In your example, you are taking the mean of positive numbers. For example, if you're looking at an investment that increases by 10% one year and decreases by 20% the next, the simple rates of change are 10% and -20%, but that's not what you're taking the geometric mean of.

At the end of the first year you have 1.1 times what you started with (the original plus another tenth of it). At the end of the second year you have 0.8 times what you started the second year with (the original minus one fifth of it). So, the numbers you are taking the geometric mean of are 1.1 and 0.8. This mean is approximately 0.938.

This means that, on average, your investment is being multiplied by 0.938 (= 93.8%) each year, a 6.2% loss.

So, the compound anual growth rate is (approximately) -6.2%.

For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant quantity is the geometric mean of these three numbers.

The question about finding the average rate of return can be rephrased as: "by what constant factor would your investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and 1.20 the third?" The answer is the

geometric mean . If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you).

Any time you have a number of factors contributing to a product, and you want to find the "average" factor, the answer is the geometric mean. The example of interest rates is probably the application most used in everyday life.

For example, if you are looking for a mean amount of rainfall, younote that the total amount of rain, which affects crop growth, etc.,is found by ADDING the daily numbers; so if you add them up and divide by the number of days, the resulting ARITHMETIC mean is the amount of rain you could have had on EACH of those days, to get the same total.

If you have several successive price markups, say by 5% and then by6%, and want to know the mean markup, you note that the net effect is

to first MULTIPLY by 1.05 and then by 1.06, equivalent to a singlemarkup of 1.05*1.06 = 1.113; taking the square root of this, if youhad TWO markups of 5.499% each, you would get the same result. This is the GEOMETRIC mean. In general, you use it where the product is an appropriate "total"; another example is when you combine several enlargements of a picture.

If you want the mean speed of a car that goes the same distance (nottime!) at each of several speeds, then the net effect of all thedriving (the total time taken) is found by dividing the commondistance by each speed to get the time for that leg of the trip, andthen adding up those times. The constant speed that would take thesame total time for the whole trip is the HARMONIC mean of the speeds. This amounts to the reciprocal of the arithmetic mean of the RECIPROCALS of the individual speeds. In general, we use the harmonic mean when the numbers naturally combine via their reciprocals. Another example is combining resistances in a parallel electrical circuit.

The profit of Company A, SYZO Ltd., has grown over the last three years by 10 million, 12 million, and 14 million dollars. It is appropriate to say that it has grown by an average of 12 million dollars yearly, for which we use the arithmetic mean.

The profit of Company B, OZYS Ltd., has grown the over last three years by 2.5%, 3%, and 3.5%. Here we cannot use the arithmetic mean and say that the average growth was 3%. Why not?

Suppose that Company B, OZYS Ltd., started with a 100-million-dollar profit. Three years later it will have become:

$100,000,000 * 1.025 * 1.03 * 1.035 = $109,270,125

This is less than a yearly increase of 3% would yield, since:

$100,000,000 * 1.03 * 1.03 * 1.03 = $109,272,700

Here we see that we should use the geometric mean of the growth factors 1.025, 1.03, and 1.035 to find the average percentage. That is always less than the arithmetic mean would yield.

Different means are used depending on what we mean by "combining" the numbers. In Dr. Floor's example of a company growing annually by 10,12, and 14 million dollars, the "combined" growth is the sum of thethree numbers. We want the total growth (the sum of the annualgrowths) to equal the total if the company grew by N million dollarseach year. The mean we seek, therefore, is the number N such that

10 + 12 + 14 = N + N + N

The number that works is the arithmetic mean:

N = (10 + 12 + 14)/3

In Dr. Floor's second example, of a company growing annually by 2.5%,3%, and 3.5%, the growth rate over the three years is computeddifferently:

Profit in year 1 = 1.025 times profit in year 0

Profit in year 2 = 1.03 times profit in year 1 = 1.03 * 1.025 times profit in year 0

Profit in year 3 = 1.035 times profit in year 2 = 1.035 * 1.03 * 1.025 times profit in year 0

Thus the ratio of the profit in the third year to the profit in thebase year is

1.025 * 1.03 * 1.035

If the growth rate had been the same each year, the ratio would be

N * N * N

For the growth over 3 years to be the same in both cases, we must have

1.025 * 1.03 * 1.035 = N * N * N

The solution is

N = (1.025 * 1.03 * 1.035)^(1/3) = 1.029992

That is, the cube root of the product of the annual growth factors.This is the geometric mean.

Note: The mean growth rate comes out to

(1.029992 - 1) * 100% = 2.9992%

which is very close to the arithmetic mean (as seen in Dr. Floor'sanswer: compare $109,270,125 to $109,272,700). The difference would begreater if the growth rates were greater.

Applications

Statisticians use arithmetic means to represent data with no significant outliers. This type of mean is good for representing average temperatures, because all the temperatures for January 22 in Chicago will be between -50 and 50 degrees F. A temperature of 10,000 degrees F is just not going to happen. Things like batting averages and average race car speeds are also represented well using arithmetic means.

Geometric means are used in cases where the differences among data points are logarithmic or vary by multiples of 10. Biologists use geometric means to describe the sizes of bacterial populations, which can be 20 organisms one day and 20,000 the next. Economists can use geometric means to describe income distributions. You and most of your neighbors might make around $65,000 per year, but what if the guy up on the hill makes $65 million per year? The arithmetic mean of the income in your neighborhood would be misleading here, so a geometric mean would be more suitable.

Besides being used by scientists and biologists, geometric means are also used in many other fields, most notably financial reporting. This is because when evaluating investment returns as annual percent change data over several years (or fluctuating interest rates), it is the geometric mean, not the arithmetic mean, that tells you what the average financial rate of return would have had to have been over the entire investment period to achieve the end result. This term is also so called the Compound Annual Growth Rate or CAGR. Population biologists also use the same calculation to determine average growth rates of populations, and this growth rate is referred to as the Intrinsic Rate of Growth when the calculation is applied to estimates of population increases where there are no density-dependent forces regulating the population.

Suppose you have this beach monitoring data from different dates:   (data are Enterococci bacteria per 100 milliliters of sample)

   6 ent./100 ml   50 ent./100 ml   9 ent./100 ml   1200 ent./100 ml

Geometric Mean = 4th root of (6)(50)(9)(1200)

   = 4th root of 3,240,000

Geometric Mean = 42.4 ent./100 ml