Other Means. The Geometric Mean The Harmonic Mean. Arithmetic and Geometric mean differences. 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.
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.
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.
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%!)
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
Taking the number of terms (n) in a setand dividing it by
The sum of the terms’ reciprocals
So with set (a1,...,an )
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.
arithmetic mean is:
the geometric mean is:
and the harmonic mean can be: