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BAM Appendix 2: Most Probable Number from Serial Dilutions

October 2010

Bacteriological Analytical Manual

Appendix 2
Most Probable Number from Serial Dilutions

Author: Robert Blodgett (retired)

For additional Information, contact: Stuart Chirtel

Revision History:

  • October 2010: Equation for most probable number (MPN) replaced with graphical version; Added expanatory note for the downloadable spreadsheet.
  • July 2003: Added Table 5. for 10 tubes at 10 ml inocula and link to spreadsheet. 

Background
References
Tables

Background

Serial dilution tests measure the concentration of a target microbe in a sample with an estimate called the most probable number (MPN). The MPN is particularly useful for low concentrations of organisms (<100/g), especially in milk and water, and for those foods whose particulate matter may interfere with accurate colony counts. The following background observations are adapted and extended from the article on MPN by James T. Peeler and Foster D. McClure in the Bacteriological Analytical Manual (BAM), 7th edition.

Only viable organisms are enumerated by the MPN determination. If, in the microbiologist's experience, the bacteria in the prepared sample in question can be found attached in chains that are not separated by the preparation and dilution, the MPN should be judged as an estimate of growth units (GUs) or colony-forming units (CFUs) instead of individual bacteria. For simplicity, however, this appendix will speak of these GUs or CFUs as individual bacteria. If a confirmation test involves selecting colonies to test, then a statistical adjustment not discussed in this appendix should be used (see Blodgett 2005a.)

The following assumptions are necessary to support the MPN method. The bacteria are distributed randomly within the sample. The bacteria are separate, not clustered together, and they do not repel each other. Every tube (or plate, etc.) whose inoculum contains even one viable organism will produce detectable growth or change. The individual tubes of the sample are independent.

The essence of the MPN method is to dilute the sample to such a degree that inocula in the tubes will sometimes but not always contain viable organisms. The "outcome", i.e., the number of tubes and the number of tubes with growth at each dilution, will imply an estimate of the original, undiluted concentration of bacteria in the sample. In order to obtain estimates over a broad range of possible concentrations, microbiologists use serial dilutions incubating tubes at several dilutions.

The MPN is the number which makes the observed outcome most probable. It is the solution for λ, concentration, in the following equation 

CFSAN BAM Equation for Appendix 1, relates MPN to lambda (concentration)

where exp(x) means ex, and

K denotes the number of dilutions,
gj denotes the number of positive (or growth) tubes in the jth dilution,
mj denotes the amount of the original sample put in each tube in the jth dilution,
tj denotes the number of tubes in the jth dilution.

In general, this equation can be solved by iteration.

McCrady (1915) published the first accurate estimation of the number of viable bacteria by the MPN method. Halvorson and Ziegler (1933), Eisenhart and Wilson (1943), and Cochran (1950) published articles on the statistical foundations of the MPN. Woodward (1957) recommended that MPN tables should omit those combinations of positive tubes (high for low concentrations and low for high concentrations) that are so improbable that they raise concerns about laboratory error or contamination. De Man (1983) published a confidence interval method that was modified to make the tables for this appendix.

Confidence Intervals

The 95 percent confidence intervals in the tables have the following meaning: 

Before the tubes are inoculated, the chance is at least 95 percent that the confidence interval associated with the eventual result will enclose the actual concentration.

It is possible to construct many different sets of intervals that satisfy this criterion. This manual uses a modification of the method of de Man (1983). De Man calculated his confidence limits iteratively from the smallest concentrations upward. Because this manual emphasizes pathogens, the intervals have been shifted slightly upward by iterating from the largest concentrations downward.

The confidence intervals of the spreadsheet and the tables associated with this appendix may be different.  The MPN Excel spreadsheet uses a normal approximation to the log (MPN) to calculate its confidence intervals.  This approximation is similar to a normal approximation discussed in Haldane (1939).  This approximation is less computationally intense so more appropriate for a spreadsheet than de Man's confidence intervals.
 

Precision, Bias, and Extreme Outcomes

The MPNs and confidence limits have been expressed to 2 significant digits. For example, the entry "400" has been rounded from a number between 395 and 405.

