Other Means

The Geometric Mean The Harmonic Mean

Arithmetic and Geometric mean differences The Arithmetic MeanThe Geometric Mean Is the sum of the observations divided by the total number of observations (a a N )/N ~ Used when several quantities are added together to produce a total. - this is the midpoint of the added numbers if those numbers are stretched out on a line * Is the nth root of the product of the observations * Can also be calculated by taking the antilog of the arithmetic mean. (a 1 ·... ·a N ) 1/N ~ Used when several quantities are multiplied by a factor to give a product. - 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 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) = and ln(1.20) = The arithmetic average of these two is ( / 2 = ) Take the antilog of the arithmetic mean: e = Multiply this by the population size each year to get a total end population of – 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 To get the geometric mean calculate: (1.10 x 1.60 x 1.20) 1/3 = 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: