# Extracts from Alice Lee's papers

Alice Lee wrote quite a few papers, some as the single author but most in collaboration. Many of the papers appear to be her work or the work of Lee and another assistant of Karl Pearson, but the paper itself was actually written by Pearson. This results in the use of "I", referring to Pearson, in papers with several authors. Let us note here that Alice Lee did outstanding work in arguing, with much well presented statistics, for women being intellectually equal to men. She did not, however, oppose Karl Pearson's racial views and many of the papers refer to civilised races and uncivilised races, terms quite standard for the time but which make uncomfortable reading today. She also seems to have accepted Pearson's views on eugenics which are totally unacceptable to the majority of people today.

1896 1897 1898 1899 1900 1901 1902 1903 1904

1905 1907 1908 1910 1914 1915 1917 1925 1927

**Click on a link below to go to the papers published in that year**1896 1897 1898 1899 1900 1901 1902 1903 1904

1905 1907 1908 1910 1914 1915 1917 1925 1927

**1896**

**1896.1:**Karl Pearson and Alice Lee, Mathematical Contributions to the Theory of Evolution. On Telegony in Man, &c.,

*Proceedings of the Royal Society of London*

**60**(1896-1897), 273-283.

The term telegony has been used to cover cases in which a female A, after mating with a male B, bears to a male C offspring having some resemblance to or some peculiar characteristic of A's first mate B. The instances of telegony usually cited are (i) cases of thoroughbred bitches when covered by a thoroughbred dog, reverting in their litter to half-breds, when they have been previously crossed by dogs of other races. Whether absolutely unimpeachable instances of this can be produced is, perhaps, open to question, but the strong opinion on the subject among dog-fanciers is at least remarkable; (ii) the case of the quagga noted by Darwin (see 'Origin of Species'), and still more recently (iii) a noteworthy case of telegony in man cited in the 'British Medical Journal' (22 February 1896).

In this latter case a very rare male malformation, which occurred in the male B, was found in the son of his widow A, by a second husband C. Here, as in the other cases cited, a question may always be raised as to the possibly unobserved or unknown occurrence of the characteristic in the ancestry of either A or C, or again as to the chance of the characteristic arising as a congenital sport, quite independently of any heredity. It seems unlikely that the observation of rare and isolated cases of asserted telegony will lead to any very satisfactory conclusions, although a well-directed series of experiments might undoubtedly do so. On the other hand, it is not impossible than an extensive and careful system of family measurements might bring to light something of the nature of a telegonic influence in mankind.

**1897**

**1897.1:**Alice Lee and Karl Pearson, Mathematical Contributions to the Theory of Evolution. On the Relative Variation and Correlation in Civilised and Uncivilised Races,

*Proceedings of the Royal Society of London*

**61**(1897), 343-357.

The following numerical data were calculated in the hope of reaching some general ideas on the comparative variation and comparative correlation in the case of civilised and uncivilised races, and further of determining, if possible, any general law connecting relative sexual variation and relative sexual correlation with the degree of civilisation, and so with what is probably inversely proportional to the degree of civilisation, namely, the intensity of natural selection.

The following two principles seem to flow from a study of variation in the organs of man:-

(a) Civilised man is more variable than uncivilised man.

(b) There is a greater equality of variation for the two sexes in uncivilised than in civilised races. Civilised woman appears, on the whole, to be slightly more variable than civilised man.

Both these principles are in accordance with the intensity of the struggle for existence - and the amount, consequently, of natural selection - being greater for uncivilised than for civilised races, and, further, greater for men than for women in the latter races.

The problem of correlation is, however, of a less simple character. While the action of selection can be shown theoretically to reduce variation, it by no means follows that it reduces correlation. Indeed, selection may increase, decrease, or reverse correlation at the very same time as it is reducing variation. We have then the following problems to guide us in our treatment of actual statistics:-

(a) Is correlation more intense among civilised than among uncivilised races?

(b) How does the relative correlation of the sexes differ in civilised and uncivilised races?

(c) Is there any marked prepotency of either sex in the matter of correlation?

These are the problems which the present calculations were designed, not to definitely solve, but to illustrate.

**1897.2:**Karl Pearson and Alice Lee, On the Distribution of Frequency (Variation and Correlation) of the Barometric Height at Diverse Stations [Abstract],

*Proceedings of the Royal Society of London*

**61**(1897), 491-493.

Although this paper contains the results of a very large amount of arithmetical work, which has been in progress during the last two or three years, it is not intended in the first place as a contribution to the meteorology of the British Isles. It is especially intended as an illustration of method. The authors believe that hitherto no exact theory of variation or of correlation has been applied to meteorological observations, and they have endeavoured to indicate that fruitful results may be obtained from such a theory when applied to one branch at least of meteorology, namely, barometric frequency. They wished to deal with a fairly extended area with an easily accessible material, and this was found in the Meteorological Observations at Stations of the Second Order for the British Isles. The "telegraph" stations would have provided better material, but it was far less accessible. The authors have accordingly only dealt with three telegraph stations. The main body of their data was drawn from twenty stations of the second order, four of which are in Ireland, and the remainder distributed round the coast of England, Wales, and Scotland, as indicated on a chart accompanying the memoir.

...

The writers hope that their paper may draw attention to the importance of rendering the large amount of barometric observations now made, available for the easy calculation of the variation and correlation coefficients. They consider that if a chain of stations round a large continental area could have their correlation for a series of intervals of time worked out, much might be done in the way of very close prediction of barometric changes.

**1897.3:**Karl Pearson, Alice Lee and G U Yule, On the Distribution of Frequency (Variation and Correlation) of the Barometric Height at Divers Stations,

*Philosophical Transactions of the Royal Society of London. Series A*

**190**(1897), 423-469.

In a memoir published in the 'Phil. Trans.', a series of generalised frequency curves are introduced, and it is shown that the asymmetry of the barometric frequency curve can probably be dealt with by one or other of these generalised curves. The importance of this conclusion lies in the fact that the distribution of barometric frequency in any locality can then be fully described by the statement of the values of three or four well defined constants.

Accordingly, in order to test this theory of barometric frequency, series of observations have been reduced and the frequency distributions for divers localities fitted with generalised probability curves. On the basis of these curves the attempt has been made to answer the following questions:

(a) Is there any one type of curve especially characteristic of barometric frequency?

(b) If so, what are the constants by which the distribution of this frequency can best be described?

(c) Does there appear to be any numerical or geographical relation between these constants,? and

(d) Does a knowledge of their values for a variety of localities enable us to make any statement with regard to the physics of atmospheric pressure?

