214 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS In sensory testing using taste, it has long been customary to use tests of this form. The most widely used is the Triangle Test in which panel members are offered three samples, two of one kind and one of another, and are requested to identify the odd sample (2-6). In such a test, which has to be carefully drawn up and executed to present every possible ordering of the samples in a balanced experimental design, so that the two samples appear as odd samples an equal number of times and in such ordering (i.e. the orderings ABB, BAB, BBA, AAB, ABA, BAA, but in random se- quences), the panel members have a 1 in 3 probability of picking the odd sample by chance even if they have no discriminating power. It is therefore necessary to rely on panels of judges to establish whether there is a distinguishable difference between two samples if three people correctly perform a triangle test, the probability of them doing so by pure chance is but («)3 or 1 in 27. Tables are available for testing the statistical significance of less conclusive results. Thus, if out of a panel of 20 members, 11 correctly identify the odd samples instead of the seven which would be expected on the null hypothesis, then the probability of so doing is less than the conventional 1 in 20 level used so often by statisticians and others when they can think of no valid reason for choosing any other level. A simple table may be found in Ostle (6). A similar procedure has been used by Harries (7) in situations involving the testing of foodstuffs. Here the panel members are offered three samples of one kind and two of the other (in random order) and the probability of a correct classification assuming a null hypothesis of chance selection is 1 in 10. Clearly such a test procedure will be more sensitive than the triangle test, and fewer panel members will be required to establish that a given sensory difference is distinguishable. Indeed, the principle involved may be generalised to an n+m test, in Table I. Probability of correct discrim[nation between samples on assumption of random selection. m n 1 2 3 4 5 6 (1.000) 0.333 0.250 0.200 0.167 0.143 (0.333) 0.100 0.067 0.048 0.036 (0.100) 0.029 0.018 0.012 (0.029) 0.008 0.005 (0.008) 0.002 (0.002)

SENSORY TESTING -- A STATISTICIAN'S APPROACH 215 which a total set of n+m samples, n being of one kind and m being of the other, are given to the panel members to sort into the two categories. Table I presents the probabilities for such tests. Table I only records half of the probabilities, but it is to be noted that (n -5 m) -- (m -5 n) . The diagonal entries printed in brackets give the pro- babilities for those cases where the total set of samples is divided into two equal groups they also refer to the case when the panel member is not required to identify either of those groups in any way. Should the panel member be asked to identify one set as being (for example) the stronger perfume, then those probabilities should be halved. Efficiency of designs for assessing whether detectable differences exist may be assessed by finding the lowest probability for a particular total number of samples. The most efficient designs for discrimination are to be found when n-m = 1 the reader may check this for himself from Table I. It should also be noted that the probability associated with a test in which n=m is the same as that for n+m where m=n-1, e.g. the probability of discrimination on the null hypothesis for a 2-52 test is «, which is exactly the same as for a 2-51 test. However, it is to be noted that the conduct of a test in which n • m is complicated by the fact that the experimenter has to make a decision about which of the two preparations is allocated to n and which to m. This is not a trivial decision because experience shows that in these two cases, the probability of discrimination may not be the same. In triangle (2-51) tests of flavour, for example, in which peppermint oils are being compared, it has been shown that it is easier to detect a stronger flavour if it is being compared with two weaker flavours than vice-versa. For this reason all possible orderings should be tested by far the simplest way of achieving this is to give the panel member equal numbers of each sample to test in whatever order he or she finds most simple to discriminate. Moreover, as we shall see, the testing procedure is frequently extended to involve questions of preference under such circumstances it can also be observed that the unbalanced designs (where n g=m) appear to bias or at least to have an effect upon the preference judgements preferences in some cases appear to go in favour of the larger number, and in other cases in favour of the smaller number by margins that cannot be due to chance. It appears desirable to eliminate this complication by making n =m. Let us therefore examine more closely those balanced test designs. The 1-51 test is clearly a nonsense we can always correctly divide up A from B without this telling us anything, hence the probability of correctly dividing