AVERAGING BACTERIAL COUNTS

A. H. ROBERTSONNew York State Department of Agriculture and Markets, Albany, New York

Received for publication, June19, 1931

When a frequency distribution diagram illustrating a normalscatter of bacterial counts is made using the actual values, thecurve assumes a distinct left-handed or J-shaped form. Thepeak of the curve and the arithmetic mean fall at quite divergentpoints. Furthermore, the standard deviation is usually muchgreater than, sometimes as much as twice as great as, the mean.Consequently a wide range of negative values is involved, in-dicating that something is radically wrong with the mathematicalprocedure. Statistical values based on an abnormal J-shapeddistribution curve are unreliable.Because the arithemtic mean does not reveal the true average

in which the mean, median and mode nearly coincide, the writerhas applied the geometric mean which is the value correspondingto the mean of the logarithms of the bacterial counts. A com-parison of the two methods of averaging is submitted herewith.The accuracy of an average is exceedingly important where the

premiums which are paid to dairymen are based on an averageof two or more bacterial counts of the milk as delivered. Like-wise scientists when preparing growth curves based on an averagecomputed from a series of similar observations may find it avaluable aid. Obviously the rate of bacterial growth is not con-stant. There are at least five phases in the normal growth curvein which the dairy bacteriologist is interested. These phases are:(1) the initial stationary or lag, (2) the positive accelerative, (3)the logarithmic or geometric, (4) the negative accelerative and(5) a stationary phase, prior to the decline when the death rateexceeds the birth rate. Notwithstanding the observation that

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geometric means among a series of samples in the logarithmicgrowth phase should approach a straight line relationship whenplotted against time, the mere fact per se that bacteria increasegeometrically, i.e., 1, 2, 4, 8, 16 and 32, instead of linearly, i.e.,1, 2, 3, 4, 5 and 6 justifies the application of the geometric mean.The range of the standard agar plate counts for most milk is

from 1,000 to 100,000,000 and undoubtedly 90 per cent of thesecounts are less than 100,000. To be sure, this extensive rangeoverlaps the upper portions of the positive and the lower portionsof the negative accelerative growth phases. However, the geo-metric mean so satisfactorily meets the requirement of a moreaccurate average as demonstrated in the charts and tables thatits value for comparing different milk supplies or the same milksupply from year to year is justified. If it more nearly ap-proaches the true average among a large number of counts, thereis no reason to doubt this same tendency among two or threecounts although its graphic demonstration is more difficult.The averaging of logarithms is regarded as an improper mathe-

matical procedure. This strict rule applies in most cases but wasformulated long before biological science was known. Certainly,it originated before attention was called to the logarithmic growthphase among bacteria. If bacterial reproduction were a linearprocess, the arithmetic mean would be adequate. Becauseour common averaging system applies to linear dimensions, suchas height, weight, length, capacity, etc., it is all the more difficultfor the layman to appreciate the geometric increases amongbacteria in their entirety. A rule 24 inches long is conceded to betwice the length of one 12 inches long. To regard a milk with a20,000 count as twice as poor as one with a count of 10,000 isincorrect. Yet, this is exactly what we are doing when we regard15,000 as the mean among the two values 10,000 and 20,000.Apparently workers in general have been slow to recognize whenaveraging bacterial counts that a bacterium reproduces byfission and makes two cells, each of which in turn splits givingfour cells, etc. Partial recognition occurs when "StandardMethods" recommends reporting only the two significant left-hand digits in the bacterial count. Probably this recognition is

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AVERAGING BACTERIAL COUNTS

wholly unintentional as the "Standard Methods" Committeehas merely intended to simplify the report.The problem then is how to transpose a geometrically increas-

ing series to a linearly increasing series so that it can be averagedcorrectly. There are some who will regard this application ashighly theoretical simply because they will not analyze mathe-matically the difference between an average among a series ofvalues resulting from an arithmetic (or linear) progression andthat resulting from a geometric (or logarithmic) progression.Table 1 illustrates some of these differences.

