means of department bacteriologyjb.asm.org/content/25/2/101.full.pdfwhile mccrady's tables and...

APPLICATION OF STATISTICS TO PROBLEMS INBACTERIOLOGY

I. A MEANS OF DETERMINING BACTERIAL POPULATION BYTHE DILUTION METHOD

H. 0. HALVORSON AD N. R. ZIEGLERDepartment of Bacteriology and Immunology, University of Minnesota

Received for publication, April 2, 1932

INTRODUCTION

Some time ago the authors had occasion to study the effect ofbacteria on the curing of meat. As a part of the work it wasnecessary to determine the number of bacteria present in thepickle at different stages of the cure. When plates were madeof the pickle in the usual manner, surprisingly low counts were ob-tained, as compared with the number estimated by direct micro-scopic observations of wet preparations. While the latter indi-cated from 500,000 to 1,000,000 organisms per cubic centimeter,the plate count obtained from the same material was 10,000 orless. The low values obtained with plating were at first attrib-uted to an unfavorable medium; consequently a solid mediumwas made by adding agar to clarified pickle. The counts obtainedwith the use of this medium were approximately the same asthose obtained with use of standard beef extract agar. Theseresults led us to believe that a considerable portion of the usualflora of curing pickle could not develop on solid media. Attemptswere then made to determine the number of bacteria present bymeans of the direct count. For this purpose a modification ofthe Breed and Brew (1925) method was used. This was notsatisfactory because of the high salt content of the pickle. Thesalt not only brought about a precipitation of the dye, but also,in crystallizing, obscured many of the organism. Furthermore,the salt prevented proper fixation of bacteria on the slide, so that

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H. 0. HALVORSON AND N. R. ZIEGLER

attempts to remove the salt by washing resulted in the removalof most of the bacteria. Because of the great variety of organismspresent, many of which were actively motile, and because of thepresence of debris, methods making use of a counting chambercould not be used.These difficulties forced us to consider the dilution method as

a means of evaluating the bacterial population.We were at once confronted with the following problems: (1)

calculation of the number of bacteria present from the number oftubes showing growth in the various dilutions, (2) evaluation ofthe accuracy of the data thus obtained, and (3), determinationof the variation in accuracy with the number of tubes used in eachdilution. A careful review of the literature was made in an at.tempt to find answers to the questions involved. We found thatmany investigators had advanced equations that would enableus to calculate the most probable number of organisms from thenumber of tubes that showed growth, and some had publishedtables to aid in the calculation. We were, however, unable tofind any satisfactory solution of the problem of variation in ac-curacy with the number of tubes used in each dilution, or, whatwould have been more desirable, a method for the determinationof the number of tubes that must be used to get a specified ac-curacy. We therefore deemed it necessary to reconsider theentire problem. This is the first of a series of articles dealing withthe various problems involved in the dilution method of deter-mining a bacterial population.

HISTORY

The use of dilution methods in bacteriology dates back to theearly days of the science. About 1875 Pasteur obtained purecultures of bacteria by diluting the original inoculum duringseveral successive transfers to a suitable culture medium. LaterMiquel, Brefeld, and Lister (Kolle, Kraus and Uhlenhuth (1930))obtained pure cultures by inoculating small amounts of dilutedbacterial suspension into a series of tubes of medium.For many years bacteriologists have been using dilution meth-

ods to give some idea of the number of orgaTisms in the material

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APPLICATION OF STATISTICS IN BACTERIOLOGY

examined. This method consists in diluting the material to beexamined, usually in powers of 10, and inoculating equal volumesof the diluted material into liquid media. If growth occurs fromthe inoculation of 1 cc. of a 1:100 dilution and not from a 1:1000dilution, the number of organisms present in the original materialis said to be between 100 and 1000 per cubic centimeter. On thebasis of chance, however, it might be possible to have only 50organisms per cubic centimeter, or more than 1000 per cubiccentimeter in the original material and still get growth from the1:100 dilution and not above that in a single test. A presumptivecoli test, utilizing varying amounts of inoculum, has been used ina similar way in the bacteriological analysis of water to give anapproximate idea of the quality of the water.

This was essentially the basis of the method introduced byPhelps (1908) to estimate the B. coli content of water from pre-sumptive test data. In his method it is assumed that the re-ciprocal of the highest dilution which shows growth representsthe most probable number of organisms present. His methodwas adopted by a Committee on Standard Methods of WaterAnalysis of the American Public Health Association (1920). Incase "skips" occurred, that is, a positive presumptive test from adilution higher than one which was negative, the result takenwas the reciprocal of the dilution next higher than the smallestone giving a positive test.A more accurate method of interpreting dilution data has been

supplied by McCrady (1915). In developing his equations hebegins with the proposition that there is only one organism foreach 100 cc. in the sample. He stated that this one organismmust obviously be contained in one of the 1 cc. volumes and thatthe probability of not getting the organism when a single cubiccentimeter is removed from the 100 cc. volume is 0.99. Thefollowing quotation from his article (p. 185) shows how he hasdeveloped his equation.Now suppose 2 B. coli are in the sample. The probability of each

organism's not being contained in the 1 cc. withdrawn for the fermenta-tion test has been shown to be (0.99). Then, by the principle just il-

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lustrated, the probability of neither organism's appearing in this 1 cc.is equal to the product of the separate probabilities, or (0.99) (0.99) =0.9801. And if a great number of such samples were examined, about98.01 -per cent of the results would be "0/1 in 1 cc."

In general if V represents the number of volumes in the sample, andx the number of B. coli in the sample, and one volume is withdrawn, theprobability that this volume will contain no B. coli is given by

[V - x

Thus when 1 cc. of the sample is withdrawn for the test, V becomes

100 and the formula becomes [ W].hen a 10 cc. quantity is with-

drawn, V becomes 10 (there are ten 10 cc. volumes in the sample), andthe formula becomes 0.9x.

With this reasoning as a foundation for his later work, McCradydeveloped the equations from which it is possible to calculate themost probable number of organisms per cubic centimeter fromdata obtained by inoculating a series of tubes with the samedilution or from several series of tubes inoculated with severaldifferent dilutions. For the special case where a series of tubesare inoculated with 10 cc., a second with 1 cc., and a third with0.1 cc., the following formula is given:

(p + q) (log 0.9) + (r + 8) (log 0.99) + (t + u) (log 0.999) p (log 0.9) r(log 0.99) t(log 0.999)IT-0.UX 1-0.99X l-0.9998

In this equation, p, r, and t represent the number of tubesshowing growth in the different series and q, s, and u representthe number of tubes showing no growth in the correspondingseries of dilutions. To simplify the use of the method, McCrady(1918) has solved the equation for all possible combinations forseveral special cases, including those when 5 and 10 tubes areused in each dilution. These solutions have been put into tablesso that the method may be used without tedious calculation.While McCrady's tables and equations may be regarded as

solving the problem of calculating, from dilution data, the mostprobable number of bacteria present, they do not offer a solution

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to the more general problem of variation in accuracy with thenumber of tubes used. In attempting to solve this problem withMcCrady's equations, we find that the mathematics become verycumbersome. McCrady's equations are each limited to a specialcase, so that to solve them one must resort to rather tedious cal-culation. A more general solution whereby all cases could beevaluated from a single equation and a single table is very desir-able. We have accordingly approached the problem from adifferent angle and have worked out tables somewhat analogousto those of McCrady.On the basis of minor assumptions, Wolman and Weaver (1917)

have simplified McCrady's formula. Their equations are con-venient for the calculation of the most probable number of bac-teria, but they do not enable one to evaluate the accuracy of thedata.

Various methods of interpreting dilution data have been con-sidered in a series of articles by Wells (1918; 1919; 1921), and inan article by Wells and Wells (1922). Their treatment does notlead to results that can be used in the general solution of theproblem.The methods advocated by Wells have been objected to by

several investigators. Notable among these is Cairns (1918).More general considerations of the dilution method have been

contributed by Stein (1922), Greenwood and Yule (1917), Fisher(1925), and Reed (1925). All of these men have shown that thenumber of tubes showing no growth when inoculated with afixed quantity of a single dilution is equal to e-a, where x repre-sents the number of organisms per cubic centimeter in that dilu-tion, and a represents the volume of the dilution used for theinoculation.

Stein not only made use of this exponential relationship tointerpret dilution data, but he attempted to simplify its appli-cation by graphical means. He also showed by graphical meanshow the accuracy of the method varied with the number of tubesused. These considerations were, however, limited to a fewspecial cases.Greenwood and Yule considered not only the special case of a

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single dilution inoculated into a series of tubes but also thegeneral case of several dilutions inoculated into a series of tubes.Although they give the equation from which the most probablenumber of organisms can be calculated from experimental datainvolving several dilutions and several tubes in each, they havemade no attempt to simplify the solution of this general case.Since the general equation is rather involved, it is necessary tosimplify the solution by means of tables in order to make it appli-cable for practical use. Greenwood and Yule show how the ac-curacy of dilution data can be evaluated. Here again, simplifi-cation is needed for general application.