**1897.4:**Alice Lee and Karl Pearson, On the Relative Variation and Correlation in Civilized and Uncivilized Races (Conclusion of a communication made to the Royal Society),

*Science, New Series*

**6**(132) (1897), 49-50.

The general conclusion would then be that, with increased civilisation, absolute size and variation tend to increase, while correlation, to judge by the males, is stationary; to judge by the females, tends to increase. It will be found somewhat difficult to reconcile these results with any simple applications of the principle of natural selection. In the first place increased variation undoubtedly suggests a lessening of the struggle for existence, and there can be no question that this increase has gone on among civilised races (See 'Variation in Man and Woman'). The lessening of the struggle has probably been greater for woman than man; hence the principle of natural selection might help to explain the preponderance of variability in civilised woman. The increase in size with civilisation seems, on the average, also incontestable. But is it the effect of lessening the struggle for existence?

**1898**

**1898.1:**Karl Pearson, Alice Lee and Leslie Bramley-Moore, Mathematical Contributions to the Theory of Evolution. VI. Reproductive or Genetic Selection. Part I. Theoretical. and Part II. On the Inheritance of Fertility in Man. And Part III. On the Inheritance of Fecundity in Thoroughbred Race-Horses [Abstract],

*Proceedings of the Royal Society of London*

**64**(1898-1899), 163-167.

The object of this memoir is twofold: first, to develop the theory of reproductive or genetic selection on the assumption that fertility and fecundity may be heritable characters; and, secondly, to demonstrate from two concrete examples that fertility and fecundity actually are inherited.

The problem of whether fertility is or is not inherited is one of very far reaching consequences. It stands on an entirely different footing to the question of inheritance of other characters. That any other organ or character is inherited, provided that inheritance is not stronger for one value of the organ or character than another, is perfectly consistent with the organic stability of a community of individuals. That fertility should be inherited is not consistent with the stability of such a community, unless there be a differential death-rate, more intense for the offspring of the more fertile, i.e., unless natural selection or other factor of evolution holds reproductive selection in check. The inheritance of fertility and the correlation of fertility with other characters are principles momentous in their results for our conceptions of evolution; they mark a continual tendency in a race to progress in a definite direction, unless equilibrium be maintained by any other equipollent factors, exhibited in the form of a differential death-rate on the most fertile. Such a differential death-rate probably exists in wild life, at any rate until the environment changes and the equilibrium between natural and reproductive selection is upset. How far it exists in civilised communities of mankind is another and more difficult problem, which I have partially dealt with elsewhere. At any rate it becomes necessary for the biologist either to affirm or deny the two principles stated above. If he affirms them, then he must look upon all races as tending to progress in definite directions - not necessarily one, but possibly several different directions, according to the characters with which fertility may be correlated - the moment natural selection is suspended; the organism carries in itself, in virtue of the laws of inheritance and the correlation of its characters, a tendency to progressive change. If, on the other hand, the biologist denies these principles, then he must be prepared to meet the weight of evidence in favour of the inheritance of fertility and fecundity contained in Parts II and III of the present memoir.

**1899**

**1899.1:**Karl Pearson and Alice Lee, On the Vibrations in the Field Round a Theoretical Hertzian Oscillator [Abstract],

*Proceedings of the Royal Society of London*

**64**(1898-1899), 246-248.

The object of this paper is to investigate the types of wave motion in the neighbourhood of a theoretical Hertzian oscillator. ... The theory given [in current textbooks of electromagnetism], is insufficient for two reasons, both of which were recognised by Hertz himself, namely, because (i) the actual oscillator has sensible extension, and (ii) the wave train it gives forth is not steady.

The present paper only attempts to remove the latter objection to Hertz's original theory; like that theory it becomes less accurate as we approach nearer to an actual oscillator. Still the range within which the damping produces a very sensible divergence from Hertz's theory, seems sufficiently large to allow of experiment being made at a considerable distance from the oscillator; certainly the chief divergences between the present and Hertz's original theory actually fall in the portion of the field, wherein his chief interference experiments were made. Besides therefore the difficulties arising from the phenomena of "multiple resonance," it seems necessary to measure the influence of damping in modifying the mathematical results for a steady wave train, which results are what Hertz made use of in interpreting his interference experiments. The four sources of divergence between theory and experiment in Hertz's case, i.e.:

(i) the damping of the wave train,

(ii) the size of the oscillator,

(iii) multiple resonance,

(iv) defect of electro-magnetic theory,

may one or all be effective, but the object of the present paper is confined entirely to a theoretical investigation of the first.

**1899.2:**Karl Pearson, Alice Lee and Leslie Bramley-Moore, Mathematical Contributions to the Theory of Evolution. VI. Genetic (Reproductive) Selection: Inheritance of Fertility in Man, and of Fecundity in Thoroughbred Racehorses,

*Philosophical Transactions of the Royal Society of London. Series A*

**192**(1899), 257-330.

I understand by a factor of evolution any source of progressive change in the constants-mean values, variabilities, correlations - which suffice to define an organ or character, or the interrelations of a group of organs or characters, at any stage in any form of life. To demonstrate the existence of such a factor we require to show more than the plausibility of its effectiveness, we need that a numerical measure of the changes in the organic constants shall be obtained from actual statistical data. These data must be of sufficient extent to render the numerical determinations large as compared with their probable errors.

In a "Note on Reproductive Selection," published in the 'Roy. Soc. Proc.', I have pointed out that if fertility be inherited or if it be correlated with any inherited character - those who are thoroughly conversant with the theory of correlation will recognise that these two things are not the same-then we have a source of progressive change, a

*vera casa*of evolution. I then termed this factor of evolution Reproductive Selection. As the term has been objected to, I have adopted Genetic Selection as an alternative. I mean by this term the influence of different grades of reproductivity in producing change in the predominant type.

If there be two organs A and B both correlated with fertility, but not necessarily correlated with each other, then genetic or reproductive selection may ultimately cause, the predominance in the population of two groups, in which the organs A and B are widely different from their primitive types - 'widely different,' because reproductive selection is a source of' progressive change. Thus this form of selection can be a source, not only of change, but of differential change. As this differentiation is progressive, it may amount in time to that degree of divergence at which crossing between the two groups begins to be difficult or distasteful. We then reach in genetic or reproductive selection a source, of the origin of species.

When I assert that genetic (reproductive) selection is a factor of evolution, I do not intend at present to dogmatise as to the amount it is playing or has played in evolution. I intend to isolate it so far as possible from all other factors, and then measure its intensity numerically.

