# Science & Research

# Volume III - 4.10 Statistics Applied to Microbiological Analysis

| DOCUMENT NO.:
| VERSION NO.:1.4 | ||

| EFFECTIVE DATE: 10/01/2003 | REVISED: 1-31-13 |

Several analyses used by ORA microbiologists call for the enumeration of microorganisms by statistical means. Two commonly used procedures for estimating the number of microorganisms in a product are the plate count and the *Most Probable Number* (MPN) tube methods. To avoid fictitious impression of precision and accuracy, only 2 significant figures are reported. Many regulatory decisions pertaining to microbial contamination or time-temperature abuse of food will be based upon the level of organisms present**.**

### 4.10.1 Introduction

Many microbiological analyses involve the counting of discrete events, for example plate and tube counts for microbial growth and isolated colonies. As in the case for radioactivity, the situation is one of random, discrete, and relatively improbable events (such as growth of a colony forming unit on an agar plate), and Poisson statistics apply.

**4.10.2 Geometric Mean**

In microbiological assays, because of the techniques used and the fact that biological systems are being measured, a variety of unique statistical situations arise. When determining, for example, the number of colony forming units on a plate from a large number of replicate inoculations, the data often does not correspond to the expected normal distribution. That is, if the frequency of a given number of colonies is plotted against the observed number of colonies, a *non-symmetrical* frequency distribution is observed (note that the normal distribution curve is completely symmetrical, centered about the arithmetic mean). Instead the distribution is *skewed*, or tailed at the higher end. This is attributed to the fact that bacterial counts tend to favor lower counts and disfavor extremely high counts. In this situation, the arithmetic mean is not the best statistical indicator; instead the *geometric mean* is most often used:

where xi are the individual counts, and Π indicates that the product of the observations is determined rather than the sum.

For example, the arithmetic mean of the individual observations 1, 2 and 3 is:

( 1 + 2 + 3 ) / 3 = 2

whereas the geometric mean of the same observations is:

( 1 x 2 x 3 )^{1/3} = 1.8

*Question:* Why would one expect lower plate counts to be more probable than higher counts, thus causing a skewed probability distribution? *Answer:* As the number of counts on a plate rises, in other words the density of colonies rises, an overcrowding error occurs from individual colonies inhibiting the formation of other colonies nearby. Another factor appears to be a "counting fatigue" error at high numbers, where the analyst may not count accurately because of the large numbers involved.

An alternative way to calculate the geometric mean, which can be easily derived from the product expression above, is to add the logarithms of individual counts rather than form the product of the counts themselves. The geometric mean is then defined as:

This formula is much easier for calculation purposes, particularly when a large number of observations are involved.

### 4.10.3 Most Probable Number

Another statistical concept unique to microbiological observations is that of Most Probable Numbe*r* (MPN). The Most Probable Number is a statistically derived estimate of the presence of microorganisms based on the presence or absence of growth in serially diluted samples. After an initial dilution, serial dilutions of the sample are made (for example, 1:10, 1:100, and 1:1000) with a number of replicate tubes (for example, 3 or 5) at each dilution. After incubation, the presence or absence of growth in each tube is tabulated. The resulting code (number of positive tubes) is compared with published tables to give the most probable number of microorganisms per unit of original, undiluted sample. Most probable number tables are published for various numbers of tubes at a number of dilutions. The statistical derivation is beyond the scope of this discussion, but is based on Poisson counting statistics. Tables are published in the *Bacteriological Analytical Manual* (BAM), the *AOAC Official Methods of Analysis*, and General Chapter <61> of the USP.

### 4.10.4 References

(Current Ed.). "Microbial Limits Tests <61>," *U. S.* *Pharmacopeia and national formulary*. Rockville, MD: United States Pharmacopeial Convention, Inc.

Tomlinson, L. (Ed.). (1998). *Bacteriological analytical manual* (8th ed., Rev. A, in hardcopy). Washington DC: R. I. Merker, Ph.D., Office of Special Research Skills, Center for Food Safety and Applied Nutrition, U.S. Food & Drug Administration. The current updated version is available online at: __http://vm.cfsan.fda.gov/~ebam__.