Systems. have been made previously to examine the interrelationship

APPLIED AND ENVIRONMENTAL MICROBIOLOGY, Feb. 1983, p. 603-69 0099-2240/83/020603-07$02.00/0 Copyright 1983, American Society for Microbiology Vol. 45, No. 2 Frequency Distribution of Coliforms in Water Distribution Systems ROBERT R. CHRISTIANt AND WESLEY 0. PIPES* Department ofbiological Sciences, Drexel University, Philadelphia, Pennsylvania 19104 Received 26 August 1982/Accepted 9 November 1982 Nine small water distribution systems were sampled intensively to determine the patterns of dispersion of coliforms. The frequency distributions of confirmed coliform counts were compatible with either the negative-binomial or the lognormal distribution. They were not compatible with either the Poisson or Poissonplus-added zeroes distribution. The implications of the use of the lognormal distributional model were further evaluated because of its previous use in water quality studies. The geometric means from 14 data sets ranged from 10-6 to 0.2 coliforms per 100 ml, and the geometric standard deviations were between 10 and 100, with one exception. If the lognormal model is representative of the coliform distribution; the arithmetic mean sample count is a poor estimator of the true mean coliform density, and the probability of water in a distribution system containing small patches with large coliform densities without detection by routine monitoring is finite. These conclusions have direct bearing on the interpretation of microbiological quality standards for drinking water. The regulations for the microbiological quality of drinking water in the United States have evolved from the treasury standards of 1914 (12) through those currently described in the National Interim Primary Drinking Water Regulations of 1976 (15). Further revisions of these standards are required by the Safe Drinking Water Act (Public Law 93-523). The minimum number of samples per month required for monitoring of coliform bacteria ranges from 1 to 500 as a function of the population served within a water distribution system (15). The determination of whether a system is in compliance with the regulations is based on maximum contamination level (MCL) criteria regarding arithmetic mean counts of total coliforms in the samples and percentages of samples above a particular density. When the membrane filtration procedure is used, the criterion for mean sample count is 1 confirmed coliform per 100 ml averaged over a set period (1 month if four or more samples are examined). Another MCL is that no more than 5% of the samples shall have more than 4 confirmed coliforms per 100 ml if 20 or more samples per month are examined (15). However, the theoretical basis for these criteria is not easily discerned. The probability of exceeding an MCL is dependent upon the number of samples, the true mean coliform density, and the frequency distribution of coliform densities. A few attempts t Present affiliation: Biology Department, East Carolina University, Greenville, NC 27834. 603 have been made previously to examine the interrelationship of these factors in drinking water (6, 8, 13). For these attempts investigators have assumed certain frequency distributions and have calculated either monitoring schedules or probabilities of compliance or noncompliance with MCLs given specified true mean densities. Frequency distributions considered have included the Poisson (8), Poisson-plus-added-zeroes (8), negative-binomial (6), and gamma distributions (13, 14). However, sufficient data on coliform occurrence in potable waters are needed to verify the assumption of any frequency distribution. Thomas (13) used monitoring data from a water system to support the gamma distribution. Data on bacterial frequencies in natural waters have been found to be compatible with either the negative-binomial or the lognormal distribution (4, 9). Here we report the first study in which the sampling program was designed to evaluate the frequency distribution of coliforms in water distribution systems and the consequences these distributions may have on the interpretation of microbiological quality of these systems. MATERIALS AND METHODS Sampling. Nine systems in Pennsylvania and New Jersey were sampled (Table 1). Seven of the systems used well water, and the other two had surface water sources. The size of the population served and the treatment provided for each system is indicated in Table 1. The results from three systems were partitioned on the basis of time, and system LB was divided into two areas served by different water

