Other Means


The Geometric Mean
The Harmonic Mean
Arithmetic and Geometric mean
differences
The Arithmetic Mean
•
Is the sum of the observations divided by
the total number of observations
(a1+ ...+aN)/N
The Geometric Mean
* Is the nth root of the product of the
observations
* Can also be calculated by taking the
antilog of the arithmetic mean.
(a1· ... ·aN)1/N
~ Used when several quantities are added together
to produce a total.
~ Used when several quantities are multiplied
by a factor to give a product.
- this is the midpoint of the added numbers if those
numbers are stretched out on a line
- this is the average of the factors that
contribute to a product.
Always less than or equal to the
arithmetic mean (only equal to it
when the components of the set are
equal)
A few examples…

Population calculations – in calculations
involving populations, the population size
must be multiplied by the factor of
increase – thus we use the geometric
mean.
Using the book example of calculations of
a population of mule deer:
The arithmetic mean tells us that in a
population of 1000 deer increasing 10%
one year and 20% the next, the average
increase is 15%. However, this gives us
1322.5 deer when the actual population
increase is to 1320 deer.
If we use the geometric mean instead, the
calculations are as follows…
10% and 20% increase is the same as 1.10 and 1.20
Take the natural log of these to get:
ln(1.10) = 0.09531 and ln(1.20) = 0.18232
The arithmetic average of these two is 0.138815
(.09531 + 0.18232 / 2 = 0.138815)
Take the antilog of the arithmetic mean:
e 0.138815 = 1.14891
Multiply this by the population size each year to get a total end
population of 1319.99 – closer to the 1320 actual deer.
Other ways to calculate
geometric mean:

There are a couple of ways to get the
geometric mean…
-
One is to take the antilog of the arithmetic mean
as we just did
-
Another is to take the nth root of the product of
the observations (or the 1/nth power of the
product of the observations which we’ll try in the
next example)
Another example…

Rates of return on investments – When
calculating the amount of return on an
investment you would again use the
geometric mean to determine what
constant factor you would need to multiply
by – this should give you the average
interest rate.

If we had an investment that returned 10%
the first year, 60% the second, and 20% the
third what is the average rate of return? (not
30%!)
To calculate this, remember 10, 60, and 20 percents are the same as
multiplying the investment by 1.10, 1.60, and 1.20.
To get the geometric mean calculate:
(1.10 x 1.60 x 1.20)1/3 = 1.283 or an average return of 28% (not 30%!)
The Harmonic Mean
One way of discussing the harmonic mean (H),
is with reference to the arithmetic mean (A)
and the geometric mean (G)…
In this way we could say that
Or…

We could get the harmonic mean by:
Taking the number of terms (n) in a set and dividing it by
The sum of the terms’ reciprocals
So with set (a1,...,an )

The arithmetic, geometric, and harmonic
means are related in the following way:
the arithmetic mean > the geometric mean > the harmonic mean
Unless the terms of the set are equal in which case the harmonic,
arithmetic, and geometric means will all be the same.
In Summary:

When calculating with a set containing only
two terms the
arithmetic mean is:
the geometric mean is:
and the harmonic mean can be: