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NOTES on the HISTORY of PAUPERISM in ENGLAND and WALES from 1850, TREATED by the METHOD of FREQUENCY-CURVES; with an INTRODUCTION on the METHOD.

By G. UDNY YULE, Assistant Professor of Applied Mathematics, University College, London.

[Read before the Royal Statistical Society, 21st April, 1896.
The Right Hon. the EARL OF VERULAM, Vice-President, in the Chair.]

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4. Method of Fitting; Moment Coefficients........................................

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5. Continuous and Discontinuous Variation; Frequency-
Curves; Normal Curve; Generalised Binomial Curve;
Hypergeometrical Series and Derived Curves

6. Criterion

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11. Contributory Causes and Component Cause-Groups............ 331 12. Homogeneity of Material ......................................................................................................................................................... 331

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13. Test of Fit

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Supplementary Note on approximate Methods of Determining the Mode 343

I.-Introduction: on the Methods used.

Sec. 1. Introductory.

CONSIDERING the amount and variety of our English poor law statistics, it seems somewhat strange that so little use has been made of them to give us information on the history of pauperism

in the kingdom as a whole. We have had, more especially of recent years, numerous instances of groupings and sub-divisions of the raw material by "in-door" and "out-door" unions, by north and south, and so on, but in few of these cases do we seem to get much grasp of the broad outlines of the statistical development, the numerical history, of English pauperism. We have had a mass of detail for single unions, hardly anything but a wretched average for the whole country; the wood seems to be hidden behind the trees.

Are rates or no? If

Now surely, to keep to the simplest facts, we might be told a little more. How do the rates of pauperism for the isolated unions group themselves round the average rate? Are they more, or less, scattered round it now than forty years ago? near the average also the most common and typical not, what is the most frequent or "fashionable" rate? How has this "fashion" or "mode" in pauperism been varying? Are not these questions that anyone might ask?

But under the guidance of the theory of probability such questions become at once more defined and more numerous. What is the numerical measure of the degree of scatter round the average, and how has it altered? What is the true mode, as opposed to the one accidentally created by our groupings of the unions? Is the distribution round the mean symmetrical? If not, what is the numerical measure of its skewness, and how has this varied? Finally, can one express immediately by some few constants the whole form of the distribution round the mean? This question we must stop and consider, for it is the fundamental one.

Sec. 2. Artificial Chance. Frequency-Polygons.

To endeavour to clear our ideas, let us revert to a case of so-called artificial chance-to a case in which we can directly compare theoretical conclusions with actual fact. Suppose we take a handful of dice, say fourteen, toss them some hundred or thousand times, and note each time the number of sixes thrown. Let us then record the results graphically. Taking a straight line as base, mark off along it a number of short equal lengths, 01, 12, 23, and so on (Fig. 1); then erect at 0 a vertical representing by its length the number of times that throws with no sixes occurred; if, e.g., there were twenty such throws, make 00' twenty millimetres long. Again at 1 erect a vertical showing on the same scale the number of throws with one six; if there were 100, make 11' 100 millimetres long. Completing the record in this way, and then joining up the tops of the verticals O', 1', 2', &c., we have a statistical "frequency-polygon." The height

VOL. LIX. PART 11.

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of each vertical, or the elementary strip of area round it, gives the frequency of occurrence of the corresponding number of sixes: the area of the whole polygon must represent on the same scale the number of observations made. The polygon will be in general more or less jagged and rough, but not badly so if the number of observations is large; the figure is drawn from an actual record of 648 throws of fourteen dice (ride the table on p. 322).

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Now in the case of dice we know à priori the data which determine the approximate form of this frequency-polygon. If any event can happen in s ways and fail to happen in r ways, then it is found (in a great variety of cases) that if the event be tried a large number of times, the ratio of successes to failures will be

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generally denoted by p and q. We know P and q approximately

for a die (assuming it to be a perfect homogeneous cube); the chance of throwing any given face is. Further we know the number of dice, n, and the distance, c, between the verticals we erected. Given P, q, n, and c, we can calculate what frequency to expect for throws with any given number of sixes, i.e., we can draw a theoretical frequency-polygon for our experimental cases and compare it with the actual one. This theoretical polygon is called a point-binomial. The point-binomial is drawn with dotted lines in Fig. 1.

Sec. 4. General Method of fitting: Moment-Coefficients.

But now suppose we had to approach the question from the other side. Suppose a friend had given us a record of dice throwings, with the remark that it was a "record of chance "distribution" but no further information. Could we make our way back from the statistical polygon to the numbers p, q, n, c, which describe it, if indeed these are sufficient? If we could do so, would the experimental numbers be likely to agree at all reasonably with the theoretical numbers? Both of these questions can fortunately be answered in the affirmative.

The method for passing back from the polygon to its constants, i.e., for fitting a point-binomial to an experimental polygon, is due to Professor Karl Pearson, to whose recent work on "Skew "Variation in Homogeneous Material" I shall have frequent occasions to refer. As the procedure is extremely general, and applicable to the fitting of all polygons or curves mentioned in this paper, I shall endeavour to describe it briefly; the description must not be taken too literally, but is correct in principle.

Taking the statistical polygon of Fig. 1, or the numerical results it represents—

(1) Determine the mean or average value of the variable, i.e., the mean number of sixes thrown.

(2) Determine the mean (deviation) from the mean, μ2; i.e., multiply the length of each observed vertical by the square of its distance from the mean, add up all the figures so obtained, and divide by the total number of observations.

(3) Similarly determine the mean (deviation) and (deviation), μg, and μ4. These quantities u may be called the moment-coefficients.

Equating 2, 3, and 4 to the corresponding moment-coefficients of the theoretical point-binomial expressed in terms of n, p, q, and c, and remembering that

p + q = 1,

we have sufficient equations to determine all the unknowns.

"Phil. Trans.," vol. clxxxvi (1895), A, pp. 343—414.

Let us revert to experiment to see how close this method will work in practice. The results of five cases are given in full in the following table, treating the experimental figures in three slightly different ways. The moment-coefficients were worked out twice, (1) by treating the observed ordinates as simply loaded proportionally to their lengths, (2) by treating the whole polygon as a system of trapezia, just as it is drawn. These different methods lead to the figures given under "trapezia" and "loaded ordinates" respectively.

Table showing the Results of some Chance Experiments.

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Now in both of these cases we assume that nothing whatever is known of the work to which the figures refer, not even whether it be a case of artificial chance or no. The numbers might have been obtained by measuring the heights of a lot of men, and counting how many there were between 5 ft. and 5 ft. 1 in., how many between 5 ft. 1 in. and 5 ft. 2 in., and so on. The unit c is consequently quite unknown, we cannot assume that it corresponds to our arbitrary groups; c must be determined like p, q, and n, from the moment-coefficients. But if we know that the figures

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