604 CHRISTIAN AND PIPES TABLE 1. Characteristics of water systems sampled System No. of Sample Raw Av pumpage (population served) Data set Sampling dates water samples vol (ml) Treatmenta source (gallons6 day x 10-6) per CV (20,000) CV 1 February to 15 March 1979 225 100 Surface C,F,S 3 WH (3,600) WHI 24 April to 3 May 1979 66 200 Well C 0.3 WHII 8 to 17 May 1979 154 200 WHIII 22 May to 4 June 1981 170 200 LB (3,000 winter) BBI 7 to 21 June 1979 92 200 Well C,F 0.3 winter (45,000 summer) BBII 10 to 19 July 1979 99 200 TPI 7 to 21 June 1979 122 200 3 summer TPII 10 to 19 July 1979 146 200 DT (9,800) DT 27 December 1979 to 12 174 200 Surface C,F,S 1.2 February 1980 BL (2,700) BLI 4 March to 24 April 1980 236 200 Well C,F 0.2 BLII 16 to 25 June 1981 169 200 MI (125) 6 May to 5 June 1980 207 200 Well None 0.005 SR (550) 25 September to 19 November 144 200 Well C 0.03 1980 MW (300) 8 October to 11 December 99 200 Well C 0.03-0.04 1980 BG (750) 8 October to 2 December 1980 46 200 Well C 0.02-0.03 a C, Chlorination; F, filtration; S, sedimentation. b One gallon is ca. 3.79 liters. sources. In system WH a change in water flow pattern and microbiological quality occurred between 3 and 8 May 1979. Thus the data were so divided (WHI and WHII). Also, this system was sampled again in 1981, and these data were treated separately (WHIII). System BL was sampled during both 1980 and 1981, and the data were divided (BLI and BLII). The LB system is a long grid of streets in which the northern and southern halves are served by separate treatment plants which provided the same type of treatment. Those data were divided according to the two service areas (BB and TP). Also, this system served several resort communities, and water use was much greater in July than in June. Samples from June (BBI, TPI) were considered separately from those in July (BBII, TPII). Samples were taken from locations throughout each system. Water was allowed to run for no less than 3 min before sampling. Samples for bacteriological analyses were collected in sterile bottles containing thiosulfate for dechlorination (2). Samples were stored in ice chests during transit to the laboratory and maintained at <10 C with coolant packs (3). The time from sampling to processing was less than 8 h with an average time between 3 and 4 h. Bacterial analysis. The membrane filtration procedure for the determination of total coliforms was used exclusively in this study (2). The volume filtered per sample was 100 ml. Duplicate samples per location were examined from all but one system (Table 1). The mean count per 100 ml of the duplicate samples from each location was used for statistical analyses. Care was taken to avoid subsequent contamination of samples with coliforms during handling and processing of samples (3). Typical coliform colonies were counted as presumptive coliforms after growth on m-endo medium for 22 to 24 h at 35 ± 0.5 C. All typical coliform colonies on a filter were picked for confirmation if five or fewer were APPL. ENVIRON. MICROBIOL. found. At least five representative colonies were chosen when more than five were counted on a filter. Confirmation was accepted as growth plus gas production in lauryl tryptose and brilliant green bile broths. The confirmed coliform densities were used for statistical analyses. Statistical analyses. Frequency distributions are models which describe the manner in which the bacteria are dispersed. Thus the confirmed coliform counts were tested for compatibility with the Poisson, Poisson-plus-added-zeros, negative-binomial, and lognormal distributions. Compatibility with the Poisson distribution was tested with the Fisher index of dispersion (D2) (7) and the Kolmogorov-Smirnov test (Dmax) (11). The Poisson-plus-added-zeroes