Non Accounting Home Page

Arithmetic Mean page

Median and Mode page

Means, Medians and Modes of Grouped Data

Introduction

The arithmetic mean is relevant when several quantities add together to produce a total. The arithmetic mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same total?"

In the same way, the geometric mean is relevant when several quantities multiply together to produce a product. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?" A useful page from Canada!

So, we use the geometric mean (GM) whenever we are looking to evaluate the mean of a rate of growth, ratios, indices, percentages.

which is the same as

GM and all values must be positive in order to use the GM. (See below GM and Negative Values for a hint here)

Let's apply the GM. Imagine that you were trying to evaluate whether to sell your house and you were only prepared to do that if it had already earned you a rate of return that you liked! You have found out that over the last three years, house prices in your area have increased as follows:

Year	Increase %
1999	10%
2000	20%
2001	25%

The arithmetic mean rate of increase would be 18.3333% but because we are dealing growth and percentages, we can't use the arithmetic mean.

, so the average percentage return over the three years is (1.181666-1)*100 = 18.1666%

What we have found is the answer to the question "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, 1.20 the next, and 1.25 the third?" Let's prove that by imagining that my house was worth 25,000 at the beginning of 1999:

Using the actual rates of increase we were given:

Start of year	Rate of increase	End of year
10000	1.1	11000
11000	1.2	13200
13200	1.25	16500

Make sure you can follow the logic of that table: column 1 multiplied by column 2 and enter the result in column 3 … now carry that result down into column 1 of the next year …

Now applying the average, geometric mean, rate:

Start of year	Rate of increase	End of year
10000	1.181666	11816.66
11816.66	1.181666	13963.34
13963.34	1.181666	16500

This table uses the same logic as the previous one.

See, the final values at the end of 2001 are equal using both methods. As a matter of interest, let's see what the arithmetic mean rate of increase would give us:

Start of year	Rate of increase	End of year
10000	1.183333	11833.33
11833.33	1.183333	14002.78
14002.78	1.183333	16569.95

See ... WRONG!

Try this yourself, now. MyCo has recently carried out four different projects and the rates of return are for projects 1 to 4 respectively 3%, 4%, 2% and 6%. Calculate the geometric mean of these returns.

Do you agree with this?:

%. Note that all we needed to do here was to treat the percentages as numbers: 3, 4, 2 and 6 and that keeps the work simple.

Geometric Mean Extension

You might already have thought of this, but the geometric mean can be used to work backwards! For example, take the house we thought about selling earlier: it has a value of 16,500 at the end of 2001 and a value of 10,000 at the beginning or 1999. If we assume that we know nothing about how the value of this house has changed over the last three years, we can still work out the GM return as follows:

GM and Negative Values

We said at the beginning that GM cannot cope with negative values and that is true: we cannot find the roots of negative numbers. However, if we take the latter approach and take tend and beginning values and determine the rate of change for a period overall, it may help us to overcome the negative value problem.

I don't believe that Microsoft Excel has its own geometric mean function. (See below for update) However, since it is basically a mathematical operation, it is not difficult to set up a spreadsheet that will cope with it. For example, the evaluate the example we have just done, we would enter our formula as follows:

=(16500/10000)^(1/3) - 1 = 0.18166575 … 18.166575%

which means 1 taken away from (evaluate 16,500 divided by 10,000 and then find the answer to that raised to the power of one third).

Alternatively, using logarithms in Excel you could use this approach: =10^(LOG(16500/10000)/3) - 1 = 0.18166575 … 18.166575%

This means, working from the inside out, find the logarithm of 16500/10000 and divide it by 3 then find the anti logarithm of the answer to that by raising 10 to the power of that answer … and then take 1 away from it.! For information, if this were a four year problem, the 3 would be replaced by 4 and by 7 if it were a 7 year problem.

For you to do: devise a function for Excel to find the GM of the increase in the value of a piece of jewellery whose current value is 1,500 but that was worth only 300 eight years ago.

Did you get either or both of the following?:

=(1500/300)^(1/8) - 1= 0.222845 … 22.2845%
=10^(LOG(1500/300)/8) - 1 = 0.222845 … 22.2845%

Marvellous isn't it? There's more! How about =EXP(LN(1500/300)/8)-1 … find out what that is by using Excel's Help files to define EXP and LN.

Update

Michael Bell has kindly written to me to tell me that since I wrote this page, either Excel XP has added the function =GEOMEAN(value1,value2 ...) or I missed it in 2002 when I wrote this page.

This means that you can now use the Excel function =GEOMEAN(value1,value2 ...) directly to find the geometric mean values.

Taking the example from Excel's Help file, here is a table of data that includes the geometric mean calculated from them. For comparison I have included the arithmetic mean, the median and the mode using the relevant in built Excel functions for them.

1	Data
2	4
3	5
4	8
5	7
6	11
7	4
8	3
geomean	5.48
arithmean	6.00
median	5.00
mode	4.00

Now you have a choice:

Return to the Arithmetic Mean page

Rework the Median and Mode page

Shoot on to the Means, Medians and Modes of Grouped Data page

Duncan Williamson
24 February 2002