RE: Blade article

["Taylor, Christopher" <Christopher.Taylor@ci.toledo.oh.us>]

This IS a great explanation of what a geometric mean is, and why it is used
in microbiological statistical studies. Makes perfect sense to me. However,
to a non-scientific/non-math oriented consumer it is very confusing. Many
people have no idea what a standard deviation, a mean or a log
transformation is. How then, do you easily explain to this person what a
geometric mean is(when they don't understand logs or means)? I know this is
basic to those of us in the field, but to the average consumer, it's very
advanced voodoo statistics!  :-)

The author I was referring to in my previous post was the author of the
Blade article, Tom Henry. I know he understands what a geometric mean is,
but he will probably never be able to get the column space to explain it..
Therein lies the problem.

I had also mentioned that the only thing I could think of that would be
"other factors" was Tom Henry's way of implying the geometric mean's
"elevating low numbers and reducing high numbers"(quoted from below). Not
sure that's what he was really trying to say, but it was all I could think
of!

Chris Taylor
Chief Chemist
Toledo Water Plant
419-245-1717



-----Original Message-----
From: Richard L Whitman [mailto:rwhitman@usgs.gov]
Sent: Wednesday, September 01, 2004 10:52 AM
To: Shannon Briggs
Cc: beachnet@great-lakes.net
Subject: Re: Fwd: RE: Blade article



I am not the author of the article referred to paper, but I can explain why
one uses geometric means.

A simple arithmetic means, also called an average, is an estimate of the
population and assumes that the samples are  representative of the true
population.  Since this almost never completely true, we invented
statistics to help us estimate of the characteristics of the population
(range, variation, distribution, mean, trends).

E. coli, like most biological populations, are not normally distributed,
they are most often clumped (clustered, contagious, patchy, etc).  In
statistical terms, this means the variance exceeds the mean.  There are
several solutions to this.  One way is to take the median.  This is the
sort of thing they do with people's incomes or home prices.  That is
because while most of us are average, those mega-rich really skew the mean
upwards.  There are two more ways that we use to make the sampled
population more normal.

Composite sampling from many places on the beach will give you a better
representation than single samples.  This is not without cost though, you
lose information on the variation which is important for most statistical
testing.   Here is a seemingly unlikely example, but I've seen this sortof
data often. You took 5 samples each reading 50, 50, 100, 100, 1000.  The
compositing of those samples would  yield 260 cfu/100ml, a beach closure.
If you had taken the samples individually you would have found that the
standard deviation was 414 and you would know that there was a problem with
your estimate of the average.  We look at sampling strategy closely in the
August issue of Env. Sci and Tech.  Julie Kinzelman and Al Dufour have
worked with composite samplings a lot and have some actual data on this.

The second way to deal with the extremes, is to log transform the data.
Microbiologist traditionally use 10 based logs.  This has the effect of
elevating low numbers and reducing high numbers.  In effect, this allows
for a more bell shape distribution, a population characteristic that is
necessary for most traditional statistical testing.
                                          
 The geometric mean can be computed by:   
    1. taking the logarithm of each       
       number                             
    2. computing the arithmetic mean of   
       the logarithms                     
    3. raising the base used to take the  
       logarithms to the arithmetic mean  
                                          


                                          
 Here is an example                       
                                          
                                          
                                          
  X                                       
            Log(X)                        
  1                                       
            0.0000                        
  2                                       
            0.30103                       
  3                                       
            0.47712                       
  10                                      
            1.00000                       
  Geometric mean = 2.78                   
            Arithmetic mean = 0.44454.    
            10.44454 = 2.78               
                                          
                                          
                                          
 If any one of the scores is zero then    
 the geometric mean doesn't make any      
 sense and cheat by adding a constant to  
 every number.  The geometric mean for    
 the example I gave intially is 120,      
 swimmable.                               
                                          
                                          
                                          
                                          
                                          
 The geometric mean is always lower than  
 the arithmetic mean, so the criteria is  
 different.  EPA can explain how they     
 derived the 126 CFU/100 ml.  Hopes this  
 helps a bit.                             
                                          
                                          
                                          
                                          


  Richard Whitman
Chief, Lake Michigan Ecological Research Station
219-926-8336 Ext. 424

1100 North Mineral Springs Road
Porter, IN 46304



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