Numerous articles have noted a bias toward over-estimation of microbial concentrations by the MPN. Garthright (1993) has shown, however, that there is no appreciable bias when the concentrations are expressed as logarithms, the customary units used for regressions and for combining outcomes. Therefore, these MPNs have not been adjusted for bias.

The outcome with all positive tubes in each dilution gives no upper bound on the concentration. The tables in this appendix list the MPN for this outcome as greater than the highest MPN for an outcome with at least one negative tube. Similarly, the outcome with all negative tubes is listed as less than the lowest MPN for an outcome with at least one positive tube.

Cautionary Notes

Improbable Outcomes

Several potential problems may cause improbable outcomes. For example, there may be interference at low dilutions or selecting too few colonies at low dilutions for a confirmation test may overlook the target microbe. If the problem is believed limited to the low dilutions, then using only the high dilutions with positive tubes might be more reliable. If the cause of the problem is unknown, then the estimate may be unreliable.

When excluding improbable outcomes, de Man's (1983) preferred degree of improbability was adopted. The outcomes included are among the 99.985 percent most likely outcomes if their own MPNs were the actual bacterial concentrations. Therefore, among 10 different outcomes, all will be found in these tables at least 99 percent of the time.

Inconclusive Tubes

In special cases where tubes cannot be judged either positive or negative (e.g., plates overgrown by competing microflora at low dilutions), these tubes should be excluded from the outcome. The resulting outcome may have different numbers of tubes than any of the tables in this appendix. Its MPN can be solved by computer algorithms or estimated by Thomas's Rule below. Haldane's method can find the confidence limits as described below Thomas's rule.

Using Tables

Selecting Three Dilutions for Table Reference

An MPN can be computed for any positive number of tubes at any positive number of dilutions, but often serial dilutions use three or more dilutions and a decimal series (Each dilution has one tenth as much of the original sample as the previous dilution.) The tables in this appendix require reducing an outcome to three of its decimal dilutions. This procedure for selecting three dilutions was developed for the designs (numbers of tubes per dilution and ratio of dilutions) in these tables. They all have decimal dilutions and a fairly small number of tubes per dilution. For other designs, other procedures may be needed. When the MPN model holds, the three decimal dilutions are chosen to give a good approximation to the MPN of the entire outcome. Otherwise, the reduction may remove interference (possible from another species of microbe or a toxic substance) that can be diluted out. The remainder of this section tells how to select the three dilutions.

First, remove the highest dilution (smallest sample volume) if it and the next lower dilution have all negative tubes. As long as this condition holds and at least four dilutions remain, continue removing these dilutions.

Next, if only three dilutions remain, use them as illustrated in example A. In each example there are five tubes in each dilution. In example A, removing the two highest dilutions (0.001 and 0.01 grams) leaves three dilutions.

If more than three dilutions remain, then find the highest dilution with all positive tubes. There are three cases. In the first case, the highest dilution with all positive tubes is within the three highest remaining dilutions. Then use the three highest remaining dilutions. In example B, the first step removes the highest dilution (0.001 grams.) Since the highest dilution with all positive tubes (1 gram) is within the three highest remaining dilutions, (1, 0.1, and 0.01 grams,) use them. In example C, the highest dilution with all positive tubes (0.01 g) is within the three highest remaining dilutions (0.1, 0.01, and 0.001.)

In the second case, the highest dilution with all positive tubes is not within the three highest remaining dilutions. Then select the next two higher dilutions than the highest dilution with all positive tubes. Assign the sum of the positive tubes of any still higher dilutions to the third higher dilution. In example D, the highest dilution with all positive tubes has 1 gram. Select the two dilutions immediately higher which have 0.1 and 0.01 grams. There is only one higher dilution whose positive tubes are assigned to form the third dilution with 0.001 grams.

In the third case, there is no dilution with all positive tubes. Then select the two lowest dilutions. Assign the sum of the positive tubes of any higher dilutions to the third dilution. In example E no dilution has all positive tubes. The two lowest dilutions have 10 and 1 grams. The sum of the positive in the dilutions with 0.1, 0.01 and 0.001 grams is assigned to form the third dilution with 0.1 grams.