**1899.3:**Karl Pearson and Alice Lee, Mathematical Contributions to the Theory of Evolution. VII. - On the Application of Certain Formulae in the Theory of Correlation to the Inheritance of Characters Not Capable of Quantitative Measurement [Abstract],

*Proceedings of the Royal Society of London*

**66**(1899-1900), 324-327.

Many characters are such that it is very difficult if not impossible to form either a discrete or a continuous numerical scale of their intensity. Such, for example, are skin, coat, or eye-colour in animals, or colour in flowers. In other cases as in the amount of shading, degree of hairiness, &c., it might be possible by counting scales or hairs to obtain a numerical estimate of the character, but the labour in the case of several hundreds or a thousand individuals becomes appalling. Now these characters are some of those which are commonest, and of which it is generally possible for the eye at once to form an appreciation. A horse-breeder will classify a horse as brown, bay, or chestnut; a mother classify her child's eyes as blue, grey, or brown without hesitation and within certain broad limits correctly. It is clear that if the theory of correlation can be extended so as to readily apply to such cases, we shall have much widened the field within which we can make numerical investigations into the intensity of heredity, as well as much lessened the labour of collecting data and forming records.

The extension of theory required for such investigations is provided in a separate memoir. It is found that the sole conditions for applying this theory are: (1) that an order of intensity must exist even if there be no quantitative scale; (2) that the correlation must be supposed normal. If these assumptions are made, individuals may even be classified into only two groups of less and greater intensity, and the correlation still found. For example, the correlation between stature and hair-colour could be found by classifying all individuals simply into short and tall, light and dark haired, although for convenience of judgment a medium class in each case might be introduced. For the purpose of ascertaining the relative variability of the characters involved, this third or medium class at least must be introduced and a nine-fold division made of the correlation table. In the introduction to the present memoir the probable errors of all the quantities involved are considered, and illustrations given of their values for selected cases.

**1900**

**1900.1:**Karl Pearson and Alice Lee, Mathematical Contributions to the Theory of Evolution. VIII. On the Inheritance of Characters not Capable of Exact Quantitative Measurement. Part I. Introductory. Part II. On the Inheritance of Coat-Colour in Horses. Part III. On the Inheritance of Eye-Colour in Man,

*Philosophical Transactions of the Royal Society of London. Series A*

**195**(1900), 79-150.

A certain number of characters in living forms are capable of easy observation, and thus are in themselves suitable for observation, but they do not admit of an exact quantitative measurement, or only admit of this with very great labour. The object of the present paper is to illustrate a method by which the correlation of such characters may be effectively dealt with in a considerable number of cases. The conditions requisite are the following:

(i) The characters should admit of a quantitative order, although it may be impossible to give a numerical value to the character in any individual.

Thus it is impossible at present to give a quantitative value to a brown, a bay, or a roan horse, but it is not impossible to put them in order of relative darkness of shade. Or, again we see that a blue eye is lighter than a hazel one, although we cannot a priori determine their relative positions numerically on a quantitative scale. Even in the markings on the wings of butterflies or moths, where it might be indefinitely laborious to count the scales, some half dozen or dozen specimens may be taken to fix a quantitative order, and all other specimens may be grouped by inspection in the intervals so determined. We can even go a stage further and group men or beasts into simply two categories-light and dark, tall and short, dolichocephalic and brachycephalic - and so we might ascertain by the method adopted whether there is, for example, correlation between complexion and stature, or stature and cephalic index.

(ii) We assume that the characters are a function of some variable, which, if we could determine a quantitative scale, would give a distribution obeying - at any rate to a first approximation - the normal law of frequency.

The whole of the theoretical investigations are given in a separate memoir, in which the method applied is illustrated by numerical examples taken from inheritance of eye-colour in man, of coat-colour in horses and dogs, and from other fields. We shall not therefore in this paper consider the processes involved, but we may make one or two remarks on the justification for their use. If we take a problem like that of coat-colour in horses, it is by no means difficult to construct an order of intensity of shade. The variable on which it depends may be the amount of a certain pigment in the hair, or the relative amounts of two pigments. Much the same applies to eye-colour. In both cases we may fail to obtain a true quantitative scale, but we may reasonably argue that, if we could find the quantity of pigment, we should be able to form a continuous curve of frequency. We make the assumption that this curve-to at any rate a first approximation-is a normal curve. Now if we take any line parallel to the axis of frequency and dividing the curve, we divide the total frequency into two classes, which, so long as there is a quantitative order of tint or colour, will have their relative frequency unchanged, however we, in our ignorance of the fundamental variable, distort its scale. For example, if we classify horses into bay and darker, chestnut and lighter, we have a division which is quite independent of the quantitative range we may give to black, brown, bay, chestnut, roan, grey, &c.

**1900.2:**Karl Pearson and Alice Lee, On the Vibrations in the Field Round a Theoretical Hertzian Oscillator,

*Philosophical Transactions of the Royal Society of London. Series A*

**193**(1900), 159-188.

Although Hertz realised very fully that his oscillator did not give "perfectly regular and long continued sine-oscillations," and although Berknes determined so long ago us 1891 the general form of the damping, it does not appear that Hertz's original investigation of the nature of the vibrations in the field round one of his oscillators has hitherto been modified. Indeed, his diagrams of the wave mot ion have been copied into more than one textbook, and have usually been taken to represent what actually goes on in the surrounding field. Actually not only the diagrams, but a good deal of Hertz's original theory of interference requires modification, if we are to obtain quantitative accordance between theory and experiment. The object of the present paper is to give a fuller theory of the nature of the vibrations in the field round a typical Hertzian oscillator.

...

Conclusions. - We may draw the following general conclusions. - (i) The effect of damping makes itself very sensible in modifying the form of the wave-surface as propagated into space from a theoretical oscillator. The typical Hertzian wave-diagrams require to be replaced by the fuller series accompanying this memoir. (ii) Three waves of electro-magnetic force may be considered as sent out from the oscillator, and these waves we believe capable of physical identification. ... (iii) The velocities of these waves undergo remarkable changes in the neighbourhood of the oscillator, but still at distances such as Hertz experimented at, and which seem indeed to some extent within the field of possible physical investigation. (iv) The point of zero phase for both transverse and axial component electric waves does not coincide with the centre of the oscillator, so that these waves appear to start from a sphere of small but finite radius round the oscillator. A fourth wave dealt with by Hertz, the wave of magnetic induction, does not, as he supposes, start from the centre of the oscillator with zero phase, but ill the case of a damped wave train with a small but finite phase. (v) Our analysis of these waves and of their singular points in the neighbourhood of the oscillator appears to add something to Hertz's discussion; it is possible that it may throw light on the difficulties which arise in connecting with some of his interference experiments.