distribution was tested by D2 for the mean and variance of the samples with coliforms present. For a more developed discussion of these distributions with respect to coliforms see Pipes and Christian (8) and El-Shaarawi et al. (4). The negative-binomial distribution is described by three interrelated parameters: mean (,u), variance (cr2), and coefficient of aggregation (k) (5). The,. and cr2 were estimated by the sample mean (x) and variance (S2). The parameter k was estimated by using the formula nfn(l + xclk) = I [Ax/(k + x)] where n is the number of samples, Ax is the number of locations with counts greater than x, and x is the coliform count at a particular location. The estimate of k was obtained by trial and error through repeated iterations of the formula by computer. Comparisons of data with derived negative binomial distributions were by the Kolmogorov-Smirnov test (13). The two parameters used to describe the lognormal distribution are the geometric mean (GM) and the geometric standard deviation (GSD). These were derived from regression analyses. Truncation of the data occurred at both high and low densities. Samples with no coliforms were recorded as <1/100 ml. When both

VOL. 45, 1983 DISTRIBUTION OF COLIFORMS 605 TABLE 2. Tests for agreement of data sets with Poisson and Poisson-plus-added-zeros distributions Tests for No. of % of agreement with Locations with Poisson plus locations locations Variance of Poissonb coliforms zeroes" sampled with >4 Mean count' counts' Data set (locations coliforms (CFU/100 (CFU/100 Mean Variancea Fraction Index of with per 100 Ml) mi)2 counta without dispercoliforms) ml D Dmx CF10 (CFU/ coli- sion 100 ml) mi)2 forms (D2) CV 225 (10) 0.4 0.11 0.67 1,364 0.956 2.52 10.03 0.96 36 WHI 66 (6) 4.5 0.40 3.62 588 0.909 4.42 25.84 0.91 29 WHII 154 (61) 9.1 >1.52 >25 2,516 0.677 >3.84 >55 0.60 859 WHIII 170 (26) 5.9 >1.76 >73 7,009 0.869 >11.49 >337 0.85 733 BBI 92 (11) 1.1 0.51 15.41 2,750 0.880 4.27 122.52 0.88 287 BBII 99 (41) 23.2 >9.38 >439 4,587 0.789 >22.65 >768 0.58 1,356 TPI 122 (41) 9.8 >2.58 >59 2,767 0.766 >7.67 >138 0.66 720 TPII 146 (53) 16.4 >6.16 >275 6,473 0.801 >16.99 >579 0.64 1,772 DT 174 (8) 1.1 >0.30 >9 5,190 0.954 >6.56 >185 0.95 197 BLI 238 (8) 0.4 >0.35 >27 18,283 0.986 >10.44 >790 0.95 530 BLII 169 (18) 0.6 >0.50 >23 7,728 0.893 >4.67 >209 0.89 761 MI 207 (7) 0.5 >0.40 >31 15,965 0.986 >11.89 >902 0.97 455 SR 144 (36) 3.4 >0.93 >26 3,998 0.840 >3.73 >94 0.75 882 MW 99 (19) 2.0 0.41 3.41 815 0.818 2.15 14.58 0.81 122 BG 46 (10) 8.7 >2.15 >70 1,465 0.813 >9.90 >263 0.78 239 a The > symbol has been used when one or more samples were TNTC. The counts for such samples were considered >80/100 ml. b All data sets had probabilities of having Poisson distributions of <0.005 by Fisher's index of dispersion and of <0.01 by the Kolmogorov-Smirnov statistic. c All data sets had probabilities of having Poisson-plus-added-zeros distributions of <0.005 by Fisher's index of dispersion. 100-ml samples were negative, low-range truncation occurred at 0.5 coliforms per 100 ml. Samples too numerous to count (TNTC) were considered as >80 coliforms per 100 ml (2). When one 100-ml sample was TNTC and the other was negative, high-range truncation occurred at 40 coliforms per 100 ml. Coliform counts were ordered from least to most, and the cumulative percentages of samples less than or equal to each density were computed based on n + 1 as 100%o. Each percentage less than or equal to a particular density was converted to standard deviation units to linearize the coordinates of the log density (I) versus standard deviation (X). Least-squares regression was performed on all counts between the truncation limits. With such transformations the Y intercept occurs at the 50%o value. Thus the antilog of the Y intercept was the median or GM, and the antilog of the slope was the GSD. Goodness of fit between the data and the derived lognormal distribution was determined from the correlation coefficient and F statistic obtained from the regression. Also, the variances and confidence limits about the GM and GSD were obtained from the computations. The arithmetic mean (I,) of data