Geometric Mean - How is it useful?

[MathHelp]

So I read that the geometric mean is a good indicator of central tendency when data is in logarithmic form. But to me, it would seem that this is a contradiction. Doesn't logarithmic data inherently not have central tendency?

I should clarify that my understanding of "logarithmic" is that it means that changes are exponential.

So if I have a data set of 1, 10, and 100, how could a geometric mean give me any useful data about the central tendency of this data set? There does not seem to be a central tendency....

[studiot]

20 hours ago, MathHelp said:

So I read that the geometric mean is a good indicator of central tendency when data is in logarithmic form. But to me, it would seem that this is a contradiction. Doesn't logarithmic data inherently not have central tendency?

I should clarify that my understanding of "logarithmic" is that it means that changes are exponential.

So if I have a data set of 1, 10, and 100, how could a geometric mean give me any useful data about the central tendency of this data set? There does not seem to be a central tendency....

Taking an average is choosing or calculating one number to stand for a whole bunch of data.

One sort of average is a statistical mean. There are three common types of 'mean'  in use.

1)

The most common is the arithemtic mean and is used when we add things together.

It answers the question

If all the numbers we added together had the same value individually, what would that value be.

Suppose we had a rectangle with sides A and B, So the perimeter = 2A +2B.

The arithmetic mean answers the question what number C represents a square with the same perimeter 

[math]C = \frac{{A + B}}{2}[/math]

So the square with the same perimeter is the square with sides   [math]C = \frac{{A + B}}{2}[/math]

Where the primeter = 4C

 

2) The geometric mean is used when we have the result of numbers being multiplied together, instead of being added. It is eqaul to the nth root of n numbers being so multiplied.

The most usual place to find this one used to be in finance where you wanted to know an average rate of return on an iaccumulating investment, where the return rate might vary from year to year.

More recently such geomtric means have become important in population growth studies, where the growth equations are similar and involve multiplication of numbers.

The geometric example that follows from the rectangle square example works if you ask for the area of the square that equals the area of the given rectangle.

That is the rectangle has the same area as a square of sides = geometric mean of A and B which is  [math]\sqrt {AB} [/math]

3)

The harmonic mean is also used iin finance.

It is known that the 3 means appear in the order

 

Arithmetic Mean >=  Geometric Mean  >=  Harmonic Mean

[math]\frac{{A + B}}{2} \ge \sqrt {AB}  \ge \frac{{2AB}}{{A + B}}[/math]

With equality only occurring when A = B

 

you can check these with the square - rectangle example  putting A = 2 and B=4  and then A = 2 and B = 2

 

 

Edited November 3, 2022 by studiot

[MathHelp]

Thanks for your response.

The trouble I am having is understanding what the geometric mean can tell me that is useful. You gave a definition:

Quote

The geometric mean is used when we have the result of numbers being multiplied together, instead of being added. It is eqaul to the nth root of n numbers being so multiplied.

But why would that be useful to anyone?

You mentioned it would be useful in finance and population growth studies but I still can't see what meaningful information it would give to someone. 

In my head it is like working out a way to calculate the number of branches on the third tree to the right in a hedge of trees. I know how to do it, I can do it, and I know what the answer would tell me (the number of branches on the tree) but I am not sure what insight this would offer... any help on this?

Can you give me an example of a real-life problem where the geometric mean would offer someone meaningful insight into a data set and what that meaningful insight would be? 

[studiot]

3 minutes ago, MathHelp said:

Thanks for your response.

The trouble I am having is understanding what the geometric mean can tell me that is useful. You gave a definition:

But why would that be useful to anyone?

You mentioned it would be useful in finance and population growth studies but I still can't see what meaningful information it would give to someone. 

In my head it is like working out a way to calculate the number of branches on the third tree to the right in a hedge of trees. I know how to do it, I can do it, and I know what the answer would tell me (the number of branches on the tree) but I am not sure what insight this would offer... any help on this?

Can you give me an example of a real-life problem where the geometric mean would offer someone meaningful insight into a data set and what that meaningful insight would be? 

Don't give and certainly don't count trees.

Did you do the simple sums I suggested ?

How about this one ?

Say you bought £1,000  to invest for three years so you bought some shares in a company.

In the first year the value increased 30%.
In the second year the value increased another 10%
In the third year the value increased another 50%

Based on these figures what would you expect the value to be after another 2 years.

Well you would need a (geomtric) average increase for the first three years and project that average forwards two more years.

So in the first year the 1,000 increase by 1.3 to 1300
In the second year this increased to 1.1(1300)
In the third year this increased to (1.5)(1.1)(1300) or  (1.5)(1.1)(1.3)(1000)

So the geometric mean is the cube root of (1.5)(1.1)(1.3) and this represents the yearly expected increase based on those figures

It is the number that if you kept multiplying by would keep the same average increase in the future.

[John Cuthber]

A geometric mean is the best measure of central trend for a log normal distribution.
https://en.wikipedia.org/wiki/Log-normal_distribution#Occurrence_and_applications

[MathHelp]

Quote

Did you do the simple sums I suggested ?

No, sorry I did not see any suggestions to try... or did you mean the triangle/rectangle calculation?

The main issue I am having is recognising where it is useful. You have essentially answered this question by giving interest rate and population growth examples.

However, I am now curious about when it might be useful for comparing the area of a rectangle with a square. Is that something a Civil Engineer would need to know?

A final point of clarification, can someone explain why it is called the geometric mean when it does not seem to have anything in common with the arithmatic mean? As an example, the arithmatic mean is clearly a measure of central tendency - I can intuitively understand what it is trying to tell me about the central tendency of data. Based on the explanations given of the geometric mean, I can understand how it could be used... but it does not seem to have anything to do with central tendency. It can help me estimate the rate of population growth, but that is about estimating and is not related to central tendency.

Is the geometric mean relevant in medicine?