TABLE 1

A comparison of linear and logarithmic increases

UNIT OF TIME

01234567.8910'111213

LINEAR OR LOGARITHMIC OR LONARITH CORRT -

ARITHMETIC INCREASE GEOMETRIC INCREASE GESPONDING TO TSE

LU, J

20,00030,00040,00050,00060,00070,00080,00090,000100,000110,000120,000130,000140,000

10,00020,00040,00080,000160,000320,000640,000

1,280,0002,560,0005,120,00010,240,00020,480,00040,960,00081,920,000

4.000004.301034.602064.903095.204125.505155.806186.107216.408246.709277.010307.311337.612367.91339

If one can visualize the processes of bacterial increase duringthe logarithmic phase with respect to the-unit of time (generationtime) changes in table 1, one can readily see that during the firstperiod the geometric increase was only 10,000. During thesecond period the actual increase was 20,000, during the third itwas 40,000, during the fourth it was 80,000, etc ... . andduring the thirteenth it was 40,960,000. Contrasted with thisis the constant linear increase of 10,000 for each unit of time.With these seemingly enormous succeeding increases in the geo-metric series, is it any wonder that the bacterial counts are not

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A. H. ROBERTSON

more inaccurate than they are when it is realized that the in-crease from 40,960,000 to 81,920,000 theoretically requires justas long as the initial increase from 10,000 to 20,000? Robertsonand Frayer (1930) have shown that these 13 units of time basedon observations on 17 samples of milk held continually at 60°F.,correspond to a 52 hour keeping quality interval, two hours afterwhich all samples were considered as unfit as a sweet milkbeverage.

If now, the bacteriologist will look at the last column in table 1,he will see listed the logarithms corresponding to values of thegeometric increase. Since logarithms are powers of numberscalculated to the base 10, they are essentially linear values ofmeasures representing a geometrically increasing series. Theconstant increase of log. 0.30103 represents a generation time andisequivalent to each 100 per cent increase in actual numbers ofbacteria over the preceding count. Both the initial 10,000 in-crease during the first time unit and the 40,960,000 during the13th time unit are represented by a logarithmic increase of log.0.30103. The logarithms themselves, therefore, increase linearlywhile the values which they represent increase geometrically.

Table 2 illustrates a satisfactory system for the transpositionof bacterial counts to logarithms and vice versa. Because of thecharacter of the logarithmic spreads, it is necessary to establishtwo separate tables for these two transpositions. A certaindegree of accuracy has been sacrificed in order to limit thelogarithm to the second or third place following the decimal andthe bacterial count to the first three significant left-hand digits.Therefore, the bacterial counts in the first part of the transposi-tion table are the lowest values which can be read as equivalentto the logarithms. For instance 989 is the lowest value whichcan be read as equivalent to log. 2.995 and 1,010 as equivalent tolog. 3.005, etc. When reading these logarithms in the seconddecimal place only, they become log. 3.00 and. 3.01 respectively,etc. Hence all counts between 989 and 1,000 correspond tolog. 3.00, all between 1,010 and 1,030 correspond to log. 3.01, . .all between 9,440 and 9,590 correspond to log. 3.98 and all between9,600 and 9,890 correspond to log. 3.99. Higher or lower bac-

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TABLE 2The transposition of counts to logarithms and vice versa

BACTERIAL COUNT ANDCORRESPONDING LOGARITHM

989 3.001,010 3.011,040 3.021,060 3.031,080 3.041,110 3.051,140 3.061,160 3.071,190 3.081,220 3.091,240 3.101,270 3.111,300 3.121,330 3.131,360 3.141,400 3.151,430 3.161,460 3.171,500 3.181,530 3.191,570 3.201,600 3.211,640 3.221,680 3.231,720 3.241,760 3.251,800 3.261,840 3.271,880 3.281,930 3.291,970 3.302,020 3.312,070 3.322,110 3.332,160 3.342,210 3.352,260 3.362,320 3.372,370 3.382,430 3.392,480 3.402,540 3.412,600 3.422,660 3.432,720 3.442,790 3.452,850 3.462,920 3.472,990 3.483,060 3.49

