Extra-Sensory Perception: Part I. General Introduction: Appendix to Chapter 2. Mathematics of Probability Used in Evaluation

APPENDIX TO CHAPTER 2

Mathematics of Probability Used in Evaluation

From the beginning of the scientific period of parapsychology, the subject has had the aid of mathematical methods in its technique of evaluation. Professor Richet first introduced the mathematics of probability into this field in his treatment of the results of his earlier work on "suggestion mentale" or "telepathy", in 1884. 2 And since then the names of Edgeworth and R. A. Fisher of England and of Hawkesworth in America have appeared frequently in connection with probability estimation in the parapsychic branch of the field.

I am no mathematician and must rely upon methods already developed, when they can be found. But in this work it is fortunately possible to make experimental method conform to easy computation of significance of results and this I have done. I have been able, by adhering to the use of five simple card-figures, to keep the probability of success by pure chance at 1/5 for each trial. Where a straight run of consecutive

1 I shall cite the odds against chance for the smaller values of D/p.e. (Deviation divided by the probable error.)

D/p.e.	Odds against a chance-theory.
1	1 to 1
2	4.6 to 1
3	22 to 1
4	142 to 1
5	1,300 to 1
6	20,000 to 1

2 I get approximately,

7	100,000 to 1
8	nearly 1,000,000 to 1
9	over 100,000,000 to 1

Now these values of X for particular groups of results have a progressive effect upon the mind. That is, if there are three groups, each with an X-value of 6, we can agree that these are more impressive than only one group with an X-value of 6. How much more? And how determine this? I have searched in vain for authority on this point, and have finally attempted a solution which I submit here and use in this report. It is tentatively offered and may be later rejected for a better method, if such is pointed out to me. I have made certain that this method errs, if it errs at all, on the safe side. And it is not at all necessary to any major issue of this report to use it. The reasons for using it are: first, there is needed an easy way of summating the "anti-chance" significance of many groups of results, instead of pooling them all together and getting the value of X after each addition through the report. But, second and more important, in such pooling together the results made by the high scorers are merged with perhaps a greater number of the poor scorers, so losing the greater contribution they made in the general assumption of equal distribution over the whole lot. A short series of 1000 trials by a good subject may well reach a higher figure for X than a poor scorer (only a little above mean expectation) over a series of 10,000 trials. For some purposes it is proper to pool these but for others it is proper to summate their joint effect against the chance-hypothesis by another method which gives proper weight to the scoring rates for each group. And, third, there is the reason that I have in some cases to deal with negative deviations, under conditions in which I tried to secure low results and succeeded. These, too, have their statistical significance and add, quite as well as the positive deviations, to the general weight of the conclusions. But if these were to be pooled with the totals, they would of course only detract from the total value. (Even this, however, would not at all destroy any of our conclusions, because of the large margin of safety.)

One may see the propriety of combining these values of X by remembering that each such value has a corresponding value (See Normal Probability Tables) representing the probability that the deviation it represents was due to chance alone; for example, for X = 3, this is 1/22; for X = 4, 1/142; for X = 6, 1/20,000. Now, three such values of X (for results given under conditions that permit generalization) can be combined by multiplying the three probability fractions and thus the total odds against chance be computed. (This is simple for low values but the needs of this report take in large values of X as well as small; and I have not found tables for the probabilities for large values of X.) Now, with the smaller

1 If larger X-values can thus be calculated from smaller in this case (as was demonstrated) and if X-values are independently usable values (as they logically have to be), the method must stand as checked, to the extent of accuracy claimed, which is all that is needed for this work.

The formula has, therefore, been used in this report and is in any case safe from exaggerative effect on the general results. And it will, I hope, serve at the same time to raise the problem for those readers who may be on better terms with the "Queen of the Sciences".

Footnotes

31:2 Richet, Charles, La Suggestion Mentale et le calcul des Probabilités. Rev. Phil., 1884. For a full review in English see Gurney, Proc. S.P.R., II: pp. 239-256, 1884.

32:1 McGraw-Hill, New York, 1925, p. 180.

32:2 3rd Ed., Oliver and Boyd, London, 1930.

34:1 There is a similar practical check of the formula in Table XLIII, in the final chapter, in which the X-value is given for the results reported in the various chapters. That value for the results reported in Chapter 8 is almost the same for both ways of computing the X-value (81.9 for the formula. and 82.1 for the computation based on the pooling together of all the results). Now, here the evenness of distribution of scoring-rates for the five major subjects makes the pooling together do no violence to the resulting values. They would not have checked had the individual differences been great. Then the formula would have given the more correct value, as it does for the other chapters represented in Table XLIII.