for the above cases with the corresponding terms for the cases that arise in the geometric series method shows that we have in the proposed test a finer subdivision of the scale of pollution. We have, however, shortened our yardstick, so that it is not as suitable for measuring a wide range of pollution as is the geometric series of tests. A question of greatest importance in determining the method to be used is that of the probable error of the results. In testing tubes GRAPH III. CURVES FOR SINGLE SAMPLES. 5 TUBES OF 10 cc. EACH of equal size, the most probable pollution and its probable error arc easily determined by using the formulae below. If N tubes of 10 cc. each are tested, n of these tubes giving negative results and m giving positive results, then the most probable number of B coli per 10 cc. is given by These formulae are given by Greenwood and Yule in the article previously mentioned. Variations of them have been stated by several other writers. Expressions for the most probable number of B. coli per unit of water, and for its probable error, cannot be easily obtained in the case of the geometric series. We may, however, obtain some idea of the relative variability of the two methods of sampling by comparing single samples in two similar cases. In the case of 1 tube of 10 cc. positive and 4 tubes of 10 cc. each negative, the most probable pollution is 22 B. coli per 1000 cc. When, in testing by the geometric series, we have 100 cc. positive and the remainder of the portions negative, the most probable number of B. coli per 1000 cc. is 23. Since the densities are practically identical in these two GRAPH IV. CURVES FOR SINGLE SAMPLES DIFFERENT METHODS OF SAMPLING cases, we may compare their probability curves to determine which one has the greater variability. Turning to Graph IV, we see that the two curves have their modes, as stated, at 22 and 23. The curve for the samples taken according to the proposed method is much higher at its mode, and is much less broad in general than is the curve for the samples taken by the geometric series method. Thus the two samples indicate equal degrees of pollution, but the probable error to be ascribed to this degree of pollution is much less in the case of the proposed standard of five 10-cc. portions than in the case of the more commonly used geometric series. This fact may also be brought out by deriving from each curve the probability that the number of B. coli per 1000 cc. is not greater than 80. For the proposed standard this probability is 0.869, whereas for the geometric series it is only 0.543. The proposed standard The proposed standard places a limit on the mean pollution and also on the variability. It will, therefore, be of interest to examine both of these factors from the mathematical point of view. Concerning the mean pollution the standard specifies that not more than 10 per cent of all the 10-cc: standard portions examined shall show the presence of organisms of the bacillus coli group. GRAPH V. PROBABILITY CURVES FOR 10 PER CENT OF PORTIONS POSITIVE For this limiting value we have the following equation expressing the probability that the water is polluted to any specified degree. = and N is the total number of portions tested. The curves for the cases, N 10, and N = 50, are plotted in Graph V. Most of the characteristics of this probability curve depend upon N. The position of the mode, that is, the most probable value of the pollu tion, is, however, independent of N, with a value of 10.5 B. coli per 1000 cc. This means that, when 10 per cent of the portions tested are positive the most likely pollution of the water is that expressed by a density of 10.5 B. coli per 1000 cc., or 1.05 per 100 cc. Although the position of the mode is independent of N, the height of the ordinate at the mode is not. This ordinate is a measure of the reliability of the predicted value of the density, the reliability increasing as the ordinate increases. The relationship between this ordinate and N, the number of portions, is shown in Graph VI. GRAPH VI. HEIGHT OF MODE FOR DIFFERENT NUMBERS OF PORTIONS TESTED -10 PER CENT OF PORTIONS POSITIVE The curve indicates that it is well to have at least 100 portions in order that we may be out of the region of sharpest increase on this curve. Another view of the increase in the reliability of the determination of the pollution with increasing N may be obtained by examining the distributions for the two cases shown on Graph V. It will be seen that the curve for N 10 is much more widely spread than = Considering variability from another point of view, we may ask the following question: Assuming that the density of B. coli remains constant at the maximum limit set by the standard (10.5 B. coli per 1000 cc.), with what frequency should we expect to obtain, on the basis of simple sampling, the different results which may arise in a sample consisting of 5 portions of 10 cc. each? Since the probability that a portion of 10 cc. will be negative is given by e-10 = 0.9, the required frequencies are given by the expansion of the binomial (0.9 + 0.1).5 These frequencies are shown in the following table: This table furnishes the basis of the second part of the proposed standard. We see that under the above assumption only 0.856 per cent of a given series of samples should by chance show three or more positive portions. The proposed standard specifies 5 per cent for this condition, thus allowing more variability than would be expected to arise from simple sampling. Bibliography (1) Greenwood, J. Junr., and Yule, G. Udny: On the Statistical Interpretation of Some Bacteriological Methods Employed in Water Analysis. Jour. Hyg., vol. 16, no. 1, July, 1917. (2) McCrady, M. H.: The Numerical Interpretation of Fermentation-Tube Results. Jour. Inf. Dis., vol. 17, no. 1, July, 1915. Wolman, Abel, and Weaver, H. L.: A Modification of the McCrady Method of the Numerical Interpretation of Fermentation-Tube Results. Jour. Inf. Dis., vol. 21, no. 3, September, 1917. McCrady, M. H.: Tables for Rapid Interpretation of Fermentation- 1919. |