having a lognormal distribution can be calculated from the GM and GSD. The relationship is elp = exp[engm + 0.5 (engsd)2] (reference 1). We calculated. for comparison with the sample arithmetic mean (i). RESULTS Table 2 shows the sample arithmetic means and the percentage of locations with >4 coliforms per 100 ml. These are the parameters associated with compliance with the microbiological MCL (15). For 6 of the 15 data sets the mean density was >1 coliform per 100 ml, and more than 5% of the locations had >4 coliforms per 100 ml. In these six cases the water system was in violation of both rules of the microbiological MCL. The 15 data sets were tested for compatibility with the Poisson distribution. All were found to be nonrandom according to both the D2 and Dmax tests (Table 2). The probabilities of fitting a Poisson distribution were all less than 0.005 for the index of dispersion (D2) tests and less than 0.01 for the Kolmogorov-Smimov (Dmax) tests. Thus in no case could the one-parameter Poisson distribution be applied. The next step was to determine which multiparameter distribution could be applied. The Poisson-plus-added-zeroes distribution assumes that some fraction of the water is not contaminated and that the coliforms within the fraction of the water which is contaminated are randomly distributed. The fraction of uncontaminated water was estimated from the proportion of locations with no coliforms. This fraction ranged from 0.58 to 0.97 (Table 2). Thus even in the most contaminated systems studied, over one-half of the locations showed no coliforms. The frequency distribution of the coliforms for the locations where they were present was not compatible with the Poisson distribution accord-

606 CHRISTIAN AND PIPES APPL. ENVIRON. MICROBIOL. TABLE 3. Applicability of the negative-binomial distribution to MF coliform counts in samples from water distribution systems Mean counta Variance of Coefficient of Kolmogorov- Data Data setnt ~ ~ Cftsa Smimoven o (CFU/100 ml) (CFU/100 mn)2 aggregation (k) statisticb (Dmax) CV 0.11 0.67 0.017 0.005 WHI 0.40 3.62 0.028 0.019 WHII >1.52 >25 0.160 0.056 WHIII >1.76 >73 0.036 0.025 BBI 0.51 15.41 0.037 0.047 BBII >9.38 >439 0.103 0.082 TPI 22.58 >59 0.101 0.041 TPII >6.16 >275 0.090 0.045 DT >0.30 >9 0.012 0.009 BLI >0.35 >27 0.005 0.013 BLII >0.50 >23 0.032 0.032 MI >0.40 >31 0.006 0.016 SR >0.93 >26 0.072 0.069 MW 0.41 3.41 0.064 0.037 BG >2.15 >70 0.054 0.041 a The > symbol has been used when one or more samples were TNTC. The counts in such samples were considered >80/100 ml. b All data sets had probabilities of having negative-binomial distributions of >0.2 by the Kolmogorov-Smirnov statistic. ing to the D2 test (Table 2). Hence this twoparameter model did not describe the distribution of coliforms in any system. The negative-binomial distribution is described by three interdependent parameters (mean, variance, and coefficient of aggregation). These parameters for 15 data sets are shown in Table 3. The coefficient of aggregation (k) ranged from 0.005 to 0.16. The fact that all estimates of k were <1 is indicative of clumped or contagious distributions. The Kolmogorov- Smimov test was used to test compatibility of the data with this model, and it was found that the negative-binomial distribution could be accepted in all 15 cases. The lognormal distribution was also tested (Table 4). Statistical analyses of compatibility were based on correlation and analyses of variance. All but one data set had coefficients of determination of 0.803 or greater. These correlations were all significant at the 0.01 level. All F values from the analyses of variance demonstrated deviation from randomness at the 0.02 level. BLI was the one data set which did not conform to this model. Seven of the eight positive locations for this system had counts of 0.5 coliforms per 100 ml, and the eighth was TNTC. These data failed to provide a significant correlation because of truncation at a count of 40 per 100 ml. Excluding BLI the GMs ranged from 1.4 x 10-6 to 0.20 coliforms per 100 ml. Thus the median coliform densities (GMs) were all less than 1 per 100 ml. The GSDs for these 14 data sets