Their equation for the general case is as follows:

(ain+,+am2......a6em,-eaz a+m2 -az anmn - anx(allass*@ -- an^)= e aa2 + 1-e aa _,-ae~

In this equation x is the most probable number of bacteria percubic centimeter that will give ni negative and ml positive resultswhen N1 tubes are inoculated with a, cc. each, and will give n2negative and m2 positive results when N2 tubes are inoculatedwith a2 cc. each, etc.

Reed's contribution is very helpful in the special cases whereone tube in each of several dilutions or where several tubes ofthe same dilution are used, but his solution is not extended to themore general problem of several tubes in each of several dliutions.The dilution method has been used by Cunningham (1915),

by Cutler (1919a), and by Cutler, Crump, and Sandon (1922)to evaluate the number of protozoa in soil. In the first two ofthese publications the dilution data were interpreted by themethod of Phelps (1908), while Cutler, Crump, and Sandon used atable calculated by Fisher (1925). This table by Fisher is usefulonly for a very special case, so that it is not of any material valuein general application. Fisher's table appears to be calculatedfrom the exponential function e-X.A theoretical consideration of the dilution method has also

been contributed by Clark (1927), who applied the method todetermine the numbers of bacteriophage in a suspension.

After a careful review of the literature on the subject we still

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feel it is necessary to reconsider the entire problem. General aswell as special equations should be developed that are based onthe reasoning used by Stein (1922), Greenwood and Yule(1917), Fisher (1925), and Reed (1925). These equations shouldbe solved for all the special cases that are in common use, and thesolutions should be arranged in tabular form, as McCrady didfor his equations. Furthermore, tables should be made availablewhich will aid in the solution of any special case not in commonuse.A more detailed consideration of the accuracy of the method

is needed. The calculation of the accuracy should be simplifiedso that anyone using the dilution method will be able to deter-mine, to a reasonable degree, the limits of accuracy of his data.This must be made simple enough so that it can be applied gen-erally.The mathematics involved in the dilution method should also

be applied to other problems in bacteriology, such as the deter-mination of the percentage of insects that may be infected withcertain viruses, or the interpretation of data obtained when aseries of animals are injected with a single dilution or severaldilutions of a given pathogenic bacterium or virus. It is alsodesirable to investigate the effect produced on dilution data ifwe accept the theory that single cells cannot develop.A consideration of these and other probability problems will

be published in a series of articles on the subject. It has beendeemed necessary to verify by experimental data some of themathematical considerations, and data thus obtained will alsobe presented in this series.

This first paper is confined to the development of equationsto be used for the evaluation of bacterial populations by thedilution method. Tables are included which aid in the solutionof these equations, as well as special solutions which simplifytheir general application.

THEORY

In order to determine the number of bacteria in a sample ofliquid material by the dilution method it is necessary to dilute

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the material to such an extent that when a sample is removed,bacteria may or may not be present. The problem at hand, then,is to determine the probability of getting bacteria in a certainsample. This probability will depend upon the number of or-ganisms present.A number of earlier investigators have shown how this probabil-

ity is related to the number of organisms present. It is believeddesirable, however, to develop these relationships from funda-mental reasoning so that it will be possible for those not familiarwith Poisson's Series to understand the derivation of the equationswithout having to make a special study of the mathematics onwhich the Poisson's Series is based. An understanding of thederivation of the equations is necessary for their proper appli-cation.There are several ways in which the bacterial population can

be determined by the dilution method. One may determine thenumber present by inoculating a large number of tubes of mediawith an equal volume of the sample to be tested, and determinethe number present by the percentage of tubes that show growth.Or one may inoculate a series of tubes of media with a givenvolume, another set with a smaller volume, and a third set witha still smaller volume, and then determine the number of organ-isms present by the number of tubes showing growth in the dif-ferent series.We are interested, therefore, in several cases: (1) the probabil-

ity of getting growth in a single tube, when it is inoculated witha definite volume of the sample; (2) the probability of getting acertain number showing growth out of a series of tubes, all ofwhich are inoculated with the same volume of a given sample;(3) the probability of getting a certain combination of tubesshowing growth out of several series of tubes when the tubes ineach series are inoculated with different sized samples.

Case I we can designate as a single tube of a single dilution;case II, as several tubes of a single dilution; and case III, asseveral tubes of each of several dilutions.