**1900.3:**Alice Lee and Karl Pearson, Data for the Problem of Evolution in Man. VI. - A First Study of the Correlation of the Human Skull [Abstract],

*Proceedings of the Royal Society of London*

**67**(1900), 333-337.

The substance of this paper was a thesis for the London D.Sc. degree; it was shown to Professor Pearson, at whose suggestion considerable modifications were made, and a revision undertaken with a view to publication.

In order to deal exactly with the problem of evolution in man it is necessary to obtain in the first place a quantitative appreciation of the size, variation, and correlation of the chief characters in man for a number of local races. Several studies of this kind have been already undertaken at University College. These fall into two classes, (i) those that deal with a variety of characters in one local race, and (ii) those which study the comparative value of the constants from a variety of races.

...

In the last place we turn to the third problem: the reconstruction of the capacity of the living head. The memoir contains tables of the skull capacity of some sixty men, and also of some thirty women, whose relative intellectual ability can be more or less roughly appreciated. It would be impossible to assert any marked degree of correlation between the skull capacities of these individuals and the current appreciation of their intellectual capacities. One of the most distinguished of Continental anthropologists has less skull capacity than 50 per cent of the women students of Bedford College; one of our leading English anatomists than 25 per cent of the same students. There will, of course, be errors in our probable determinations, but different methods of appreciation lead to sensibly like results, and although we are dealing with skull capacity, and not brain weight, there is, we hold, in our data material enough to cause those to pause who associate relative brain weight either in the individual or the sex with relative intellectual power. The correlation, if it exists, can hardly be large, and the true source of intellectual ability will, we are convinced, have to be sought elsewhere, in the complexity of the convolutions, in the variety and efficiency of the commissures, rather than in mere size or weight.

**1901**

**1901.1:**Alice Lee and Karl Pearson, Data for the Problem of Evolution in Man. VI. A First Study of the Correlation of the Human Skull,

*Philosophical Transactions of the Royal Society of London. Series A*

**196**(1901), 225-264.

**Note.**

The substance of this paper was presented by Miss Lee as a thesis for the London D.Sc. in March, 1899. After its presentation Miss Lee asked me to criticise and revise it with a view to publication. Illness in the spring of 1899 and later pressure of other work prevented my completing this revision until now. When Miss Lee started her work practically nothing had been published on the correlation of the parts of the skull; since then an interesting paper has appeared by Dr Franz Boas. To this reference is made in the footnotes at points where there is agreement or disagreement with his conclusions. The subject is of such great scientific interest, and anthropologically of such importance, that I urged Miss Lee to somewhat enlarge her original thesis by a series of additional investigations now incorporated in this paper. I have further rearranged a good deal of her material and reworded some of her conclusions, but the reduction of the material and the inferences drawn from it are substantially her work. My task has been that of an editor, who wished to mould the author's researches into a component part of a wider series dealing generally with the quantitative data for the problem of evolution in man. Such is the limit of my revision. I have passed of course nothing which did not seem to me valid, and have suggested to the author some lacunae which could be filled up by a consideration of additional data. - Karl Pearson

**Introduction.**

The reconstruction of an organism from a knowledge of some only of its parts is a problem which has occupied the attention of biologists for many years past. Cuvier was the first to introduce in his 'Discours sur les Révolutions de la Surface du Globe,' 1812, the idea of correlation. He considered that a knowledge of the size of a shoulder blade, leg, or arm might make it possible to reconstruct the whole individual to which the bone had belonged. The conception was taken up by Owen, but has fallen into discredit owing to the many errors made in attempts from a wide but only qualitative knowledge of the skeleton, to reconstruct forms the appreciation of which depends really on quantitative measurement and an elaborate quantitative theory. Such a theory having now been developed, and anatomists having provided large series of measurements, it has become possible to reconsider the problem on a sounder basis, and to determine more closely the limits under which our modern methods may be safely applied. The three fundamental problems of the subject are

(i) The reconstruction of an individual, of whom one or more organs only are known, when a series of organs for individuals of the same local race have been measured and correlated. ...

(ii) The reconstruction of the mean type of a local race from a knowledge of a series of one or more organs in that race, when a wide series of these and other organs have been measured in other races. ...

(iii) The reconstruction of an organ in the living individual not measurable during life, from a determination of the size of accessible organs, and a knowledge of the correlation between these organs and the inaccessible organ obtained from measurements made on individuals of the same race after death.

**1901.2:**Karl Pearson, Alice Lee, Ernest Warren, Agnes Fry and Cicely D Fawcett, Mathematical Contributions to the Theory of Evolution. IX. - On the Principle of Homotyposis and Its Relation to Heredity, to the Variability of the Individual, and to That of the Race. Part I - Homotyposis in the Vegetable Kingdom [Abstract],

*Proceedings of the Royal Society of London*

**68**(1901), 1-5.

If we take two offspring from the same parental pair, we find a certain diversity and a certain degree of resemblance. In the theory of heredity we speak of the degree of resemblance as the fraternal correlation, while the intensity of the diversity is measured by the standard deviation of the array of offspring due to given parents. Both correlation and standard deviation are determined for any given character or organ by perfectly definite well-known statistical methods. Passing from the case of bi-parental to asexual reproduction, we may still determine the correlation and variability of the offspring. This ultimately leads us to the measurement of the diversity and likeness of the products of pure budding, or, going still one stage further, we look, not to the reproduction of new individuals, but to the production of any series of like organs by an individual. Accordingly one reaches the following problem:- If an individual produces a number of like organs, which so far as we can ascertain are not differentiated, what is the degrees of diversity and of likeness among them. Such organs may be blood-corpuscles, hairs, scales, spermatozoa, ova, buds, leaves, flowers, seed-vessels, &c., &c. Such organs I term homotypes when there is no trace to be found between one and another of differentiation in function. The problem which then arises is this:- Is there a greater degree of resemblance between homotypes from the same individual than between homotypes from separate individuals? If fifty leaves are gathered at random from the same tree and from twenty-five different trees, shall we be able to determine from an examination of them what has been their probable source? Are homotypes from the individual only, a random sampling, as it were, of the homotypes of the race?

By the examination of very few series from the animal and vegetable kingdoms I soon reached the result, that homotypes, like brothers, have a certain degree of resemblance and a certain degree of diversity; that undifferentiated like organs, when produced by the, same individual, are, like types cast from the same mould, more alike than those cast by another mould, but yet not absolutely identical. I term this principle of the likeness and diversity of homotypes homotyposis. It soon became clear to me that this principle of homotyposis is very fundamental in nature. It must in some manner be the source of heredity. It does not, of course, "explain" heredity, but it shows heredity as a phase of a much wider process - the production by the individual of a series of undifferentiated-like organs with a certain degree of likeness. My first few series seemed to show that the homotyposis of the vegetable and animal kingdoms had approximately the same value, and it occurred to me that we had here the foundation of a very widespread natural law.