ranged from 10 to 441, with all but one being less than 100. Thus the lognormal distribution was sufficient to describe the data in all but one case. The arithmetic means (,u) associated with the lognormal distribution were computed from the respective GMs and GSDs (Table 4) and ranged from 0.55 to 693.3 coliforms per 100 ml, excluding BLI. All of these were larger than their respective sample arithmetic means (x) (Table 2). The ratios of,u to xi ranged from <1.4:1 to <393.7:1, where the < sign refers to uncertainty in the xi because of TNTC values. DISCUSSION It is clear that coliforms were not randomly dispersed in any of the water distribution systems. In all systems studied most locations (up to 96.6%) yielded no coliforms. If the bacteria were randomly dispersed the few positive samples would have had low counts (8), and the probabilities of obtaining TNTC samples would be almost infinitesimal. For example, if the mean density were 1 coliform per 100 ml, the probability of obtaining a TNTC sample from a random dispersion (Poisson distribution) would be much less than 10-100. But TNTC samples were found in 11 of 15 data sets with sample means between >0.30 and >9.38 coliforms per 100 ml. The probabilities of obtaining samples with more than a few coliforms per 100 ml were much greater than would be predicted by either the Poisson or the Poisson-plus-added-zeroes distributions. Previously, we (8) used these two distributions in developing a sampling model for

VOL. 45, 1983 TABLE 4. DISTRIBUTION OF COLIFORMS 607 Parameters of the lognormal distribution of coliforms in water distribution systems as computed by least-squares regression Coefficient of Probability Calculated Data set GM coliforms per 100 ml GSD (range of determination F of arithmetic (range of 1 SE) 1 SE) (R2) randomness mean (p.) CV 1.9 X 10-3 (1.1 x 10-3-3.2 x 10-3) 29 (22-38) 0.951 156.4 <0.0001 0.55 WHI 1.3 X 10-3 (3.8 x 10-4-4.3 x 10-3) 88 (43-181) 0.906 38.7 0.0034 29.31 WHII 0.20 (0.19-0.21) 10 (9.6-10.7) 0.974 1,878.9 <0.0001 2.83 WHIII 3.5 X 10-3 (2.0 x 10-3-6.3 X 10-3) 92 (63-134) 0.899 142.4 <0.0001 96.39 BBI 3.7 x 10-3 (1.4 x 10-o-9.5 x 10-3) 39 (21-71) 0.803 36.6 0.0002 3.04 BBII 0.21 (0.18-0.23) 56 (48-65) 0.960 762.1 <0.0001 693.03 TPI 0.15 (0.13-0.16) 19 (17-21) 0.948 687.2 <0.0001 11.45 TPII 8.4 x 10-2 (7.4 x 10-2-9.5 x 10-2) 67 (59-77) 0.963 1,059.7 <0.0001 79.85 DT 6.9 x 10-4 (2.6 x 10-4-1.9 x 10-3) 53 (31-143) 0.919 57.0 0.0006 1.83 BLI 0.50 0.07 0.000 >105 <0.0001 0.50 BLII 8.1 X 10-3 (4.4 x 10-3-1.5 X 10-2) 20 (14-29) 0.822 69.4 <0.001 0.72 MI 1.4 x 10-6 (9.7 x 10-7-1.9 x 10-6) 441 (375-518) 0.999 1,438.1 0.0168 157.46 SR 9.1 x 10-3 (7.8 x 10-3-1.1 X 10-2) 35 (31-40) 0.960 747.5 <0.0001 5.06 MW 8.7 x 10-3 (5.9 x 10-3-1.3 X 10-2) 35 (27-51) 0.934 182.5 <0.0001 4.83 BG 1.2 x 10-2 (6.4 x 10-3-2.1 X 10-2) 97 (59-160) 0.923 83.4 <0.0001 420.37 coliforms in water distribution systems and relating potential sampling results to the microbiological MLCs for drinking water (15). Given the nonrandom dispersion of coliforms, our previous conclusions as to the probabilities of exceeding MCLs for various numbers of samplings must be considered unrealistic. Either the negative-binomial or the lognormal distribution is an adequate representation of the coliform data obtained. This was also found to be the case for coliforms from the raw water supply of a treatment plant on Lake Michigan (9). El-Shaarawi et al. (4) found that the negative-binomial distribution described bacterial data from Lake Erie, but they did not test the lognormal distribution. Of the two, the lognormal distribution has been used more extensively in water quality evaluations (9). The true test of which distribution may best describe overall coliform dispersion in drinking water resides in knowledge of the frequency distributions of coliforms below and above the data truncation limits. Large-volume samples can be used to detect very low densities, but there may be interference due to particulate matter or large numbers of noncoliform bacteria (10). TNTC samples have >80 coliforms per 100 ml, and large numbers of very small volume samples would be needed to give accurate