If the three dilutions selected are not in the tables, then something in the serial dilution probably was unusual. This is a warning that the outcome is sufficiently improbable that the basic assumptions of the MPN may be questionable. If possible, redoing the test may be the most reliable procedure. If an MPN value is still desired, use the three highest remaining dilutions. In example F, the three highest dilutions are used. If these dilutions are not in the tables, then use the highest dilution with any positive tubes. The section entitled 'MPN for a single dilution with any positive tubes' shows how to calculate the MPN.

Table of Examples
Examples10 g1 g.1 g.01 g.001 g 
A41000410xx
B55100x510x
C45451xx451
D45431xx431
E43011432xx
F43321xx321

Conversion of Table Units

The tables below apply to inocula of 0.1, 0.01, and 0.001 g. When different inocula are selected for table reference, multiply the MPN/g and confidence limits by whatever multiplier makes the inocula match the table inocula. For example, if the inocula were 0.01, 0.001, and 0.0001 with three tubes per dilution, multiplying by 10 would make these inocula match the table inocula. If the outcome were (3, 1, 0), multiply the Table 1 MPN/g estimate, 43/g, by 10 to arrive at 430/g.

Bounds and approximations for a design without a table

The MPN for a serial dilutions not addressed by any table (e.g., resulting from accidental loss of some tubes) may be computed by iteration or bounded as follows.

Equation of iteration

Where W and Q are two disjoint sets of dilutions that together contain all the dilutions. The lower bound allows low dilutions with all positive tubes to be deleted from the bound. Blodgett (2005b) introduces these and other bounds.

The following gives an estimate of the MPN. First, select the lowest dilution that doesn't have all positive tubes. Second, select the highest dilution with at least one positive tube. Finally, select all the dilutions between them. Use only the selected dilutions in the following formula of Thomas (1942):

MPN/g = (∑ gj) / (∑ tjmj ∑ (tj-gj)mj) (½)

where the summation is over the selected dilutions and

∑ gj denotes the number of positive tubes in the selected dilutions,

∑ tjmj denotes the grams of sample in all tubes in the selected dilutions,

∑ (tj-gj)mj denotes the grams of sample in all negative tubes in the selected dilutions.

The following examples will illustrate the application of Thomas's formula. We assume that the dilutions are 1.0, 0.1, 0.01, 0.001, and 0.0001 g.

Example (1). For outcome (5/5, 10/10, 4/10, 2/10, 0/5) use only (–,–, 4/10, 2/10,–); so ∑ tjmj = 10*0.01 + 10*0.001 = 0.11. Where * means multiplication. There are 6 negative tubes at 0.01 and 8 negative tubes at 0.001, so ∑ (tj-gj)mj = 6*0.01 + 8*0.001 = 0.068. There are 6 positive tubes, so

MPN/g = 6/(0.068 * 0.11)(½) = 6/0.086 = 70/g

Example (2). For outcome (5/5, 10/10, 10/10, 0/10, 0/5) use only(–, –, 10/10, 0/10,–), so by Thomas's formula,

MPN/g = 10/(0.01 * 0.11)(½) = 10/.0332=300/g

These two approximated MPNs compare well with the MPNs for (10, 4, 2) and (10,10,0) (i.e., 70/g and 240/g, respectively).

Approximate confidence limits for any dilution test outcome can be calculated by first estimating the standard error of log10(MPN) by the method of Haldane. We describe the method for 3 dilutions, but it can be shortened to 2 or extended to any positive number.

Let m1, m2, m3 denote the inoculation amounts at the largest to the smallest amounts (e.g., m1 = 0.1 g, m2 = 0.01 g, m3 = 0.001 g in these tables).

Let g1, g2, g3 denote the numbers of positive tubes at the corresponding dilutions. For legibility, we denote yx by "y**x" and "y times x" by "y*x".

Now we compute

T1 = exp(-mpn*m1), T2 = exp(-mpn*m2), etc. 

Then we compute

B = [g1*m1*m1*T1/((T1 - 1)**2] + ... + [g3*m3*m3*T3/((T3 - 1)**2)].

Finally, we compute

Standard Error of Log10(mpn) = 1/(2.303*mpn*(B**0.5))

Now the 95 percent confidence intervals, for example, are found at

Log10(mpn) ± 1.96*(Standard Error).

MPN for a Single Dilution with any Positive Tubes

If just one dilution has any positive tubes, then a simpler expression gives its MPN.