**1901.3:**Karl Pearson, Alice Lee, Ernest Warren, Agnes Fry and Cicely D Fawcett, Mathematical contributions to the theory evolution. - IX. On principle of homotyposis and its relation, the variability of the individual, and to that of the race. Part I. - Homotyposis in the vegetable Kingdom,

*Philosophical Transactions of the Royal Society of London. Series A*

**197**(1901), 287-299.

The present paper endeavours to deal with a problem upon which I have long been occupied, adopting the widest basis compatible with the time and means at my disposal. In the first place, I have often been impressed with the small reduction in variability which can be produced by selection. The offspring of a single parent while diverging in character, possibly very widely from the average character of the race, will still have a variability in that character only slightly reduced, say at most 10 per cent, below the racial variability. Even if we select the ancestry for an indefinite number of generations, the offspring will have a variability upwards of 89 per cent, of that of the original race. Now this capacity in the parent for producing variable offspring must be in some manner related to the degree of resemblance in those offspring. We have thus the two fundamental divisions of our subject: (i) What is the ratio of individual to racial variability? (ii) How is the variability in the individual related to inheritance within the race? I must endeavour to explain my meaning a little more fully and clearly. The individual puts forth a number of like organs, corpuscles in the blood, petals of the flower, leaves of the trees, scales on the wing. These may or may not be divided up into differentiated groups. Special forms of leaves occur in the neighbourhood of the fruit; florets may be differentiated according to their position on the flower, scales according to their position on the wing; there may be two or more classes of blood-corpuscles.

**1902**

**1902.1:**Alice Lee, Marie A Lewenz and Karl Pearson, On the Correlation of the Mental and Physical Characters in Man. Part II,

*Proceedings of the Royal Society of London*

**71**(1902-1903), 106-114.

In a first paper on this subject we gave a brief account of our material - Miss Beeton's copies of the Cambridge anthropometric measurements with degrees added at the University Registry, and the school measurements carried out by assistance from the Government Grant Committee. This material will take years to exhaust, but the present notice gives further conclusions to be drawn from Dr Lee's and Miss Lewenz's later reductions from this great mass of raw statistics.

In the first place we may refer to certain matters which arise directly from the first paper. In the discussion which followed the reading of that paper it was suggested that we ought not to correlate intelligence with absolute measurements on the head, but with their ratio to the size of the body. The answer made on that occasion was based on data not then published, namely, that there is no sensible correlation between intelligence and the absolute size of the body. The answer made on that occasion was based on data not then published, namely, that there is no sensible correlation between intelligence and the absolute size of the body. Hence the correlation between intelligence and any ratio of body lengths must also be small. ...

...

Since our school measurements were started, MM Vaschide, and Pelletier have published in the 'Comptes Rendus' a statement that although unable to find any relation between intelligence and length or breadth of head, they consider a relationship to hold between intelligence and the auricular height of head. Their process was of the following kind. They asked the school teacher to select ten intelligent and ten non-intelligent children, and then measured the heads of these two sets, and found their means. This was done for groups of three ages in boys and two ages in girls. The probable errors of the difference of the means of ten observations are not considered, and by exactly the same process that they reason that the auricular height is greater for the more intelligent children they might have deduced from their statistics that intelligent girls of 11 years have lower heads than intelligent girls of 9 years, and non-intelligent boys of 11 years lower heads than the same class of 9 years! Frankly, we consider that the memoir is a good illustration of how little can be safely argued from meagre data and a defective statistical theory.

**1902.2:**Alice Lee, Dr Ludwig on Variation and Correlation in Plants,

*Biometrika*

**1**(3) (1902), 316-319.

A number of points arise from Dr Ludwig's paper in the October number of

*Biometrika*which deserve to be considered from the standpoint of statistical theory. I have accordingly worked out the statistical constants of the material given by him ...

**1902.3:**Cicely D Fawcett and Alice Lee, A Second Study of the Variation and Correlation of the Human Skull, With Special Reference to the Naqada Crania,

*Biometrika*

**1**(4) (1902), 408-467.

The present investigation was commenced in 1895, but the long series of measurements involved and the elaborate numerical calculations necessary, have delayed the completion of the work until the present time. It forms part of a more general scheme for determining the size, variability and correlation of the chief organs and characters in man, which has been in progress at University College for some years past. When this scheme was started but little had been done to obtain a scientific measure of the variability and correlation of the parts of the human body. Innumerable anthropometric, including craniometric measurements, had been made and published but very little had been done in determining scientifically their statistical constants. In fact there was considerable danger that the want of proper statistical theory would bring the science of craniology into discredit with archaeologists. The manner in which variation is dealt with even in such a classical work as Rütimeyer and His's

*Crania Helvetica*is astonishing to the statistician who has realised the nature of the distribution of any character in a homogeneous population. A considerable population can be measured and we can determine whether or no it is sensibly differentiated from a second statistically defined population. But to classify a few individuals into different races by means of two or three measurements, such as the cephalic index, the length, or the facial angle, - before the correlation and the variation of these characters have been determined for even a single race - is a very dangerous proceeding, and calculated to bring craniometry into discredit.

It was with a view accordingly of providing anthropologists with the needful constants for determining racial differences that the scheme spoken of was started. It consisted partly in the reduction of existing published measurements, and partly in the measurement of new and large series, where such were not already available. A fairly comprehensive series of determinations of variability in man were made by Dr Alice Lee, Mr G U Yule, and Professor K Pearson, and published by the latter in his

*Chances of Death and other Studies in Evolution,*Vol. I. Further a considerable quantity of new material was collected and reduced in a series of papers entitled:

*Data for the Problem of Evolution in Man*, published by the Royal Society in their

*Proceedings*and

*Transactions*.

**1903**

**1903.1:**Alice Lee, On the Relation Between Rates, Expenditure on Remunerative Works, and Rate of Increase of Population in Fifty-Eight British Municipalities,

*The Economic Journal*

**13**(51) (1903), 424-429.

The statistics on which the following results are based were taken from Burdett's

*Official Intelligence*(now called the

*Stock Exchange Official Intelligence*), Vol. VI, 1888, pp. 26-7, and the Stock Exchange Official Intelligence, Vol. XX, 1902, pp. cxxiv-cxxv. The statistics there published are of value, but no sound judgment can be based upon them until they are quantitatively reduced, and a calculation made of the actual coefficients of correlation between the quantities involved.

It is also of interest to ascertain how far the rate of increase of population affects the relationship between expenditure on remunerative works and the magnitude of the rates. The data for rate of increase of population are taken from the Census Report of the Registrar-General for 1901, pp. xii-xiii, The investigation is confined to towns of over 50,000 inhabitants.