3,130 3.503,200 3.513,270 3.523,350 3.533,430 3.543,510 3.553,590 3.563,670 3.573,760 3.583,850 3.593,940 3.604,030 3.614,120 3.624,220 3.634,320 3.644,420 3.654,520 3.664,620 3.674,730 3.684,840 3.694,950 3.705,070 3.715,190 3.725,310 3.735,430 3.745,560 3.755,690 3.765,820 3.775,960 3.786,100 3.796,240 3.806,380 3.816,530 3.826,680 3.836,840 3.847,000 3.857,160 3.867,330 3.877,500 3.887,670 3.897,850 3.908,040 3.918,220 3.928,410 3.938,610 3.948,810 3.95

LOGARITHM AND CORRESPONDINGBACTERIAL COUNT

3.00 1,0003.01 1,0203.02 1,0503.03 1,0703.04 1,1003.05 1,1203.06 1,1503.07 1,1803.08 1,2003.09 1,2303.10 1,2603.11 1.2903.12 1,3203.13 1,3503.14 1,3803.15 1,4103.16 1,4503.17 1,4803.18 1,5103.19 1,5503.20 1,5803.21 1,6203.22 1,6603.23 1,7003.24 1,7403.25 1,7803.26 1,8203.27 1,8603.28 1,9103.29 1,9503.30 2,0003.31 2,0403.32 2,0903.33 2,1403.34 2,1903.35 2,2403.36 2,2903.37 2,3403.38 2,4003.39 2,4503.40 2,5103.41 2,5703.42 2,6303.43 2,6903.44 2,7503.45 2,820

9,020 3.96 3.46 2,8809,230 3.97 3.47 2,9509,440 3.98 3.48 3,0209,600 3.99 3.49 3,0909,890 4.00

3.50 3,1603.51 3,2403.52 3,3103.53 3,3903.54 3,4703.55 3,5503.56 3,6303.57 3,7203.58 3,8003.59 3,8903.60 3,9803.61 4,0703.62 4,1703.63 4,2703.64 4,3703.65 4,4703.66 4,5703.67 4,6803.68 4,7903.69 4,9003.70 5,0103.71 5,1303.72 5,2503.73 5,3703.74 5,5003.75 5,6203.76 5,7503.77 5,8903.78 6,0303.79 6,1703.80 6,3103.81 6,4603.82 6,6103.83 6,7603.84 6,9203.85 7,0803.86 7,2403.87 7,4103.88 7,5903.89 7,7603.90 7,9403.91 8,1303.92 8,3203.93 8,5103.94 8,7103.95 8,9103.96 9,1203.97 9,3303.98 9,5503.99 9,7704.00 10,000

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A. H. ROBERTSON

terial counts than those listed in table 2 may be transposed to theproper logarithms simply by increasing or decreasing the charac-teristic of the logarithm. For instance, if the count is 202 thelog. is 2.31; if 20,200 it is log. 4.31; if 202,000 it is log. 5.31; etc.

In the second half of table 2, the values nearest the mean foreach logarithm are listed. Because of the wide spread of valuescorresponding to these logarithms when shortened to the secondor third1 decimal place, it is necessary to employ the mean valuecorresponding to each logarithm.Table 3 illustrates the use of the geometric mean when two

counts are averaged. Because bacteria increase geometrically,a proportionate instead of a linear relation exists between countson all samples of milk.Where the difference between the counts is not great, the two

means approach each other as in example 1, but where thedifference is great, the geometric mean gives a lower averagecount, due to the proportionate relations. The linear relationsin example 2 indicate that the 20,000 count milk is 20 times aspoor as the 1,000 count milk. Counts from duplicate platesshould always be averaged arithmetically.