MPN/g = (1/m)*2.303*log10((∑ tjmj)/(∑ (tj-gj)mj))

Where m denotes the amount of sample in each tube in the dilution with a positive tube.

Special requirements and tables included

The attached spreadsheet should be able to handle most specialized designs. Garthright and Blodgett (2003) discusses this spreadsheet. Requests for special computations and different designs will be honored as resources permit. Designs may be requested with more or less than 3 dilutions, uneven numbers of tubes, different confidence levels, etc. (Telephone 301-436-1836 or write the Division of Mathematics, FDA/CFSAN, 5100 Paint Branch Parkway, HFS-205 Rm 2D-011, College Park, MD 20740) The most-published designs, three 10-fold dilutions with 3, 5, 8, or 10 tubes at each dilution, are presented here.

 References

  1. Blodgett, R. J. 2005a. Serial dilution with a confirmation step. Food Microbiology 22:547-552.
  2. Blodgett, R. J. 2005b. Upper and lower bounds for a serial dilution test. Journal of the AOAC international 88 (4):1227-1230.
  3. Cochran, W. G. 1950. Estimation of bacterial densities by means of the "Most Probable Number." Biometrics 6:105-116.
  4. de Man, J. C. 1983. MPN tables, corrected. Eur. J. Appl. Biotechnol. 17:301-305.
  5. Eisenhart, C., and P. W. Wilson. 1943. Statistical methods and control in bacteriology. Bacteriol. Rev. 7:57-137.
  6. Garthright, W. E. and Blodgett, R.J. 2003, FDA's preferred MPN methods for Standard, large or unusualtests, with a spreadsheet. Food Microbiology 20:439-445.
  7. Garthright, W. E. 1993. Bias in the logarithm of microbial density estimates from serial dilutions. Biom. J. 35: 3:299-314.
  8. Haldane, J.B.S. 1939. Sampling errors in the determination of bacterial or virus density by the dilution method. J. Hygiene. 39:289-293.
  9. Halvorson, H. O., and N. R. Ziegler. 1933. Application of statistics to problems in bacteriology. J. Bacteriol. 25:101-121; 26:331-339; 26:559-567.
  10. McCrady, M. H. 1915. The numerical interpretation of fermentation-tube results. J. Infect. Dis. 17:183-212.
  11. Peeler, J. T., G. A. Houghtby, and A. P. Rainosek. 1992. The most probable number technique, Compendium of Methods for the Microbiological Examination of Foods, 3rd Ed., 105-120.
  12. Thomas, H. A. 1942. Bacterial densities from fermentation tube tests. J. Am. Water Works Assoc. 34:572-576.
  13. Woodward, R. L. 1957. How probable is the most probable number? J. Am. Water Works Assoc. 49:1060-1068.
     

 Tables


 

Table 1. For 3 tubes each at 0.1, 0.01, and 0.001 g inocula, the MPNs per gram and 95 percent confidence intervals.
Pos. tubesMPN/gConf. lim.Pos. tubesMPN/gConf. lim.
0.100.010.001LowHigh0.100.010.001LowHigh
000<3.09.5220214.542
0013.00.159.6221288.794
0103.00.1511222358.794
0116.11.218230298.794
0206.21.218231368.794
0309.43.638300234.694
1003.60.1718301388.7110
1017.21.3183026417180
102113.638310439180
1107.41.3203117517200
111113.63831212037420
120113.64231316040420
121154.5423209318420
130164.54232115037420
2009.21.43832221040430
201143.642323290901,000
202204.542330240421,000
210153.742331460902,000
211204.54233211001804,100
212278.79433>1100420

 


 

