portion. It is assumed, therefore, that the cities in Group I may vary from 91 to 104 per cent from the average proportion spent for instruction by all such cities without such costs being considered either high or low. In Table 58 it is shown that the average proportion spent for instruction in the cities of Group I is 75 per cent. Applying the percentages (91 and 104) to this proportion it is found that a city must spend for instruction from 68.25 to 78 per cent of its total cost if it comes within the zonal bounds of the middle half. The cities which spend more than 78 per cent for instruction, i. e., the cities which have a geometrical deviation greater than 1.04, have been indicated with the letter H in figure 37. Those cities spending less than 68.25 per cent for instruction, i. e., those cities with a geometrical deviation less than 91, have been indicated by the letter L in the figure. When these facts are considered, it is found in figure 37, Part A, that Seattle comes within the middle zone on the proportion spent for administration but stands low (L) on instruction costs. Cleveland stands high (H) on general control costs but low (L) on instruction costs. In this curve it is clearly seen that greater deviations (from 70 to 100) are permitted for overhead costs than for instruction costs (from 91 to 104)—the range in the former being 30 points and in the latter only 13 points. The range variations explain why Seattle is marked low (L) on instruction costs and not marked at all on administration costs which are even less than instruction costs (1.08 for general control and 1.21 for instruction) when compared with the average amounts spent for these respective functions. With these explanations in mind, it is seen that Seattle spends for auxiliary agencies and for fixed charges more than the corresponding proportion so spent by the middle 50 per cent of the cities and for instruction less than the average percentage so spent by these 23 cities. It would have been incorrect to have inferred that the proportion spent for general control in Seattle was its most commendable expenditure simply because the index (1.08) for this purpose was lower than that for any other function of cost. The proportions spent for the other functions, operation and maintenance, come within middle bounds. These considerations relative to a distribution of expenditures do not vitiate the deductions made above, that the total unit cost in Seattle exceeds that of any other city represented and that the absolute unit cost for each function exceeds the average for the group on every count. Only on the cost for administration does this city entail a unit cost falling within the middle half of the cities. In Cleveland the total cost per pupil is just above the middle half. For administration it spends considerably more than the average amount; for instruction a little less; for operation, maintenance, and auxiliary agencies more than the average; and for fixed charges considerably more than the average. The high proportion spent for administration and the low proportion spent for instruction are outstanding, when judged by the practice of the middle half of the cities. This study does not attempt to say, however, that these distributions are unwise. If better results are secured by such asymmetrical distributions, the practice is not to be condemned. In fact, progress in school administration often evolves out of the extraordinary. This study attempts to point out where high or low expenditures are likely to be found; not to say why they exist or whether the expenditures are to be commended or condemned. A closer analysis of local conditions is necessary for sane judgments regarding a wise distribution of city school expenditures. It is hoped that the graphic presentations will stimulate interest in an analysis of city school expenditures, and form a standard with which city school costs can be compared. Any wide variations from the general practice should receive the most careful scrutiny from those charged with administering city school finances. Unusually low costs also should challenge analysis. THE NORM AND THE INDIVIDUAL CITY. It is not practicable to study and to present graphically the expenditures of all city school systems. Those chosen probably represent the average conditions for all cities in their group. With these norms established, it becomes an easy matter to ascertain the location of any city not included in the graphic presentations. First, secure for the city in question data corresponding to that found in Table 58. Second, compute the average annual cost per pupil in average daily attendance for all purposes and for each function separately. Third, divide these average costs in the given city by the respective average costs for all cities shown in Table 22. The quotients thus obtained are used in locating the city on the proper graphs for its class—the quotient for the total cost determining its rank among the 45 cities, and the functional quotients, the respective points on the supplementary curves. Fourth, divide each ratio to the average for each function of expense by the corresponding ratio for the total cost for the city in question. The quotients correspond to those given in the paired columns 6 and 7, 10 and 11, 14 and 15, 18 and 19, 22 and 23, and 26 and 27 of Table 59. Fifth, mark with "H" all those quotients exceeding the respective third quartile points and with "L" all those quotients less than the respective first quartile points shown in the last line of each section of Table 59. In this manner any city may be superimposed on the graphs by inserting a new horizontal line at the proper place. Thus, with a minimum of work the expenditures of any city not included in the diagrams can be analyzed and compared with 45 other cities of the same class. The only questionable phase of this procedure is that the additional city studied does not form a part of standard by which it is measured as do the other cities included in the graphs. The standard, however, would not be materially affected by the addition of a single city. In fact, it may be more desirable to compare a city with a standard of which it is not a part. THE AVERAGE VERSUS THE MEDIAN. By using the arithmetical average, it has been possible to incorporate in the same series of charts a study of the proportional distribution of expenditures as well as a comparative analysis of absolute costs. Had the median been used instead to indicate the central tendency a study of proportion distribution could not have been included, since it does not possess the necessary mathematical properties. The middle half and the quartile points have been indicated in each accompanying chart. A sort of hybrid has, therefore, been employed the average and the middle half. One limitation of this method needs clarification. Each city contributes its "per pupil" weight in determining the basal arithmetical averages-the larger cities drawing the averages unduly near to them. The result is most pronounced in bunching the ratios either above or below the average of 1.00 as shown in columns 5 and 13 and in the paired columns 6 and 7, and 14 and 15 of Table 59, Group I. This asymmetrical apportionment of ratios does no particular harm since the range of the middle half is "skewed" accordingly. By classifying the cities according to size this undesirable tendency is practically nullified in the cities of the last four groups, since no city in any one group is more than three times the size of another city in the same group. In Group I the cities vary greatly in size and some exert undue influence in displacing an average. |