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APPLICATION OF STATISTICS IN BACTERIOLOGY 109

CASE I

To calculate the probability that an organism will or will notbe contained in a certain sample, let us assume that a large vol-ume (N cc.) of the material to be sampled be at hand, and that inthis material there are x organisms per cubic centimeter. Let usassume further, that the volume of an organism is v cc. (unit vol-ume), that all organisms have the same volume, and that thewater present be divided up into particles of the same unit volumeas a bacterium. If we imagine now that we remove one of theseparticles, and let P equal the probability that our selection willnot be an organism, then

N_ N$

p V

Nv

This will be true because N represents the total number ofparticles in a volume N. Nx is the number which are bacteria,and N - Nx, the number which are water. The probability Q

Nxthat the selection will be an organism is N. These expressions

v

can be simplified to P = 1 - vx; Q = vx.Let us assume that enough of these small particles are removed

so that the aggregate is 1 cc. The total probability will be ex-

pressed by the binomial [(1-vx) + vx]i. The first term of the1

binomial will be (1 - vx)', which represents the probability thatno organisms will be contained in a 1 cc. sample.

P - (1 - vX).

This can be simplified as follows:

In P-- 2 I- 2,4'2 3 4 .


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Since v is very small, the second and following terms will bevery smnall in comparison with the first term as long as relativelysmall values of x are considered.

ThereforeIn P- -z

or

P e~ $ (1)

If instead of taking out 1 cc., we remove a cc., thena

P - (1 - vx)'

lnP=a[-xvz' v2xIn P - a-x 2 ..'

- ax

or

P e- 2s (2)

andQ e1-ea2 (3)

CASE II

If several tubes are to be inoculated with the same dilution thenumber of tubes which show growth will depend upon the numberof organisms present and upon the element of chance.

If n tubes are inoculated, the total probability will be expressedby the binomial [e-a + (1 - eax)] n. By expanding this binomial,the following series is obtained:

(e a)n + n(e)-a (1- e-) + - 1) (e a)n(- - ax

2 I

The first term of this series gives an expression for the probabilitythat none of the tubes will show growth, the second term for theprobability that only one of the tubes will show growth, etc.

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If we let p equal the number of tubes that show growth and qthe number that show no growth, in which case p + q = n, theseries becomes:

(e az)Q + n(e- as)f (1-e ax)p + n(n -

1) (eG az)Q" a(1 - . ..

If we let P be the probability, then the following equation willhold for any combination of p and q:

- (p + q) a(e)q ( az)P (4)p I q I

If p, q, and a are kept constant in this equation, P will vary onlywith x. From the equation and the nature of the problem, it isevident that there must be a maximum value of P correspondingto some particular value of x. This optimum value of x can befound by differentiating equation (4), which gives the following:

- axdP 1 paedx P --ax

By putting the derivative equal to 0, it is possible to find thevalue of x that corresponds to the maximum value of P. If welet x be the optimum or most probable value of x, its value may befound by substituting x for x in the above equation, when thederivative is placed equal to 0. The resulting equation simplifiesto

f Ilnn (5)

a q

In the case of several tubes in a single dilution, the most prob-able number of organisms per cubic centimeter is given by thenatural logarithm of the ratio of the total number of tubes to thosethat show no growth. Changing from base e to base 10, theexpression becomes

2.3026 nf = 23026log- (6)

a q

In deriving this equation, it has been assumed that in bacterialsuspensions, all values of x are equally likely to occur between

illl


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O and a number determined by the maximum bacterial popula-tion that can occur in a sample. By means of this formula it iseasy to calculate the most probable number of organi presentin a sample from the number of tubes that show no growth. Thiscan also be obtained by means of equation (5) and table 1 (ap-pendix). From equation (5), we have

q -az- en

The function e-ax then gives the fraction of tubes which show nogrowth. By means of a table showing values of e-1 for differentvalues of x, it is possible to determine the most probable number

of organisms present corresponding to any value of q. Suchn

tables have been calculated by other investigators (L. von Bort-kewitsch (1898) and H. E. Soper (1915)). The table of Bort-kewitsch is carried out to only four places and that of Soper tosix places. Whereas these tables may be adequate for the cal-culation of the most probable number from dilution data, wefound that they did not suffice for the calculation of the fre-quency distribution of experimental values that might be ob-tained on a single suspension. Such calculations are essentialin order to show a convenient relationship between the accuracyof the data and the number of tubes used in each dilution. Forthis reason we found it necessary to calculate these tables to ninedecimal places. They are included in the appendix. Values ofthe logarithm of (1 - e-z) are included in this table to simplify thesolution of other problems which will be discussed later.Examples: Suppose that in an experiment 0.100 of the tubes

inoculated with a cc. each showed no growth, then from the tablewe find that the value of x corresponding to e-x = 0.100 is 2.30.This number is therefore the most probable number of organismspresent in the size sample used for the inoculation.By means of this table it is also possible to determine the

percentage of tubes showing growth which are inoculated withdifferent sized samples. Let us assume that a liquid contained0.650 organism per cubic centimeter, or 65 organisms in a 100 cc.