Table 2. For 5 tubes each at 0.1, 0.01, and 0.001 g inocula, the MPNs and 95 percent confidence intervals.
Pos. TubesMPN/gConf. lim.Pos. tubesMPN/gConf.lim.
0.10.010.001LowHigh0.10.010.001LowHigh
000<1.86.8402216.840
0011.80.096.8403259.870
0101.80.096.941017640
0113.60.710411216.842
0203.70.710412269.870
0215.51.815413311070
0305.61.815420226.850
10020.110421269.870
10140.710422321070
10261.8154233814100
11040.712430279.970
1116.11.815431331070
1128.13.4224323914100
1206.11.8154403414100
1218.23.4224414014100
1308.33.4224424715120
131103.5224504114100
140113.5224514815120
2004.50.7915500236.870
2016.81.815501311070
2029.13.4225024314100
2106.81.8175035822150
2119.23.4225103310100
212124.1265114614120
2209.33.4225126322150
221124.1265138434220
222145.9365204915150
230124.1265217022170
231145.9365229434230
240155.93652312036250
3007.82.12252415058400
301113.5235307922220
302135.63553111034250
310113.52653214052400
311145.63653318070400
3121763653421070400
320145.73654013036400
321176.84054117058400
322206.84054222070440
330176.840543280100710
331216.840544350100710
332249.8705454301501,100
340216.84055024070710
341249.8705513501001100
350259.8705525401501700
400134.1355539202202600
401175.93655416004004600
 555>1600700

 


 

Table 3. For 10 tubes at each of 0.1, 0.01, and 0.001 g inocula, the MPNs and 95 percent confidence intervals.
Pos. tubesMPN/gConf. lim.Pos. tubesMPN/gConf. lim.
0.10.010.001LowHigh0.10.010.001LowHigh
000<.903.1820177.734
0010.90.043.182119934
0021.80.335.1822211039
0100.90.043.6823231144
0111.80.335.183019934
0201.80.335.1831211039
0212.70.87.2832241144
0302.70.87.2833261250
1000.940.055.1840221039
1011.90.335.1841241144
1022.80.87.2842261250
1101.90.335.7843291458
1112.90.87.2850241144
1123.81.49851271250
1202.90.87.2852291458
1213.81.49853321562
1303.81.49860271250
1314.82.111861301458
1404.82.111862331562
20020.377.2870301458
20130.817.3871331773
20241.49872361774
21030.827.8880341773
21141.49881371774
21252.111900177.531
22041.49.190119934
22152.111902221039
2226.1314903241144
2305.12.11191019939
2316.1314911221040
2406.1314912251144
2417.23.115913281458
2507.23.115914311458
3003.20.99920221044
3014.21.49.1921251146
3025.32.111922281458
3104.21.410923321458
3115.32.111924351773
3126.4314930251250
3205.32.112931291458
3216.4314932321562
3227.53.115933361774
3306.5314934402091
3317.63.115940291458
3328.73.617941331562
3407.63.115942371774
3418.73.617943412091
3508.83.617944452091
4004.51.611950331773
4015.62.212951371774
4026.8314952422091
4105.62.212953462091
4116.83149545125120
41283.617960381774
4206.8315961432091
42183.6179624721100
4229.23.7179635325120
4308.13.617970442091
4319.34.5189714921100
432105209725425120
4409.34.5189736026120
441115209805025120
450115209815525120
451125.6229826126120
460125.6229836830140
50062.5149905725120
5017.23.1159916330140
5028.53.6179927030140
5039.84.5181000231144
5107.33.1151001271250
5118.53.6171002311458
5129.84.5181003371773
513115211010271257
5208.63.6171011321461
5219.94.5181012381774
522115211013442091
530104.51810145225120
531115211020331573
532135.6231021391779
540115211022462091
541135.62310235425120
5421472610246330140
550136.3251030401791
5511472610314720100
5601472610325625120
6007.83.11710336630140
6019.23.61710347734150
6021152010358939180
603125.62210404921120
6109.23.71810415925120
6111152110427030150
612125.62210438238180
6131472610449444180
62011521104511050210
621125.62210506226140
6221472610517430150
623157.43010528738180
630125.623105310044180
63114726105411050210
632157.430105513057220
64014726105614070280
641157.43010607934180
6421793410619439180
650167.430106211050210
65117934106312057220
65219934106414070280
66017934106516074280
66119934106618091350
67019934107010044210
700104.520107112050220
70112521107214061280
702136.325107315073280
703157.228107417091350
71012522107519091350
711136.3251076220100380
712157.2281077240110480
713177.731108013060250
720136.426108115070280
721157.228108217080350
722177.731108320090350
723199341084220100380
730157.2301085250120480
731179341086280120480
732199341087310150620
7332110391088350150620
74017934109017074310
74119934109120091380
7422110391092230100480
7432311441093260120480
750199341094300140620
7512110391095350150630
7522311441096400180820
7602110391097460210970
7612311441098530210970
76225124610996102801300
77023114410100240110480
77126125010101290120620
800135.62510102350150820
8011572610103430180970
802177.530101045402101300
80319934101057002801500
810157.128101069203501900
811177.7311010712004802400
812199341010816006203400
8132110391010923008105300
 101010>23001300

 

 


 

Table 4. For 8 tubes each at 0.1, 0.01, and 0.001 g inocula,
the MPNs per gram and 95 percent confidence intervals.

