Stations of the Cross

Graphs.

It is frequently of advantage to set out in diagram matic form such tabulated results as those referred to, as at least the broader features of the comparison of two or more series of figures can of ten be seen more clearly in this form. Such simple graphical comparisons are familiar through their use in meteoro logical reports where readings are shown by the upward and downward movements of a line crossing from left to right a series of vertical lines marking the hours of the day or the days of the week. The particulars of the consumption, per head of the popula tion of the United Kingdom, of sugar and of tea in each of the 5o years from 1864 to 1913, are thus shown in Fig. (I). A neces sary precaution in planning such graphical comparisons is also illustrated in the same diagram. The figures which are plotted relating to sugar show the average consumption year by year in lb. per head. The consumption of tea in lb. per head yields the line at the foot of the diagram, the variations of which are so slight as to suggest no similarity with those of the line showing sugar consumed. If, however, the consumption of tea per head is expressed in ounces, it becomes clear that, though in individual years the variations shown are by no means similar for sugar and for tea, over the whole period the increase was in approximately the same proportion for these two commodities'. The fluctuations were more considerable in the case of sugar, and the diagram suggests that, up to 1901, the quantities of sugar increased some what more rapidly than, and thereafter failed to maintain as great an increase as, those of tea.

Even though graphs fall short, in the matter of precision of statement, of the numerical tables, they have the great advantage of enabling a large mass of figures to be grasped as a whole much more readily than is possible when those figures are presented in one or more tables.

Averages and Dispersion.

A second illustration of statistical series is seen in the following summary of the heights of a number of men :— The table may be interpreted as meaning that 126 were found between 5 ft. 71 in. and 5 ft. 81 in. and similarly for other heights. In tables such as this some method of dealing with cases falling exactly on the dividing line between two adjacent groups must be laid down, and a common method is to assign half the number found on the dividing line to each of the classes of which it forms the limit. This procedure may, of course, result in numbers in some of the groups which are not integers. The true average height of the 73o men covered by the table can be ascertained only by reference to a more detailed statement showing the height of each man exactly, instead of in a number of groups. It will be observed that 297 were not taller than 5 ft. 71 in., while 307 were 5 ft. 81 in. or more, the distribution being not quite sym metrical on both sides of the numerically largest group. The approximate average height was about kin. less than 5 ft. 8 in.

A table of this kind tells us, however, something more than the average height of the individuals represented. We note that 191 were not more than 5 ft. 61 in. in height and 198 were not less

than 5 ft. 91 in. in height. By calculation from the figures shown, assuming that the individual heights were distributed with ap proximate regularity along the intervals from inch to inch, the points representing 5 ft. 6.3 in. and 5 ft. 9.7 in. would divide the series of heights so that one-quarter of the whole number fell below the former, and one-quarter of the whole number above the latter. A similar calculation gives, as the point dividing the series into two equally numerous groups, 5 ft. 8.04 in. This last point is called the median of the series, and the two others are the lower and upper quartiles, the three points serving to divide the whole number examined into four groups of equal numbers of cases. The distance between the upper and lower quartiles, in the case in question 3.4 in., gives the range within which the mid dle half of the instances recorded lay. This distance expresses much more definitely the degree of concentration of the indi viduals in the neighbourhood of the median height than, for ex ample, the whole range (14 in.) within which all the measurements lie. If a more exact description of the nature of the distribution than is afforded by the specification of the median and quartiles, but of the same general character, is desired, the group may be divided into a number of parts, e.g., into ten equally numerous parts, the points of division being then known as deciles.

In many varieties of statistical problems it is found that the observations are distributed in a manner similar to that shown in the above illustration, and the question arises whether the form of the distribution is of a recognizably definite character, the determination of which can be of use in the interpretation of the results obtained. It is found that, in numerous cases, the 'Cf. Jubilee Vol. of the Royal Statistical Society, p. 257, where the late Professor Marshall used this illustration. The figures are based on imports less exports of sugar, excluding sugared goods.

manner in which the individual observations are distributed is in close accordance with that of events dependent on pure chance. Such a case is presented by the following. A number of balls, indistinguishable in size, weight or form, are placed in a bag, half of the balls being white and half black. If one ball be drawn from the bag, its colour, whether white or black, may be noted. The ball being replaced and the bag shaken, another drawing will give a result wholly independent of the first. The repetition of such drawings will furnish a record of runs of white and of black balls, some short, some long. The numbers of cases (a) of a change of colour in consecutive drawings (b) of sequences of the same colour of two, three, four, etc., in number, being noted, the material for a table is furnished, and this table would have a general similarity with that of the men's heights used above for illustration. It is possible to determine theoretically the relative frequency with which the various sequences would recur in a series of trials indefinitely extended.