measurements of high coliform densities. We were not able to sample effectively beyond the limits normally associated with 100- or 200-ml volumes. We have chosen to consider the lognormal distribution for further analysis because of its general acceptance in water quality studies. If the systems which we studied are representative of small water distribution systems, our parameters of the lognormal distribution may be taken as guidelines for future regulations. For those data sets which did not exceed the MCLs, the GMs ranged from 10-6 to 10-2 coliforms per 100 ml (excluding BLI). When MCLs were exceeded the GMs ranged from 3 x 10-3 to 0.2 coliform per 100 ml (Table 4). In all cases less than half of the samples showed the 0G) E C cn 0.99 0 a- 0.90 0-J va 0 0.70 0 0.50 CP._ C 0.30 -a o 0.10 0 0.0-0 0. 0-000l *. 10 20 30 40 50 60 70 80 Samples per month FIG. 1. Theoretical probabilities for the lognormal distribution of obtaining at least one positive sample as a function of the number of samples when the arithmetic mean is 1/100 ml and the GSD is 10 or 100.

608 CHRISTIAN AND PIPES presence of coliforms. As much as 25% of the samples could be positive without the sample mean exceeding 1 coliform per 100 ml, if TNTC is counted as 80 coliforms per 100 ml (SR in Table 2). However, a TNTC result is indeterminate, and it would be reasonable to assume that the sample mean for any set with one or more TNTC results is greater than 1 coliform per 100 ml. Generally when MCLs were exceeded the percentages of locations positive were larger than when they were not (15 to 42% compared with 4 to 25%, respectively), and the GMs were larger than when violation did not occur. However, overlap existed within these two criteria between violation and compliance. No distinction could be made on the basis of the GSD. The range for data sets in compliance was 20 to 441. For sets in violation the range was 10 to 97. Excluding BLI, all but one GSD were between 10 and 100. Thus by inference it might be expected that coliforms in small water distribution systems will usually have a GSD between 10 and 100 regardless of their GM. The sample arithmetic mean is not a good estimator of the true arithmetic mean of coliforms in water distribution systems (8a). There are methodological reasons for this, but part of this poor accuracy is based on the frequency distribution. The calculated,u for a lognormal distribution is based on both the GM and GSD. Of these two parameters the GSD provides the larger component in determining the value of,u. A more heterogeneous dispersion (greater GSD) of bacteria gives rise to larger,u, and heterogeneity is not a direct component in the calculation of x. The sample arithmetic mean is based on a truncated data set (normally 1 to 80 coliforms per 100 ml when the MF technique is used). No knowledge of densities above or below these limits is obtained by routine sampling. The estimates of,. from the lognormal parameters with extrapolations above 80 coliforms per 100 ml lead to small but finite probabilities of samples with very high densities (i.e., thousands per 100 ml). These high densities dominate the calculation of,u. Thus the p.s derived from the lognormal distribution were always larger than i and in some cases were orders of magnitude larger. If these extrapolations are reasonable and can be validated, we suggest that a finite probability exists that a person may consume water with large densities of coliforms (and perhaps pathogens) without violation of MCLs as determined by a monitoring program. Systems with large GSDs would have a greater likelihood of this occurring than those with smaller GSDs. However, for a given p., the systems with small GSDs would be more likely to yield positive coliforms samples (Fig. 1). Both distributions represented APPL. ENVIRON. MICROBIOL. have a p. of 1, but their GSDs are 10 or 100. There is a higher probability of obtaining a positive sample when the GSD is 10 than when it is 100. With the p. fixed at 1 for both, the system with a GSD of 100 contains fewer positive samples than with a GSD of 10, but these samples must contain more bacteria per sample. The same