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sample. In this case there will be 0.0065 organism in 0.01 cc.,0.065 organism in 0.1 cc., and 0.65 organism in 1 cc. Referringto the table, we see that for

x - 0.0065 e- = 0.9935x = 0.065 e = 0.9371x- 0.650 e = 0.5222x = 6.500 e-z = 0.0015

This means that for every 10,000 tubes inoculated with each ofthese dilutions, on the average, 15 of those inoculated with 10cc., 5222 of those inoculated with 1.0 cc., 9371 of those inoculatedwith 0.1 cc., and 9935 of those inoculated with 0.01 cc. would showno growth; or if 100 tubes were inoculated with each dilution,one would expect that all the tubes receiving 10 cc., 48 receiving1 cc., 6 receivng 0.1 cc., and 1 receiving 0.01 cc. would showgrowth. If a person wished to determine the most probable num-ber of organisms by the result of inoculating a series of tubes witha single dilution, the best size of inoculum to use would in thatcase be 1 cc.When a person inoculates a series of tubes with a single dilution

and finds the percentage which show growth, he will also be able,by means of this table, to determine what size inoculum wouldhave produced growth in 50 per cent of the tubes.

CASE III

In this case several dilutions are used, and several tubes areinoculated with each dilution.For this development the following terms will be used:

wI, (1 - wi) - the probability of a success and failure respectively, in asample of size al,

Wo2, (1 - W2))- the probability of a success and failure respectively, in asample of size a2,

tWI, (1 -ws) = the probability of a success and failure respectively, in asample of size a,,

n= the number of samples of size a, that are taken,n2 the number of samples of size a, that are taken,n, = the number of samples of size a, that are taken,

pi, q, - the number of failures and successes obtained out of ni trials.P2, q,2 the number of failures and successes obtained out of n12 trials,ps, q, = the number of failures and successes obtained out of n, trials.

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The total probability will therefore be the product of severalbinomials, and by expanding each of them and multiplying theexpanded forms together, the different terms in the product willbe the probability of getting any set of combinations of failuresand successes, as

plql; p2q2; psqs;.

The general formula for the probability of any one of these termswill then be:

(piq+ql) I(1, Po,)Wl (P2 +q2) I! (1 - P) (P, +qs) I (2w),W( )A. (7)p l ql I p, !q, I ps Iqs I

This equation is obtained by multiplying together several equa-tions, as (4), each being derived from a different binomial.Now if the probabilities wl, w2, W8, etc., are functions of x, it

is posible to find the most probable values of x by differentiatingthe above equation with respect to x- and putting the derivativeequal to zero. This equation can be differentiated most readilyby taking the logarithms of both sides, thus:

InP l (pi+ql) (+n (p2 + q2) + In (ps +q+)+qllnwi+pil q ! P q2 p!q!

q2ln ws + q3lnws ..... + pil n(l-wl) + p2ln(1-W2) + P31n(1-W).....

Differentiating and substituting n - pi for q1, etc., we getdPi ln _i Pi 1 lnW2 _ 2 1 d [n" ns 1_dx P dx l1 +dzWILd 1-J+--dx 1- WJ .....

Now if x represents the percentage of a certain kind that arepresent in a unit, and a,, a2, a3, etc., represent the number of theseunits that are selected, then

toi = (1 - )1 0W2 = (1 - x)a2; t (1 x)G; etc.

and

d In w, -a d In W2 -a2 dlnw, -a,csdx 1-x' dx 1-x' dx 1-2'

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Substituting these values in the above differential equation,putting it equal to zero, and factoring out all common multipliers,we get:

a, n1 - + Ch71-2D2+ as ns- )1

+WI,- -

_a + p2s+ P3 3 ., ain, + a2n + asn3 . . . . .(8)

