.4774 4.7285 5.0147 5.3408 5.7128 6.1375 6.6232 7.1797 7.8188 8.5543 9.4033 10.386 11.527 12.856 14.408 16.230 18.373 20.905 23.908 27.482 31.755 36.883 43.063 50.544 59.632 70.729 84.335 101.09 121.81 147.55 179.68 219.99 270.75 334.99 416.72 521.03 3.1462 3.1623 3.1900 3.2298 3.2821 3.3477 3.4273 3.5220 3.6332 3.7624 3.9113 4.0823 4.2779 4.5011 4.7555 5.0454 5.3759 5.7528 6.1832 6.6755 7.2397 7.8877 8.6338 9.4952 10.492 11.651 13.000 14.578 16.428 18.607 21.183 24.237 27.876 32.227 37.451 43.748 51.374 60.644 71.968 85.852 102.97 124.14 150.46 183.32 224.55 276.51 342.25 426.01 532.78 10