Demystifying Geometric Means
George Shook
University of Wisconsin
Dairy Science Department
Madison, Wisconsin
Summary
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The NMC submitted a proposal to the 2011 National Conference on Interstate Milk Shipments to
lower the somatic cell count (SCC) regulatory limit in the US to 400,000 cells/ml. In addition, the
NMC proposal recommends basing regulatory action on a three-month rolling geometric mean SCC.
Geometric means are always less than arithmetic means.
Geometric averaging diminishes the influence of high SCC values.
The greater the range among SCCs, the greater the difference between the geometric mean and
arithmetic mean.
Geometric means are more stable from month to month than arithmetic means.
It is easier for herds to stay in compliance with SCC regulations when geometric means are used.
Calculating geometric means is easily accomplished with computer software or scientific
calculators.
Why is the Geometric Mean an Issue?
The NMC has recommended using a rolling three-month geometric mean for regulating SCC standards in
the US. For many years, geometric means have been used for regulatory purposes in most other
countries that have developed dairy industries. Adopting the geometric mean in the US would
harmonize standards for international trade.
Because geometric means are so uncommon, this article explains the whys and hows of geometric
means for SCC.
Why Use a Geometric Mean Instead of an Arithmetic Mean?
Means (or averages) and medians (the value that separates the top 50% of data from the bottom 50%)
are used to characterize the center of data sets. SCC data are characterized by many low values and
relatively few high values. Because of this, geometric means for SCC have better performance
characteristics (statistical properties) than arithmetic means (or simple averages).
Arithmetic means for SCC are strongly influenced by those few observations at the high end.
Consequently, arithmetic means of SCC are often much higher than the median. The process of
geometric averaging diminishes the influence of those few high SCC values; and the geometric mean
SCC is typically close to the median SCC.
Using a rolling three-month mean evens out the month-to-month variation in bulk tank SCCs. A herd
may sometimes spike above the 400,000 SCC threshold and continue to remain in compliance with the
requirement.
Because high SCCs have less influence on the geometric mean, geometric means are more stable from
month to month than arithmetic means.
Examples
Table 1 lists examples from five herds showing three months of bulk tank SCCs and their corresponding
arithmetic and geometric means. These herds were chosen because they had wide month-to-month
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variations in SCC. All the herds in this example (Table 1) had 1 or 2 sample months with SCC greater
than 400,000. The table illustrates several characteristics of arithmetic and geometric means.
Table 1. Monthly bulk tank SCC, arithmetic mean, geometric mean and difference for five example
herds.
Sample month SCC
Arithmetic
Geometric
Herd
June
July
August
Mean
Mean
Difference
A
82
400
580
354
267
87
B
370
100
700
390
296
94
C
640
180
440
420
370
50
D
518
690
343
517
497
20
E
730
810
270
603
542
61
The most striking comparison between the two kinds of means in Table 1 is that the geometric mean is
less than the arithmetic mean. In fact, it is always true that the geometric mean is less than the
arithmetic mean.
Although none of the herds in Table 1 has consistent SCC values from month to month, herds C and D
have the smallest range. This illustrates a second characteristic: The smaller the range among SCCs,
the smaller the difference between the geometric mean and arithmetic mean. Consequently, a
herd with one (or in some circumstances two) bulk tank SCCs above 400,000 could have a geometric
mean below 400,000 and continue to remain in compliance. Herds A, B, and C have this characteristic.
Notice that herds A, B, and C in Table 1 all have one or two months with SCC greater than 400,000.
Both the arithmetic and geometric means for herds A and B are below the threshold of 400,000.
However, for herd C the arithmetic mean is greater than 400,000 while the geometric mean is below
400,000. It is easier for herds to stay in compliance with SCC regulations when geometric means
are used.
How to Calculate a Geometric Mean
Calculation of the geometric mean of three monthly SCC values is straightforward on a scientific
calculator. Suppose the monthly SCCs are 100, 200, and 400. First, multiply the three values: 100 x
200 x 400 = 8,000,000. The geometric mean of three numbers is the cube root [i. e., the one-third
power] of the product: 8,000,0001/3 = 200. To check the result, multiply 200 x 200 x 200 = 8,000,000
which is equal to the product of the three values in the geometric mean. By comparison, the arithmetic
mean of these numbers is (100 + 200 + 400)/3 = 233.
Reprinted from the National Mastitis Council newsletter, “Udder Topics”. Vol. 34, No. 1 & 2 (2011)
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