1WI 1-W2 I - 1

This is the same type of equation as that developed by Green-wood and Yule (1917), but is a little more general. With thisformula a1, a2, a3, etc., ni, n2, n8, etc., are constants for any givenexperiment. Pl, P2, ps, etc., can be determined by experimentaldata. Since w1, w2, w3, etc., are functions of x, it is then possibleto solve for x the most probable value of that variable. The solu-tion of the equation can be simplified if tables are worked out thatgive the relationship between x and w1, w2, W3, etc.To apply this to the problem of determining the most probable

number of bacteria per cubic centimeter, let us assume that aset of tubes are inoculated with various amounts of the sample,one set inoculated with 10 cc. each, another set with 1 cc. each,and a third set with 0.1 cc. in each tube. In this case,

WI =e- lW2 - e Ws = e71

a= 10 a2 = 1 a4 = 0.1

Let us assume that ten tubes are used in each set, Thenn-= n2=-n = 10

The general formula then becomes10 pi P2 P3

1Oz+ + =1111-e 1-e~~l ( -e-1

in which pi = the number of tubes receiving 10 cc. that showgrowth,

P2 = the number of tubes receiving 1 cc. that showgrowth,

p3 = the number of tubes receiving 0.1 cc. that showgrowth.

In the appendix, table 2, will be found values of l for1 - c1

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116 H. 0. HALVORSON AND N. R. ZIEGLER

the different values of x. With the aid of these tables it is aneasy matter to solve the equation by trial and error by selectingthe value of 1 1 1

1 - e13' 1 e-r-' 1 -e-' X that will make theleft hand side of the equation equal to 111. The values of x

corresponding to these values of 1 , etc., will be the most1 - e x

probable number of organisms per cubic centimeter.To demonstrate the method of calculation, a specific example

is included.Suppose in an actual experiment 160 tubes were inoculated

with each of a series of dilutions. Suppose that the followingresults were obtained:

DILIJTION NUMBER SBROING GROWTH NUMBEX SHOWING NO GROwT

1' 160 0106 160 0106 158 2107 61 99108 8 152199 0 160

The critical dilutions are therefore 106, 107, 108. The most prob-able number of organisms present in the 107 dilution can beobtained from the general equation:

ap + 2p + a-ps ainl-e 1-e 1-em

In this experiment, a, is 10, a2 is 1, and as is 0.1; n1 = 2=n = 160; p1 = 158, p2 = 61, ps = 8. Substituting these valuesin the above we get

10-158 1*61 8-1Oz+ + -160 (10+1 + 0.1)1-e 1-e 10(1-eii°)

1580 61 8l-ox1- + -1776.01-e 1~~~-x 10(1 z°

An approximate idea of the value of x can be obtained by con-sidering the middle dilution in which the proportion of tubes


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showing no growth is - = 0.6188. From table 1 (appendix)160

the value of e-x = 0.6188 is found to correspond to a value of x= 0.4800. By obtaining the reciprocals of 1 - e-10X, 1 - e-x,and 1 - e- from table 2 (appendix) and substituting in the aboveequation, we can obtain a more accurate value of x. Substitutingthe proper reciprocals in the left hand side of the equation andsolving, we get

1776.898 for x - 0.46501774.598 for x - 0.47001770.194 for x - 0.4800

By interpolation, we find that the function becomes1776.00 for x -0.4669

In the case of the ten tubes inoculated with each of the dilu-tions this equation has been solved for all the combinations thatare likely to occur. Table 3 (appendix) shows the most probablenumber of organisms which correspond to each of the combina-tions.When 10 tubes are used in each of these dilutions, and when

a1 = 10 cc., a2 = 1 cc., and as = 0.1 cc., the general equation (7)becomes

10! 10 10 (e los)x (e-)? (es1)q (l-ez io)P(1se0z)p (1 - e s

pi qIg P2 I q2 IAp I q,!

Taking the log to the base 10 of both sides of the equation, weget

log P= log 10l + log! I + log l-!-z10Oq+ q2+-I loge+p1!ql p!q2! p,!q,! 10,

lOpi log (1- e- lo" + P2 109 (1-e6-) + pS 109 (1-e- 10)By means of this equation it is possible to solve for the value

of P for any given value of x and pi, p2, and p3. This has beendone for all the combinations where P has a value greater than0.01 per cent. These values show how often a certain combinationcan be expected if the number of organisms in the solution are asindicated by the most probable values of x. These frequencies

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118 H. 0. HALVORSON AND N. R. ZIEGLER

are helpful in interpreting data. If certain combinations arefound to occur more often than indicated by these values of P,one may become suspicious that something is wrong either withthe technic or with the medium.