In order to compare the two averaging systems to a betteradvantage, the standard agar plate counts and their correspond-ing logarithms for the 174 counts in table 4 are averaged. Thesecounts, listed in the order of sampling one dairyman's night'smilk four times monthly, were obtained through the courtesy ofa New York State milk plant for the period beginningFebruary1, 1926, and ending December 31, 1929.The arithmetic mean for the bacterial counts is 20,500 and the

mean corresponding to the average log. 4.01 is 10,200. Themean 20,500 is far from the value corresponding to the peak ofthe curve. It is evident in chart 1 that 27 observations fall at

1 When averaging two counts only, it is advisable to use the counts correspond-ing to the additional logs. 3.005, 3.015, 3.025, 3.035, etc., because the sum of thelogarithms frequently terminates with an odd digit. Failure to use this causesmany of the geometric mean values to exceed the arithmetic means in cases wherethe two counts are relatively near each other. Obviously, they will exceed themin some cases but thus far the writer in applying the method has found no casewhere the dairyman has been placed at a premium disadvantage.

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the peak of the curve which is somewhere between 4,000 and6,000. However, the long tail containing the five counts 96,000,133,000, 136,000, 320,400 and 346,500, is of sufficient weight tobring the average up to 20,500. The mean is more than fourtimes as great as the mode.Now, if the logarithms of these same counts are used, there are

23 counts falling at the peak of the curve which is somewhere

TABLE 3

Application and comparison of averages secured by the arithmetic and geometric mean

10,00011,000

2)21,000Mean 10,500

1,00020,000

2)21,000Mean 10,500

1,36055,300

2)56,660Mean 28,400

15,0002)20,000

Mean 10,000

CORRESPONDING LOGARITHM ANDGEOMETRIC MEANS

log. 4. 00log. 4.04

2)8.04Mean log. 4.02 = 10,500

log. 3. 00log. 4.30

2)7.30Mean log. 3.65 = 4,470

log. 3.14log. 4.74

2)7.88Mean log. 3.94 = 8,710

log. 3.70log. 4.18

2)7.88Mean log. 3.94 = 8,710

between 10,000 and 12,600. The distribution may be repre-sented by a normal bell-shaped curve and both the mode and themean approach each other very closely. Such being the case,does not the geometric mean reveal more accurately than thearithmetic mean the true distribution of a normal series of bac-terial counts for market milk?

In order to emphasize the accuracy of this application, the

EXtAMSPLE NUMtBER |ACTUAL BACTERIAL COUNTS ANDEXAMPLENUMBER ARITHMETIC MEANS

1

2

3

4

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130 A. H. ROBERTSON

TABLE 4

Bacterial counts and corresponding logarithms for producer's night's milk

29,400 4.47 7,300 3.86 4,000 3.60 14,100 4.1538,500 4.59 21,500 4.33 4,100 3.61 9,700 3.9914,900 4.17 10,800 4.04 2,100 3.32 7,100 3.8551,800 4.71 6,300 3.80 3,300 3.52 15,700 4.2041,300 4.62 6,700 3.83 5,800 3.76 7,300 3.8611,400 4.06 11,000 4.04 9,500 3.98 1,800 3.2611,500 4.06 25,900 4.41 5,300 3.72 12,800 4.1111,500 4.06 16,300 4.21 15,700 4.20 4,900 3.692,300 3.36 9,300 3.97 23,800 4.38 2,400 3.3814,000 4.15 2,000 3.30 500 2.70 1,600 3.2116,000 4.21 19,200 4.28 13,800 4.14 1,200 3.087,100 3.85 16,800 4.23 5,800 3.76 3,000 3.4860,900 4.78 18,500 4.27 19,900 4.30 1,700 3.232,600 3.42 11,300 4.05 4,600 3.66 28,500 4.5936,400 4.56 3,100 3.49 5,200 3.72 12,700 4.1110,800 4.04 8,900 3.95 2,200 3.34 4,700 3.676,500 3.81 10,200 4.01 65,800 4.82 42,800 4.632,300 3.36 6,900 3.84 5,500 3.74 32,200 4.513,000 3.48 6,900 3.84 33,600 4.53 18,100 4.269,700 3.99 6,900 3.84 22,900 4.36 49,700 4.703,500 3.54 14,200 4.15 10,900 4.04 4,100 3.61