The

*Official Intelligence*may fairly he looked upon as having no individualistic or socialistic bias in the preparation of its material.

I premise here that the coefficient of correlation is a statistical constant marking the degree of relationship between two quantities, and that it ranges from a value = 0, when the two quantities are quite independent, to a value = 1, when we may suppose their variations to be absolutely dependent or causal. A partial coefficient of correlation measures the relationship between two variables, when a third variable remains constant, e.g., we express what would be the degree of correlation between rates and remunerative expenditure, supposing the population to remain constant. We are thus able to remove from consideration any disturbing influence of changes of population on the relationship between rates and remunerative loans.

**1903.2:**Karl Pearson and Alice Lee, On the Laws of Inheritance in Man: I. Inheritance of Physical Characters,

*Biometrika*

**2**(4) (1903), 357-462.

About eight years ago I determined to supplement the data obtained by Mr Francis Galton for his work Natural Inheritance by a rather wider series of measurements on blood relations in man. Mr Galton had most generously placed his original data at my disposal and I had used them as far as stature was concerned in my memoir of 1895 and in a joint paper with Dr Lee in 1896. The eye-colour data of his Family Records were not reduced until after the discovery of a method for dealing with characters not capable of exact quantitative measurement, and it is only recently that the full scheme of relationships back to great-grandparents has been completed. There were about 200 families in Mr Galton's records and only one measurable character, stature. The conditions as to age of the measured, or to method of measurement were not, perhaps, as stringent as might now be considered desirable, but Mr Galton's data were amply sufficient to lead him to his great discovery of the general form of the inheritance of blending characters in a stable community. The full significance of this discovery is hardly yet understood, and one constantly notices grave misinterpretations of Mr Galton's theory in the works of non-statistically trained biologists. The constants as determined from Mr Galton's stature data did not seem to me to be final; they were to some extent irregular and were not in full accord with the more uniform eye-colour results. It therefore appeared to me desirable to obtain further data, not only for several physical characters and to compare the results for these characters with those for mental characters, but to deal with both in as wide as possible a system of blood relationships.

**1903.3:**Karl Pearson, G U Yule, Norman Blanchard and Alice Lee, The Law of Ancestral Heredity,

*Biometrika*

**2**(2) (1903), 211-236.

Alice Lee writes: From Mr Blanchard's racehorse coat-colour pedigrees, I have, paying no attention to sex, been able to extract 1155 cases of great-grandparent and offspring and 978 cases of great-great-grandparent and offspring. When it is noted that there are 16 types of great-grandparental and 32 types of great-great-grandparental relationship, so that 48 correlation tables would be required for the full working out of these cases, it will be noted why in this preliminary study, I have not differentiated between the sexes. Tables I and II reproduce my data. ...

**1903.4:**S Jacob, A Lee and Karl Pearson, Craniological Notes: Preliminary Note on Interracial Characters and their Correlation in Man,

*Biometrika*

**2**(3) (1903), 347-356.

A distinction has been drawn in this Journal between intraracial and interracial correlation, and the present preliminary note is intended to emphasise the importance of this distinction. If we take a race of which we have sufficient data and determine its type by a number of characteristics, either by forming the means or the modes for the frequency-distributions of these characteristics in the race, we shall find that within the race an individual who diverges from the type of the race for one character will probably do so for a second, and that there is for the total of individuals within the race an interrelationship between these divergences - expressible by their coefficient of correlation. This correlation within the race is an intraracial coefficient, it predicts only the probable within the race itself, and must be very cautiously extended from one race to a second without à priori justification. The coefficient of correlation thus determined varies as a rule from race to race. Because the correlation between length and height of head of Aino is .5, it does not follow that this Aino characteristic may be applied to prehistoric Egyptians or modern Germans. In fact, whereas within these races a long-headed Egyptian or Aino was probably a high-headed individual also, a long-headed German tends to be low-headed. It is accordingly very misleading to predict from observations within one race what are the probable relationships between characters in a second; still less legitimate is it to predict from the coefficient of correlation in one race what would be the probable value of a defective measurement in an individual of a second race. Our knowledge at present tends to show that correlation varies from race to race, and that only for certain special organs (related in a particular way to the manner in which selection has differentiated the two races from a common stock) will the regression coefficients on which prediction depends remain the same for the two races. Only by an extensive tabulation of the differences between regression coefficients shall we be able ultimately to predict the evolution of racial differences, and accordingly there must always be danger in extending intraracial results from one race to a second; we are leaping over the very fence we have ourselves erected when we classified them as separate races, for the source of that separation is written from the evolutionary side in the very differences of regression coefficients which we disregard when we predict from one race to a second. But it is not only in predicting from one race to an individual of a second that we need caution. How far may we even assert that what holds within one race holds for the races of the world taken as a whole? A long-headed Aino is probably tall-headed; are the long-headed races of the world tall-headed races? A platyrrhine Naqada was chamaeconchic. Is racial platyrrhiny usually associated with racial chamaeconchy? There are many such problems which can only be answered when a much more ample tabulation of racial types than we have at present has been provided. Still there is an obvious and correct method of approaching and solving such problems; we must correlate the type values of the characters for as many races as possible. Such correlation coefficients have been termed interracial coefficients of correlation, and their discovery must form the basis of an exact theory of race for any species, in particular of a theory of race in man. The present note is only preliminary. Its chief function is by an illustration or two to serve as a caution against the extension of intraracial results to interracial conclusions, or agaiost the application of intraracial results to reasoning on individual organisms belonging to different or possibly quite unknown races. We confine our attention for the present to characters of the human head.

