Volume 1

Collected papers of R. A. Fisher / edited by J.H. Bennett.

  • Ronald Fisher
Date:
1971-1974
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Collected papers of R. A. Fisher / edited by J.H. Bennett. Source: Wellcome Collection.

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    12. The fact that the mean value f of the observed correlation coefficient is numerically less than p might have been interpreted as meaning that given a single observed value r, the true value of the correlation coefficient of the population from which the sample is drawn is likely to be greater than r. This reasoning is altogether fallacious. The mean f is not an intrinsic feature of the frequency distribution. It depends upon the choice of the particular variable r in terms of which the frequency distribution is represented. When we use t as variable, the situation is reversed. Whereas in using r we cramp all the high values of the correlation into the small space in the neighbourhood of r — 1 , producing a frequency curve which trails out in the negative direction and so tending to reduce the value of the mean, by using t, we spread out the region of high values, producing asymmetry in the opposite sense, and obtain a value t which is greater than t. The mean might, in fact, be brought to any chosen point, by stretching and compressing different parts of the scale in the required manner. For the interpretation of a single observation the relation between t and r is in no way superior to that between f and p. The variable t has been chosen primarily in order to give stability of form to the frequency curves in different parts of the scale. It is in addition a variable to which the analysis naturally leads us, and which enables the mean and moments to be readily calculated, and so a comparison to be made with the standard Pearson curves, but it is not, with these advantages, in a unique position. In some respects the function, log tan ^ (a + ^) , is its superior as independent variable. I have given elsewhere* a criterion, independent of scaling, suitable for obtaining the relation between an observed correlation of a sample and the most probable value of the correlation of the whole population. Since the chance of any observation falling in the range dr is proportional to 11-1 for variations of p, we must find that value of p for which this quantity is a maximum, and thereby obtain the equation Since we have dx 1 / 3 V 1 1 (cosh x + cos [ n — 1 Vsin 6 3 Q] 6 1 2 C-f) n- 1 2~ dx (cosh x + cos 9) n 1 = 0, * R. A. Fisher, “ On an absolute criterion for fitting frequency curves,” Messenger of Mathematics, February, 1912.