346,500 5.54 2,900 3.46 28,000 4.45 9,500 3.983,600 3.56 10,000 4.00 46,300 4.67 10,000 4.003,000 3.48 34,300 4.54 96,000 4.98 4,700 3.676,900 3.84 35,000 4.54 11,700 4.07 9,000 3.956,700 3.83 10,000 4.00 54,600 4.74 3,000 3.487,300 3.86 4,600 3.66 45,600 4.66 12,000 4.087,800 3.89 25,900 4.41 4,400 3.64 133,000 5.138,900 3.95 42,700 4.63 5,900 3.77 60,000 4.781,600 3.21 35,700 4.55 6,100 3.79 6,000 3.78

25,900 4.41 4,500 3.65 6,500 3.81 4,000 3.602,200 3.34 320,400 5.51 2,300 3.36 9,000 3.9555,300 4.74 10,100 4.01 9,700 3.99 22,000 4.345,100 3.71 68,600 4.84 5,100 3.71 49,000 4.698,500 3.93 7,000 3.85 2,400 3.38 4,000 3.608,700 3.94 49,000 4.69 1,200 3.08 136,000 5.149,100 3.96 29,400 4.47 4,300 3.63 14,000 4.1525,900 4.41 63,000 4.80 2,000 3.30 43,000 4.6314,900 4.17 31,500 4.50 8,000 3.90 35,000 4.547,100 3.85 9,700 3.99 4,300 3.63 4,000 3.6017,700 4.25 7,700 3.89 3,000 3.48 3,000 3.4813,300 4.13 5,200 3.72 4,800 3.68 21,000 4.3212,000 4.08 9,600 3.98 4,300 3.63 8,000 3.9016,100 4.21 5,300 3.72

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AVERAGING BACTERIAL COUNTS 131

writer invites attention to the facts as diagramatically shown bythe respective standard deviations. The standard deviationamong a series of values may be calculated most easily by theAyer's method. This consists of extracting the square rootof the difference between the average of the sum of the squares

25.

20

15

X 109:0'

5

og0

o 0 0

80k 00 0

C; 0

Go

Standard agar plate count

(arithmetic grouping)

0

8to

o 0

o 0

o 0

i c.

8CL

CHART 1. FREQUENCy DISTRIBIUTION OF THE ACTUAL BACTERIAL COUNTS ON 174SAMPLES OF NIGHT'S MILK AS DELIVERED BY A NEW YORK STATE DAIRYMAN

AT A MILK PLANT WHERE PREMIUMS WERE PAID FOR LOW COUNT MILK

and the square of the average of the sum for any series of values.The empirical equation may be written:

For the values in table 4, the standard deviation for the actualplate counts is i 39,600 and for the geometric mean it is log.

Tho five counts greater than 80,000which are not shown in this diagram areas follows;- 96,000, 135,000, 136,000,320,400 and 346,500.

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132 A. H. ROBERTSON

0.48. In other words when using the arithmetic average 68.26per cent of all the actual values should fall between -19,100and +60,100. Could anything be more absurd than expectingfrom 10 to 15 per cent of all these bacterial counts to be less thanzero?