Pos. tubesMPN/gconf. lim.
0.100.010.001LowHigh
000<1.24.3
0011.1.0574.3
0022.3.426.7
0101.1.0584.4
0112.3.426.7
0202.3.426.7
0213.41.09.1
0303.41.09.1
1001.2.0646.7
1012.4.426.8
1023.61.09.1
1102.4.427.3
1113.61.09.1
1124.81.812
1203.61.09.1
1214.91.812
1304.91.812
1316.12.815
1406.22.815
2002.6.479.1
2013.81.09.1
2025.11.812
2103.91.09.9
2115.21.812
2126.52.815
2205.21.812
2216.52.815
2227.93.318
2306.62.815
2317.93.318
2408.03.318
2509.44.319
3004.11.212
3015.51.912
3026.92.815
3105.61.913
3117.02.815
3128.44.018
3207.02.915
3218.54.018
322104.319
3308.64.018
331104.319
332125.224
340104.319
341125.224
350125.224
4006.02.115
4017.52.915
4029.14.118
4107.62.918
4119.24.119
412114.322
4209.34.119
421114.322
422135.724
430114.322
431135.724
432146.628
440135.724
441156.629
450156.629
451167.233
460177.233
5008.33.318
501104.319
502125.224
503146.629
510104.322
511125.224
512146.629
513166.732
520125.324
521146.629
522167.233
523187.233
530146.629
531167.233
532187.233
540167.233
541187.633
542219.039
550197.633
551219.039
560219.039
600114.324
601135.725
602166.632
603187.233
610145.829
611166.632
612187.233
613219.039
620166.733
621187.433
622219.039
623231150
630197.635
631219.039
632241150
633271253
640219.040
641241150
642271253
661311369
670311369
700166.632
701187.233
702219.040
703251150
710197.939
711229.040
712251150
713291254
720229.045
721251151
722291368
723331369
730261153
731301368
732341369
733391791
740301369
741351369
742391791
7434518101
750361475
751401791
7524618101
7535221117
760411791
7614721117
7625321117
7635924146
7704821117
7715521117
7726124146
7805621119
800239.750
801281254
802341369
803411791
810291268
811351375
812431791
8135221120
8146328150
820361491
8214517100
8225521120
8236728150
8248132190
8304718120
8315821150
8327228150
8338739190
83410239190
83511850240
8406224150
8417728190
8429439190
84311044220
84413053250
84515068280
8508432190
85110039220
85212050250
85314067280
85417074340
85519074340
85621090400
86011045240
86114053280
86216068340
86319074340
86422090400
865250120490
866290120520
87016068340
87119074400
87223090490
873270116520
874310150710
875370150720
8764301901000
8775101901000
88024099490
881300120710
8823801501000
883510.01901200
8847002401700
8859803402200
88614004903100
88721007105100
888>21001000


 

 


(Added July 2003)

Table 5. For 10 tubes at 10 ml inocula,
the MPN per 100 ml and 95 percent confidence intervals.
Pos. tubesMPN/100mlConf. lim.
LowHigh
0<1.13.3
11.1.055.9
22.2.378.1
33.6.919.7
45.11.613
56.92.515
69.23.319
7124.824
8165.933
9238.153
10>2312

(Added July 2003)

 


Download an Excel spreadsheet to calculate values (zip file).

Note: The confidence intervals of the spreadsheet and the tables associated with this appendix may be different. The MPN Excel spreadsheet uses a normal approximation to the log (MPN) to calculate its confidence intervals. This approximation is similar to a normal approximation discussed in Haldane (1939). This approximation is less computationally intense so more appropriate for a spreadsheet than de Man's confidence intervals.


Hypertext Source: BAM 8th Edition, Modified from Revision A CD ROM version 1998 on 6/21/2000.