118 PART 3 Getting Down and Dirty with Data
approach. The difference is that each type of mean adds a slightly different twist
to the basic mathematical process.
INNER MEAN
The inner mean (also called the trimmed mean) of N numbers is calculated by
removing the lowest value (the minimum) and the highest value (the maximum),
and calculating the arithmetic mean of the remaining N – 2 inner values. For the
sample of seven values of DBP from study participants from the example used
earlier in this chapter (which were 84, 84, 89, 91, 110, 114, and 116 mmHg), you
would drop the minimum and the maximum to compute the inner
mean: 84
89
91
110
114
5
488 5
97 6
/
/
. .
An inner mean that is even more inner can be calculated by making an even
stricter rule. The rule could be to drop the two (or more) of the highest and two (or
more) of the lowest values from the data, and then calculate the arithmetic mean
of the remaining values. In the interest of fairness, you should always chop the
same number of values from the low end as from the high end. Like the median
(discussed earlier in this chapter), the inner mean is more resistant to extreme
values called outliers than the arithmetic mean.
GEOMETRIC MEAN
The geometric mean (often abbreviated GM) can be defined by two different-
looking formulas that produce exactly the same value. The basic definition has
this formula:
Geometric Mean
II
GM
X
N
We describe the product symbol Π (the Greek capital pi) in Chapter 2. This formula
is telling you to multiply the values of the N observations together, and then take
the Nth root of the product. Using the numbers from the earlier example (where
you had DBP data on seven participants, with the values 84, 84, 89, 91, 110, 114,
and 116 mmHg), the equation looks like this:
GM
84
84
89
91 110 114 116
83 127 648 746 160
93 4
7
7
,
,
,
,
,
.
Even with technology, this formula is computationally challenging. By using log-
arithms (which turn multiplications into additions and roots into divisions), you
can develop a numerically stable alternative formula, which is:
log(
)
log(
) ,or
antilog
log(
)
GM
X
N
GM
X
N
This formula may look complicated, but it really just says, “The geometric mean
is the antilog of the mean of the logs of the values in the sample.” In other words,