REFERENCESBORTKEWITSCH, L. VON 1898 Das Gesetz der Kleinen Zablem. Druck und Verlag

von B. G. Teubner, Leipzig, vi + 52 p.BREED, R. S., AND BREW, J. D. 1925 Counting bacteria by means of the

microscope. Circular of the New York State Agricultural ExperimentStation, Geneva, second reprint, 58, 1-12.

CAIRNS, W. D. 1918 Science, N.S., 47, 239-240.CLARK, H. 1927 Jour. Gen. Physiol., 11, 71-81.CUNNINGHAM, A. 1915 Jour. Agric. Sci., 7, 49-74.CUTLER, D. W. 1919a Jour. Agric. Sci., 9, 430-444.CUTLER, D. W. 1919b Jour. Agric. Sci., 10, 135143.CUTLER, D. W., CRUMP, L. M., AND SANDON, H. 1922 Phil. Trans. Roy. Soc.

London, Series B, 211, 317-350.FISHER, R. A. 1925 Statistical Methods for Research Workers. Oliver and

Boyd, London, vii + 239 p.GREENWOOD, M., AND YULE, G. UDNEY. 1917 Jour. Hyg., 16, 36-54.KOLLE, W., KRAUS, R., AND UHLENHUTH, P. 1930 Handbuch der Pathogen

Microorganismen. Zehnter Band, 800 pp. Gustav Fischer, Jena.MCCRADY, M. H. 1915 Jour. Infec. Dis., 17, 183-212.McCRADY, M. H. 1918 Pub. Health Jour., 9, 201-212.PHELPS, E. B. 1908 Amer. Jour. Pub. Hyg., N.S., 4, 141-145.REED, J. LOWELL. 1925 Report of Advisory Committee on Official Water Stand-

ards. Appendix III. Public Health Reports, Washington, 40, 693-721.

Standard Methods 1920 Standard methods for the Examination of Water andSewage. American Public Health Association, Boston, 4th edition,vii + 115 P.

SOPER, H. E. 1914 Biometrika, 10, 25-35.STEIN, M. F. 1917 Eng. News Rec., 78, 391-394.STEIN, M. F. 1918 Amer. Jour. Pub. Health, 8, 820-829.STEIN, M. F. 1919 Jour. Bact., 4, 243-265.STEIN, M. F. 1921 Jour. Amer. Water Works Assoc., 8, 182-187.STEIN, M. F. 1922 Engin. and Cont., 57, 445-6.WELLS, W. F. 1918a Science, N.S., 47, 46-48.WELLS, W. F. 1918b Amer. Jour. Pub. Health, 8, 904-905.WELLS, W. F. 1919a Amer. Jour. Pub. Health, 9, 664-667.WELLS, W. F. 1919b Science, N.S., 49, 400-402.WELLS, W. F. 1919 Amer. Jour. Pub. Health, 9, 956-959.WELLS, W. F. 1921 Jour. Amer. Water Works Assoc., 8, 187-190.WELLS, P. V., AND WELLS, W. F. 1922 Jour. Amer. Water Works Assoc., 9,

502-527.WOHLMAN, A., AND WEAVER, H. L. 1917 Jour. Infect. Dis., 21, 287-291.


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PROBAEILITY TAMIA ACode X P Code x p Oode x

1010 10 100.0 100.0 10 10 0 2.40 3.4 10 9 0 1.70 5.3010 10 92 0 9 9 6.0 0.O 10 8 7 3.10 0.0;10 10 1 3 10 9 85.26 0.04 10 8 b 2.78 0.191010 7 12.0 26.2 10 9 7 4.58 0.27 10 8 5 2.49 0.10 10 b 9.18 25.0 1096 3.9 0.85 10 8 4 2.21 .10 10 5702 24.2 10 95 346 2.25 10 8 3 1.96 4.5.10 10 4 542 234202g 248 10 83 1.719 .410 10 3 4.25 21.7 10 3 2.63 8.57 10 8 11.50 l.3114 54234 10 4.9 4.80 108118410 10 2 3.49 17.3 10 22. 11.77 108 Q 1. 7.2110 10 1 2.75 10.2 10 1 1.97 11.02 10 7 62.19 0.01

