Extra-Sensory Perception: Part III. Explanation and Discussion: Second Appendix to Chapter 15.

Throughout this report the values of groups of data have been presented in the form X=D/p.e. In the beginning of the experiment this ratio was important as a proof of the inadequacy of the chance hypothesis. When the values grew far beyond the limits of any available tables the X ratio was useful mainly as an arbitrary quantitative measure of differences between blocks of data and between the work of individuals. These X ratios may serve their original purpose as anti-chance ratios, however, if any reader still clings to the Chance Hypothesis.

The following table (Table XLIV) gives an approximation to the meaning in terms of chance probability of the X values listed in Table XLIII [10^-14 means 1 fourteen places to the right of the decimal point (.000 000 000 00001). This gives odds of 100 thousand billions to one against mere chance as an explanation. It is left to the reader to write out the figure corresponding to 10^-1220.]

X	P	X	P	X	P	X	P
13	10^-14	22	10^-51	57	10^-320	91	10^-820
16	10^-22	23	10^-56	65	10^-425	111	10^-1220
19	10^-34	24	10^-61	82	10^-666

The figures given are rough approximations computed by using only the coefficient of the series expansion for the probability integral. 2 As

the error of such an approximation is less than the last term used in the series, and as we have used only the first term, 1, the error in P is equal to the figure given; that is, an error of 100%. This would be a serious error in a pay envelope or cost estimate, but with these tiny ratios it is negligible. For example: if X = 20 the approximation would give P = 10^-40.744, but with an error of 100% the actual value of P may lie anywhere between 10^-44.443 and 10^-∞. As we are here concerned with only the upper limit of the error the difference between . (40 zeros)18 and . (40 zeros)36 is considerably less than the errors introduced by our using only two and three significant figures in the tabulation of the exponents. 1

And a final word of warning to the mathematical enthusiast. Mathematically the difference between X = 20 and, let us say, X = 40, is enormous beyond our capacity to experience. For that reason, for the purpose of evidence in opposition to a Chance Hypothesis the difference is simply too superfluous to be appreciated. The higher values do give the experimenter a quantitative measure of comparison and a feeling of comfortable certainty, but the skeptic who refuses to be convinced by a value of 15 or 20 will scarcely be moved by any mathematical treatment of the data whatever.

Footnotes

168:1 I am indebted to Mr. Charles E. Stuart, Graduate Assistant in Psychology, for this Appendix, as well as for much assistance in the earlier mathematics of the production. He has been, as mathematician, assistant, and subject, an extraordinarily useful and competent man.

168:2 The formula is given in Rietz, H. L.: Handbook of Mathematical Statistics. P. 15.

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X	P	X	P	X	P	X	P
13	10^-14	22	10^-51	57	10^-320	91	10^-820
16	10^-22	23	10^-56	65	10^-425	111	10^-1220
19	10^-34	24	10^-61	82	10^-666

The figures given are rough approximations computed by using only the coefficient of the series expansion for the probability integral. 2 As

Footnotes

168:2 The formula is given in Rietz, H. L.: Handbook of Mathematical Statistics. P. 15.

169:1 This paragraph may be regarded as primarily for mathematicians since it would require much space to make it clear to the layman. It simply justifies the approximation-method used in measuring the odds against chance implied by a given X-value. Since few men have ever required such odds for establishing belief in a principle, the tables are not available; and the computation by more exact methods would be a burdensome—and a uselessly burdensome—task.

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Second Appendix to Chapter 15 1

Footnotes

Footnotes