**1904**

**1904.1:**Amy Barrington, Alice Lee and Karl Pearson, On Inheritance of Coat-Colour in The Greyhound,

*Biometrika*

**3**(2/3) (1904), 245-298.

There is little doubt that if money and time were no consideration direct experiments on the breeding of dogs would lead to results of the highest importance not only for the theory of inheritance, but also for the practical guidance of dog-fanciers. To be of the most complete service such experiments would have to commence with two or three generations of in-breeding simply to insure the purity of the various stocks to be employed in the final experiments. Further, in the description of the selected characters, a classification would have to be adopted of a far more comprehensive character than appears to be usual in a number of recent experiments on hybridisation. Lastly, from the standpoint which we believe to be the correct one, that safe conclusions can only be drawn from the average of large numbers of crossings, at least 50 and probably 100 individuals of both sexes would have to be the basis of an effective experimental stud. Now the difficulty both in time and money of dealing with such a stud may not in the future be insuperable, but at present to propose it as the only means of approaching the problem of inheritance in dogs is to adjourn sine die any consideration of that problem. In certain points also the extensive breeding records which are already available for dogs possess advantages which are not to be wholly disregarded when we compare them with the special merits of a biometric stud-farm. In the first place we have all the gain which arises from dealing with literally immense numbers. For example, in the present memoir we were able to classify over 10,000 cases of parent and offspring, over 7000 cases of grandparent and offspring, and over 24,000 cases of siblings. Nothing approaching such totals could be obtained by experiment ad hoc. Further, the colour pedigrees for a number of generations were directly available. Against these advantages is to be put in the foremost place the primary value of exactitude and uniformity in record such as might be in a well organised scientific experiment. This counts for a great deal, but it does not count for everything with those who realise what are the probable errors of small series, and how inconclusive such series usually are. On the other hand also, if we admit the want of scientific exactness and the play of individual judgment in the character classifications of breeders, we have still to remember that when the breeding of a particular species has been long established a conventional scale also grows up which, owing to the contact of breeder with breeder at sales and shows, and further to the regulations of societies and judges, becomes within broad lines universally recognised and appreciated. Hence, while we fully recognise all the disadvantages of stud-book records, we still hold that highly valuable work may be done in the field of inheritance by accepting the classification of professional, if non-scientific breeders.

**1905**

**1905.1:**J Blakeman, Alice Lee and Karl Pearson, A Study of the Biometric Constants of English Brain-Weights, and Their Relationships to External Physical Measurements,

*Biometrika*

**4**(1/2) (1905), 124-160.

The purpose of this paper is to present a biometric analysis of the measurements provided by Dr R J Gladstone and published in this volume. The conclusions reached are therefore of the same order of validity as the data upon which they are based. An attempt has been made to compare them with the fuller material reduced by Dr Raymond Pearl, and in many points where comparison was possible general confirmation of his conclusions has been obtained. Gladstone's statistical material differs from that used by Pearl in two essential points. It is in the first place more meagre, but in the second place it provides additional measurements which enable us to predict with a moderate degree of accuracy brain-weight from external measurements on the living subject.

**1907**

**1907.1:**Alexandra Wright, Alice Lee and K Pearson, A Cooperative Study of Queens, Drones and Workers in "Vespa Vulgaris",

*Biometrika*

**5**(4) (1907), 407-422.

The 11,000 to 12,000 microscopic measurements on which this paper is based were made by Miss Alexandra Wright, who also determined about half the indices. The remainder of the indices and all the statistical reductions were carried out by Dr Alice Lee.

The material on which this study is based was provided by Mr 0 H Latter of Charterhouse, who most kindly sent to my Biometric Laboratory a nest of

*Vespa vulgaris*and a second of

*Vespa germanica*, the members of the latter being still under measurement. The former nest was a singularly fortunate one, it contained 129 perfect queens, upwards of 150 perfect drones, and many hundred workers. In the actual reductions the measurements on the whole of the 129 queens were used; the first 130 drones were used and the first 129 workers out of 183 actually measured up to date. In all cases the wings were mounted permanently for reference and the body preserved in a separate numbered tube, so that it will be possible some day to compare the size, variability and correlation in other than wing characters. Seven measurements were made on both right and left fore wing - it is proposed to consider later the area of the wing by planimetering the image thrown by a lantern on a screen, a process found quite feasible. The measurements were taken with an ocular micrometer and a Leitz 1b objective. This objective was found to work very well for this purpose, and the draw tubes of both objective and ocular were set at fixed points throughout. The magnification was such that one ocular unit = 1.48 mm. All measurements are, however, given in this paper in the actual micrometer readings. Every measurement was twice repeated, and if the second reading (which followed after an interval) did not agree with the first, a special control measurement was made. In this way between 11,000 and 12,000 measurements were taken and nearly two years spent over the microscopic work. This will serve to explain not only the delay in publication, but the reason why only 130 individuals of each class were dealt with.

**1908**

**1908.1:**Alice Lee, On the Manner in Which the Percentage of Employed Workmen in this Country is Related to the Import of Articles Wholly or Mainly Manufactured,

*The Economic Journal*

**18**(69) (1908), 96-101.

An argument of the following kind has been occasionally raised by the advocates of Tariff Reform:- The import of manufactured articles means the employment of foreign instead of British workmen, and protection would transfer to our own wage-fund the large sums that at present pass, owing to these imports, to foreign workmen.

It is an extremely difficult problem to confirm or to refute an argument of this kind. The older English economists would have met it by an

*à priori*reasoning which appealed only to a small extent to specific experience. The modern economist recognises far more clearly the complexity of the problem, the great part which local circumstance, environment, economic, and political development play in any real treatment of the question. It would probably be impossible to demonstrate the truth or falsehood of the argument in our own case by anything short of a gigantic and risky experiment. It is more easy, however, to show that no evidence in its favour can be obtained from any data bearing on the point available in our own country.

...

Now, of course, the data are slender, but, so far as they go, they not only lend no support to the Tariff Reform argument, but, on the contrary, they appear to stultify it. Like many other arguments used in political controversy, this appears based on a misuse of statistics, for, as Lord Goschen said some twenty years ago:- "Given a great number of figures partially unknown, given unlimited power and discretion of selection, and given an enthusiast determined to prove his case, and I will not answer for the consequences."

**1908.2:**Karl Pearson and Alice Lee, On the Generalised Probable Error in Multiple Normal Correlation,

*Biometrika*

**6**(1) (1908), 59-68.

The normal correlation surface in the case of n variables is known ... We see accordingly that the chance of an outlying or not, - the observation consisting of a complex of $n$ variates, - can be readily found, if the incomplete normal moment functions have once been tabled, and the constants of the correlation surface be known. These incomplete normal moment functions serve a variety of purposes which will be developed in later papers. The present paper merely refers to the means they provide of determining the probability of any observation lying outside a given contour ellipsoid ...

The general use of the table provided will be obvious, it enables us to tell the probability of any outlying individual really being a member of a population of which the constants are known. Thus one may look forward to the day when the biometric constants of a race being sufficiently well known, it may be possible to tell from a complex of five or six characters whether a skeleton or a skull may be reasonably supposed to have belonged to a member of that race. At present the labour of calculating the correlation coefficient-determinants and their minors stands in the way of much work in this direction, when we wish to advance beyond two or three characters.

**1910**

**1910.1:**Karl Pearson, Alice Lee and Ethel M Elderton, On the Correlation of Death-Rates,

*Journal of the Royal Statistical Society*

**73**(5) (1910), 534-539.

The discovery of possible inter-relationships between diseases by an examination of their death-rates as affected by varying environment, occupation, or race, has not been without fascination for more than one investigator. Personally I have considered the problem more than once, but always failed to make progress owing to the existence of spurious correlations, which I did not see how to meet....