10 7 51.95 0.20 10 3 1 25 0J9 10 4 0107 0110 41.74 0.80 '10 2 109 2:0-1 4O 4 0 0510 2.5 10 10.6 S. i 4 3 018 0.l10 2 1 10.1 10 0 0. 210 10 4 2 0.700 2.10 I 1.6 10.0 0 5 1 '° 02 lo 4 1 0.5810 1 5 4 1.5 0.14 10 4 .0.493 1410 6 1.75 0.01 10 5 1.02 0.9 10 340.77 0.0218 I 0.08 l 5 0 10 0. 010 14 , 0.11 10 1 0.742 10 0 .10 3 1.25 0.59 10 0 0.622 12.: 10 1 0:.4 b.931030 399 1767 10 0 2 .340.15 9 1 07.8610O 2 . Zl001 10 0 1 .26 I.4950 341710 2 3.5' 0.12 10 00 .231 7.95 4 0:21022.4 .99 89 0 449003 0.2210 2 -1 .38 53 1 4t 09102 9 4 1 .341.1310 2 0 .918,4] 0 .43501 40 2 3510 1 . 0. 2: 0.26

10 1 1 . 3 52 38 01010 .75 1. 5 .41 0. 0 .255 629 2 3 1 08.6 0 . 0.11 8 2 2 .210 0.14} 2 ~0 * :831 86i0 .20 0.04 8 20:1li .112 0. 8 0 8 2 1 .1 41.11 22 1 84 81 0 .1 9.3121 0 .19.03 8 .2 0. 8 0 2 .16 0.069 02.2 0.0 8 21 0. O

2 .18800 1.1 0. 8a 0 19 3 0 .1 06.02900.6 5.2 82 3 .23o.0 7 0 .212 0.02

O'j°'12&? °* 411 1 2 .1I

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2 ~~~~~~~~1 . 10 07 15001 0 .C> .02 .106 o.o4 52 0.086 3.36 4 3 0 .080 0.4301.092 0.83 1 2 . 4 21 0.lg1O07 0.0? 1 1 08O 2 0 2.060.1:812 O.,1 1650 . 260.2 01 09 0.3 4 1 2.0:0 0.01

I4 081 6 0.00 5

0 0218.64 121.06 0.50240.101O1.072 0. 1 0 .0 7.21D1 .11 0.0 U8 1 0.11 4 0 2 .* 0.0211.05 0.79 2020 . 0.01 0 1 802 .113 .00 0 0.0930.4 401 0o 18 92 1 .095 0.2B 4 3 1.092 0.03 3 0 .0501.02 og6 ° 2 0 18.64 1 2 1 .038 0.02

i 1cc.0eaha t 0.1cc.2 0eah 1T 0n 0oi2 0 .0 1°l 2 l2 1 I :M 8:62 1 0 1°0 421 2 22 0:. 2 1 0 .00 .01 I 0 0 *01103I I .055 0. 2 0 2 .04 O.pl1 0 2 0 .1 AOt21 0 .0 7.A 2 0 1 .03 0.1 1 1 .018 0.008 2 o.0 o 02 12 °0 0 :0o; Ao 0 1 0 o :

This table indicates the number of bacteria per cubic centimeter based on the numberof tubes showinggrowth which have been inoculated with dilutions as outlined. Ten tubes are to be inoculated with 10 cc.each, ten tubes with I cc. each, and ten tubes with 0.1 cc. each. The oode number is made up of thenumber of tubes showing growth in each case, the first number of the code representing the number oftubes showing growth that are inoculated with 10 cc. each, the second those inoculated with 1cc. each,and the last those inoculated with 0.1 cc. each. The column labeled X then gives the most probablenumber of bacteria per cubic centimeter in the material used for inoculation. The column labeled Pgives the percentage of times that the code would be obtained if an infinite number of determinationswere made of a solution containing the number of organisms indicated by X.Example: Suppe that a culture is inoculated into broth in a series of 10 dilutions in steps of 10.Assume that 9 of the tubes inoculated with 10-7 cc., that 3 of the tubes inoculated with 10-6 cc. and thatnoneof the tubesinoculatedwith 10-' cc. show growth. The codewill then be9 30. Referring tothe table,X is found to be 0.255. This means that the most probable number of bacteria in the 10-8 cc. dilutionwas 0.255 bacteria per cubic oentimeter, orthat the most probable numberin the original solution was 0.255times 108 or 25,500,000. If this experiment were repeated an infinite number of times on a solutioncontaining this number, this result would be obtained 6.23 per cent of the time.If 5 tubes are used in each dilution instead of 10, then multiply the oode obtained by two, and referto the table, as above. In this case, however, the column labeledP does not apply.121


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