will A make only 7.5 errors in the average set of 100 observations? We had p=.032 or log (1+p)=.01377, and 0-1(1—2n) will now be equal to 1 (.85)=1.02. log (1 + x)= (.01377)(1.02) =.0295, and (1+x)=1.07. Answer: 1.07. III. One can also find the ratio of error at any ratio of stimuli when the ratio of error with one ratio of stimuli is given, by the following formula. Find log (1+p) as before. Then find n in the following formula where (1+x) represents the new ratio of stimuli, .477 log (1 + x) log (1+p) n 1 일 2 Example 4.-Find the ratio of wrong answers in Example 1 when the ratio of stimuli is 1.1. We have log (1+ p)=.01377; log (1 + x)= log (1.1) =.0414. Hence IV. I will also show how a practical threshold can be obtained if desired. Example 5.-Taking the practical threshold at one error in 100 answers and the probable error within its extreme limits in the case of the writer in the experiments on pressure above referred to, viz. .05 and .016, at what ratio will this threshold occur? log (1.05).0212; log (1.016)=.00689; 0—1(1 — 2n) = 01(.98)=1.649, Answer: 100 ounces and 105.6 ounces in the first case; 100 and 118.4 ounces in the second case. Note.-Intermediate values in this table are derived by interpolation in the ordinary way. APPENDIX D. Rules for Computing the Probable Error. These rules I take from Jevons, Principles of Science, p. 387. 1. "Draw the mean of all the observed results. 2. Find the excess or defect, that is, the error in each result from the mean. 3. Square each of these reputed errors. 4. Add together all these squares of the errors, which are of course all positive. 5. Divide by one less than the number of observations. This gives the square of the mean error. 6. Take the square root of the last result; it is the mean error of a single observation. 7. Divide now by the square root of the number of observations, and we get the mean error of the mean result. 8. Lastly, multiply by the natural constant 0.6745 (or approximately by 0.674, or even by ), and we arrive at the probable error of the mean result." For illustrations of this process and methods for shortening the work see Jevons and works on "Probabilities" there referred to. It is generally advisable to divide up the observations, and find the probable error of each group and then draw a mean. It is also sometimes desirable to be able to test how closely the number of errors of each degree of deviation from the mean follows the number assigned by the probability curve. Mr. Francis Galton gives an admirable account of this in an appendix to his "Hereditary Genius," to which the reader is referred. PSYCHOLOGICAL LITERATURE. 1.-EXPERIMENTAL PSYCHOLOGY. Zur Psychophysik des Lichtsinns. Von HJALMAR NEIGLICK. Philosophische Studien, IV, 1, pp. 28-112. This is a continuation of the experimental study begun by Dr. Lehmann in the former number of the Studien. The method used is that of the "mean gradations," and consists in rapidly rotating three discs, each containing a certain amount of black and white, so that in rotating a uniform gray of a lighter or darker tint is produced, and in requiring the observer to regulate the amount of black on one of these discs so that it shall produce a gray exactly intermediate between the constant grays of the darker and the lighter discs. If the amount of black on the adjustable disc proves to be the mean proportional between that on the light and that on the dark disc, Weber's law holds. Lehmann's elaborate study brought out the many sources of error in this experiment, and above all, the enormous effect of the contrast of the disc with its background. It was found best to set each disc against a background of its own tint; this can readily be done for the two constant discs, but seems difficult to do for the medium disc without giving the observer a clue as to the tint he ought to choose. Neiglick solved this problem by having the background itself a disc much larger than the one to be adjusted, but similarly marked as to white and black, so that when both rotate on a common axis, the adjustable disc, like the others, is seen against its own background. With all these precautions it was found that in a general way Weber's law held, and seemed to hold the more rigidly the more carefully the experiment was conducted. But a new result, on which Professor Wundt, in a note to this article, lays much stress, is that the absolute difference in grayness between the extreme discs affects the validity of the law in other words, while the mean proportional between x and y is ry, and the mean proportional between 2 and 2y is also ry, yet, as a fact, the adjustment of the one pair will be nearer the mean proportional than that of the other pair. And the difference between the discs in which the law has its greatest validity corresponds to that relation of the tints of the two discs at which the researches of Lehmann showed that the maximum amount of mutual contrast occurs. For example, a setting in which the one disc is entirely white and the other 40° of black is one of the relations at which the law most closely holds. An interesting discussion of the bearing of the phenomena of contrast on Weber's law closes the article. Two remarks may be added to the account of this research: the first is that it proves the extreme intricacy of this psychophysical method, and yields an excellent instance of the way in which side effects can entirely distort the law of a series of phenomena; the |