...

It was only after reading Dr Maynard's paper in the current number of

*Biometrika*, and thinking over the difficulties to which he draws attention, that another way of tackling the problem occurred to me. We reduce all our sub-populations to a standard population, or population with a standard age-distribution. The assumption made in doing this is, practically, that the particular standard population used is immaterial.

...

Dr Alice Lee took forty cities in the Registration States, United States, having more than 90,000 inhabitants, and calculated their age correction factors for (a) cancer deaths, and (b) all deaths other than cancer and diabetes. She then found the ten correlation coefficients of the following five quantities:- (1) Deaths from cancer. (2) Deaths from all diseases other than cancer and diabetes. (3) Population. (4) Cancer age corrective factor. (5) All diseases, except cancer and diabetes, age corrective factor.

**1914**

**1914.1:**Alice Lee, Table of the Gaussian "Tail" Functions; When the "Tail" is Larger than the Body,

*Biometrika*

**10**(2/3) (1914), 208-214.

In a paper published in

*Biometrika*, Vol. VI. pp. 59-68, tables for the incomplete normal moment functions were printed, and they have since been reproduced in Tables for Statisticians and Biometricians recently issued from the Cambridge University Press. From these tables values of the Gaussian "Tail" functions were deduced and a short table of ψ1 and ψ2 appeared in Biometrika, Vol. VI. p. 68. The value of these functions being demonstrated in practice during the last few years, a more complete table of ψ1 , ψ2 , ψ3 , has appeared in the

*Tables for Statisticians and Biometricians*.

In the introduction to those tables, however, Professor Pearson indicated that it was important to have a similar table when the "tail" forms more than half the entire curve, and gave the fundamental formulae for obtaining the numerical values of the functions. The present table has been calculated to supply the want thus indicated.

**1915**

**1915.1:**Alice Lee, Tuberculosis and Segregation,

*Biometrika*

**10**(4) (1915), 530-548.

In his book

*The Prevention of Tuberculosis*(London: Methuen) Dr A Newsholme has examined the influence of segregation on Tuberculosis. ...

Dr Newsholme in the course of his chapter gives a number of very high correlations between the phthisis death-rate and the indirect forms of the segregation ratio he has selected, and he interprets these as well as a long series of graphs as demonstrating that institutional segregation has been a most important factor in the diminution of the phthisis death-rate. Now any two variates which are changing continuously with the time - say, the consumption

of bananas per head of the population and the fall in the birth-rate - will exhibit high correlation and will show graphically very high association, if plotted to appropriate scales and on a common time basis. Until the time factor has been removed, either by partial correlation or otherwise, it would be most unwise to interpret such cases as providing any causal relationship.

It seemed accordingly worth while to reinvestigate Dr Newsholme's problems with the aid of a rather more adequate statistical apparatus.

**1917**

**1917.1:**Alice Lee, Further Supplementary Tables for Determining High Correlations from Tetrachoric Groupings,

*Biometrika*

**11**(4) (1917), 284-291.

The difficulty of determining correlations between .80 and 1.00 by the tetrachoric method, owing to the slow convergency of the terms of the fundamental equation for 'tetrachoric $r$' has long been recognised. In 1912 Everitt published "Supplementary Tables for Determining High Correlations from Tetrachoric Groupings." These tables much simplified the work within the field in which it is really possible to determine accurately a high correlation - beyond certain values of '$h$ and $k$' such determination is impossible owing to the influence of random sampling on a quadrant category which in most practical cases will only contain an isolated unit or two. Everitt's tables covered the values of $r$ from + .80 to + 1.00 for values of the dichotomic planes given by $h$ and $k$ varying from + .0 to + 2.6. They admitted at once of our dealing with those cases of negative values of $r$, for which

*either*$h$ or $k$ was negative, but not with cases in which $r$ was negative and both $h$ and $k$ remained of the same sign. The present tables provide for this omitted portion of the possible field and thus complete Everitt's work.

I have followed his method of quadrature in evaluating my integrals. But I have preserved more decimal places than he has done, partly because my significant figures are thrown into higher decimal places than his by the nature of the case, and partly because recent experience in other fields has shown workers in the Biometric Laboratory, that tables are often of service for purposes other than those for which they were originally calculated, and that it is worth while preserving every reliable figure. I think my results are always correct to six figures and generally to the actual number tabulated.

**1917.2:**H E Soper, A W Young, B M Cave, A Lee and K Pearson, On the distributions of the correlation coefficient in small samples. Appendix II to the papers of "Student" and R A Fisher,

*Biometrika*

**11**(4) (1917), 328-413.

In a paper of 1908 "Student" [William Gosset] dealt experimentally with the distribution of the correlation coefficient of small samples, and gave empirical curves - in particular for the case of zero correlation in the sampled population - which have proved remarkably exact. The problem was next considered in 1913 by H E Soper who obtained the mean correlation and the standard deviation of the distribution of correlations to second approximations. ... The next step was taken by R A Fisher who gave in 1915 the actual frequency distribution of [the correlation $r$ in samples of $n$ from a population by a curve]. ... Clearly in order to determine the approach to Soper's approximations, and ultimately to the normal curve as $n$ increases we require expressions for the moment coefficients of [Fisher's curve], and further for practical purposes we require to table the ordinates of [Fisher's curve] in the region for which $n$ is too small for Soper's formulae to provide adequate approximations. These are the aims of the present paper. It is only fair to state that the arithmetic involved has been of the most strenuous kind and has needed months of hard work on the part of the computers engaged. On the other hand the algebra has often been of a most interesting and suggestive character.

**1925**

**1925.1:**Alice Lee and Karl Pearson, Table of the First Twenty Tetrachoric Functions to Seven Decimal Places,

*Biometrika*

**17**(3/4) (1925), 343-354.

The present table was computed as ancillary to a complete table for tetrachoric coefficients of correlation, which will shortly be published. It gives the tetrachoric functions from 0 to 19 to seven decimal places for argument intervals of $h$ equal to .1 and proceeds from 0 to 4.0.

... a grant from the Government Grant Committee of the Royal Society, has enabled Dr Alice Lee to devote her time to the calculation of this table and certain other tables shortly to be published.

**1927**

**1927.1:**Alice Lee, Supplementary table for determining correlation from tetrachoric groupings,

*Biometrika*

**19**(1927), 354-404.

This table enables the value of the tetrachoric coefficient of correlation to be found by simple interpolation and without the need of solving a higher order equation for all positive values of $r$ when $h$ and $k$ have the same sign, and for all negative values of $r$ when $h$ and $k$ have different signs.

Last Updated September 2021