Karl Pearson

On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling

Let x1, x2 … x n be a system of deviations from the means of n variables with standard deviations σ1, σ2 … σ n and with correlations r12, r13, r23 … r n −1,n.

Mathematical contributions to the theory of evolution.—On a form of spurious correlation which may arise when indices are used in the measurement of organs

(1) If the ratio of two absolute measurements on the same or different organs be taken it is convenient to term this ratio an index. If u = f1(x, y) and v = f2(x, y) be two functions of the three variables x, y, z, and these variables be selected at random so that there exists no correlation between x, y, y, z, or z, x, there will still be found to exist correlation between u and v. Thus a real danger arises when a statistical biologist attribntes the correlation between two functions like u and v to organic relationship. The particular case that is likely to occur is when u and v are indices with the same denominator for the correlation of indices seems at first sight a very plausible measure of organic correlation.

The Problem of the Random Walk

CAN any of your readers refer me to a work wherein I should find a solution of the following problem, or failing the knowledge of any existing solution provide me with an original one? I should be extremely grateful for aid in the matter.

Mathematical contributions to the theory of evolution. VIII. On the correlation of characters not quantitatively measurable

1. In August last I presented to the Society a memoir on the inheritance of coat-colour in thoroughbred horses, and of eye-colour in man. This memoir, which was read in November of last year, presented the novel feature of determining correlation between characters which were not capable à priori of being quantitatively measured. The theoretical part of that memoir was somewhat brief, but I showed by illustrations that the method could be extended to deal with problems like the effectiveness of vaccination and of the antitoxin treatment in diphtheria.

Mathematical Contributions to the Theory of Evolution. V. On the Reconstruction of the Stature of Prehistoric Races.

1. The object of this memoir is to illustrate the general theory by which we may reconstruct from the knowledge of one organ in a fossil or prehistoric race, the dimensions of other organs, when the correlation between organs in existing races of the same species has been ascertained. The particular illustration chosen is the reconstruction of probable stature from a measurement of the long bones.

Tables of the Incomplete Gamma-Function

I SHOULD be greatly obliged if you could allow me a little of your valuable space to state that Dr. J. F. Tocher has kindly pointed out an error in my Introduction to the above Tables. In a table on page xx the wrong argument has been inserted to the correct value of the function.

On the Mathematical Theory of Errors of Judgment, with Special Reference to the Personal Equation

1. The following investigation has been in progress for some years and led to a paper, communicated to the Society on December 29, 1896. I therein pointed out that personal judgments were frequently correlated. This correlation may be of the kind which in that paper I termed “spurious,” or it may be genuine. By “spurious” correlation I understand the quantitative measure of a resemblance in judgments, which resemblance is due solely to the particular manipulation of the observations. Very customary treatment of observations will lead to the existence of a spurious correlation, which may be and generally is entirely overlooked by the observers. For example: if the quantity to be determined by judgment were the time taken by a bright point, say a star, in travelling from a position C intermediate between spider lines A and B to the line B, and the result were to be expressed by the ratio of this time to the known time from A to B, then there would be correlation in the results obtained by two observers for a number of stars, even if their absolute judgments on the time from C to B were quite independent. Again, if the judgments of two observers be in both cases referred to a standard observer, then such relative judgments will be found to be correlated; and this is true, although if we could find the absolute errors of the two observers, we might discover that these errors were quite uncorrelated. We shall see illustrations below of the manner in which this spurious correlation almost imperceptibly creeps into any ordinary method of manipulating observations, and how very little attention has hitherto been paid to it. But apart from this spurious correlation the experiments described in this memoir seem to show that there exists almost invariably a genuine correlation between the judgments of independent observers. This may be due to two sources: (i.) Likeness of the environment in the case of each individual observation, which leads to likeness of judgment in the individual observers. One experiment may appear to be made under precisely the same conditions as a second, but really it has a certain atmosphere of its own which influences the observers in a like manner, (ii.) Likeness in the physical or intellectual characters of the observers leading to a likeness in their judgments of what took place.

Tracts for Computers

I REGRET that certain errata have been found in No. III. of the above Tracts. As they might cause confusion to any one computing from one of the formulae affected, I have had an erratum slip printed, which can be obtained by purchasers of the above series by sending a stamped and addressed envelope either to Mr. C. F. Clay, Cambridge University Press, Fetter Lane, E.G.4, or to The Secretary, Biometric Laboratory, University College, Gower St., W.C.1.

Contributions to the Mathematical Theory of Evolution. III. Regression, Heredity, and Panmixia.

The object of this paper is to develop the methods and generalise the conclusions of Mr. Francis Gallon’s work on ‘Natural Inheritance.’ It endeavours to show the wide field which a purely statistical (as distinguished from a mechanical or physiological) theory of heredity may be made to cover. In order to do this it is needful to define certain biological terms in such a manner that they are capable of quantitative measurement, the symbols in terms of which they are expressed being the standard-deviations, correlation-coefficients, and regression-coefficients already well known from the labours of Mr. Galton.

Data for the problem of evolution in man. II. A first study of the inheritance of longevity and the selective death-rate in man

1. According to Wallace and Weismann the duration of life in any organism is determined by natural selection. An organism lives so long as it is advantageous, not to itself, but to its species, that it should live.