20

15

0-

a 10100*

5'

nai WI W6 1n_ C 6 t0 Vs 1 lb iO (i slAtC4 pi 00 0tI- iQob ci- 6 Aw 03 0tQb 41 0l~ CU- 0 0

0i;Z§01 a; CCZ N; 0; a; all n; rzrmnt; t t; 1 t~ 0; WiOs'Standard agar plate count

(logarithmic grouping)

O C0 0 C-4 r4,00000000 00 000000000 00000 00 0 0 8 ° oO N i O 01 0O 0CO 004 gQ04r4 °000000000000 0000000°°t14 03e0w to 1 t0qo O CQ 50 to r4 0 tteoko co 0o40 0'uC.O O0 O O O. 4

r-40;;;tg C~; V;C; ; ; 0 C V;0; C;0 tosl-40.0) 0.0r4W-o

H~~~~~r o4 a1 C- go to to - Q o to 4 (b O 0 Snoeb0.-4 r4 -4 QC*OJ U)U U) U) Ll. 0

Counts grouped according to logarithmic relations

CHART 2. FREQUENCY DISTRIBUTION OF THE LOGARITHEMS OF THE BACTERIALCOUNTS ON 174 SAMPLES OF NIGHT'S MILK AS DELIVERED BY A NEW YORKSTATE DAIRYMAN AT A MILK PLANT WHERE PREMIUMIS WHERE PAD FOR LowCOUNT MILK.

If the standard deviation based on the logarithmic spread beused, the range is from log. 3.53 to log. 4.49. By transposing,these values become 3,390 and 30,900. Does this range notconform more closely with the known range and distribution in

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AVERAGING BACTERIAL COUNTS

chart 2? Other statistical values merely confirm the ones hereindescribed.Other scientists have recognized the fallibility of the arithmetic

mean and consequently have grouped bacterial counts instead ofaveraging them. Unfortunately, their grouping systems havenot always taken into consideration the proportionate or geo-metric relations among counts. Breed and Stocking (1920) andRobertson (1921), have measured the relationships betweencounts by comparing ratios.Frank (1927) was probably the first to apply the geometric

mean to bacterial counts. Wholly unaware of the work of theUnited States Public Health Service, Robertson and Frayer(1930) demonstrated statistically the principles involved whenthe geometric mean is used. They have applied the geometricrelations among counts in certain correlation studies also. Asearly as 1912, Dr. E. C. Levy (Commission (1912)) realizing theunreliability of the arithmetic mean, prepared a parabolic curveto register differences in bacterial counts. An arithmetic averageamong a series of class percentage values gave what he styled the"bacterial index." Kelley and Possam (1926) have recom-mended Levy's scheme to develop a more satisfactory way ofaveraging bacterial counts in milk and cream quality contests.

CONCLUSIONS

The geometric mean is a more nearly correct average than thearithmetic mean over a normal distribution of bacterial countsin market milk. In other words, the arithmetic mean whenapplied to the actual values in an exponential increase for deter-mining the true average is unreliable.

REFERENCES

BREED, R. S., AND STOCKING, W. A., JR. 1920 The accuracy of bacterial countsfrom milk samples. N. Y. (State) Sta., Tech. Bul. 75.

Commission. 1912 Report of the Commission on Milk Standards Appointedby the New York Milk Committee. Treasury Department, PublicHealth and Marine Hospital Service of the United States, Reprint 78,1-30.

FRANK, L. C. 1927 Standard Milk Control Code. Tentative Draft, November,1927, United States Public Health Service (1927).

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KELLY, E., AND PossAm, R. J. 1926 How to conduct milk and cream contests.U. S. Dept. Agr., Dept. Cir. 384.

ROBERTSON, A. H. 1921 The relation between bacterial counts from milk asobtained by microscopic and plate methods. N. Y. (State) Sta.,Tech. Bul. 86.

ROBERTSON, A. H., AND FRAYER, J. M. 1930 Variability, accuracy and adapta-bility of some common methods of determining the keeping quality ofmilk. I. Methods of comparison. Vermont Sta., Bul. 314.

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