conclusion can be reached from the negative-binomial distribution. The probability of a zero count for the negative-binomial distribution is P(O) = (1 + p./k)-k. Thus, for the same value of p., the smaller the value of k the smaller the probability of obtaining a count different from zero (positive sample). For instance, when p. = 1 the probability of a negative sample for k = 0.01 is 0.95 and for k = 0.1 is 0.79. When there are fewer positive samples, some of the positive samples must have very high counts in order for the mean density to be the same, and there is higher probability of obtaining TNTC samples. To summarize, coliform dispersion in water distribution systems was found to be compatible with the lognormal and negative-binomial distributions. The GMs and percentage positive samples were found to be higher in systems that violated MCLs than those that did not. The GSDs generally ranged between 10 and 100. If either distribution is valid, the sample arithmetic mean is a poor estimator of the true arithmetic mean. Also, the likelihood of a person consuming water with a very high coliform density is small but finite even if the system in not shown to be in violation. ACKNOWLEDGMENTS We thank the following people for their aid in sampling and laboratory and statistical analyses: H. A. Minnigh, E. M. Podgorsk, M. Goshko, M. Troy, J. E. Sioma, G. Burlingame, E. D. Becker, L. Scarino and S. Sees. This research was supported in part by U.S. Environmental Protection Agency, Office of Water Supply through grant EPA R-805-637. LITERATURE CITED 1. Aitchison, J., and J. A. C. Brown. 1957. The lognormal distribution. Cambridge University Press, Cambridge. 2. Bordner, R., J. Winter, and P. Scarpino. 1978. Microbiological methods for monitoring the environment-water and wastes, EPS-600/8-78-017. Environmental Monitoring and Support Laboratory, U.S. Environmental Protection Agency, Cincinnati, Ohio. 3. Christian, R. R., and W. 0. Pipes. 1979. Recognition of subsequent contamination of samples from distribution systems with coliforms. Ann. Conf. Proc. Am. Water Works Assoc. Part 2:645-659. 4. El-Shaarawl, A. H., S. R. Esterby, and B. J. Dutka. 1981. Bacterial density in water determined by Poisson or negative binomial distributions. Appl. Environ. Microbiol. 41:107-116. 5. Fisher, R. A. 1941. The negative binomial distribution. Ann. Eugen. 11:182-187. 6. Muenz, L. R. 1978. Some statistical considerations in water quality control, p. 49-56. In C. W. Hendricks (ed.), Evaluation of the microbiology standards for drinking water, U.S. Environmental Protection Agency, Washington, D.C.

VOL. 45, 1983 7. Pklou, E. C. 1969. An introduction to mathematical ecology. Wiley-Interscience, New York. 8. Pipes, W. O., and R. R. ChristIan. 1978. A sampling model for coliforms in water distribution systems. Ann. Conf. Am. Water Works Assoc. Part 2:33-35. 8a.Plpes, W. O., and R. R. Christian. 1982. Mean coliform density of a water distribution system. Ann. Conf. Am. Water Works Assoc. Part 2:857-870. 9. Pipes, W. O., P. Ward, and S. H. Ahn. 1977. Frequency distributions for coliform bacteria in water. J. Am. Water Works Assoc. 69:644-647. 10. Seldler, R. J., T. M. Evans, J. R. Kaufman, C. E. Warrick, and M. W. LeChevaller. 1981. Limits of standard coliform enumeration techniques. J. Am. Water Works Assoc. 73:538-542. DISTRIBUTION OF COLIFORMS 609 11. Sokal, R. R., and F. J. Rohif. 1969. Biometry. W. H. Freeman and Co., San Francisco. 12. Taylor, F. 1978. Experiences with coliform standards under the interstate program, p. 153-158. In C. W. Hendricks (ed.), Evaluation of the microbiology standards for drinking water. U.S. Environmental Protection Agency, Washington, D.C. 13. Thomas, H. A., Jr. 1952. On averaging the results of coliform tests. J. Boston Soc. Civil Engrs. 39:253-261. 14. Thomas, H. A., Jr. 1955. Statistical analysis of coliform data. Sew. Ind. Wastes 27:212-228. 15. U.S. Environmental Protection Agency, Office of Drinking Water. 1976. National interim primary drinking water regulations, EPA-570/9-76-003. U.S. Environmental Protection Agency, Washington, D.C. Downloaded from http://aem.asm.org/ on September 17, 2018 by guest