Should I use the Arithmetic or Geometric Mean?

[CycloWizard]

This may seem like a really elementary question, but I wanted to be sure that I'm using the right method for reporting the 'average' of my measurements. Basically, I've measured the diameter of an object four times in four different meridians and I want to know the 'average' or 'equivalent' diameter of the object. It turns out that, for my particular case, the values are, for all intents and purposes, identical, but I want to be sure that I report the correct method.

I did a little digging on the various uses of these means, but it's not readily apparent (to me at least 😛) which is the right one to use.

This page probably gives the most thorough description. I tried MathWorld, but it doesn't really say when to use the different means, just how to compute them.

Based on this explanation from the above link:

Suppose that we have two quantities, A and B. Taking their arithmetic mean we get the number (A+B)/2 which can be interpreted in a number of ways. One interpretation (probably the most common) is that this quantity is the midpoint of the two numbers viewed as points on a line.

Now suppose that we have a rectangle with sides of lengths A and B. The arithmetic mean can also be interpreted as the length of the sides of a square whose perimeter is the same as our rectangle. Similarly, the geometric mean is the length of the sides of a square which has the same area as our rectangle.

This leads me to believe that the arithmetic mean is the correct choice, since I am interested in the right length of the diameter that would sum to the total of my four measurements. Am I right? One paper that performed a similar experiment used the geometric mean, which is why this issue came up.

 

[f95toli]

You should use the arithmetic mean. I can't really think of any reason why one would use the geometric mean in this case since you are basically trying to improve the "precision" of your measurment by averaging.

 

[CycloWizard]

Originally posted by: f95toli
You should use the arithmetic mean. I can't really think of any reason why one would use the geometric mean in this case since you are basically trying to improve the "precision" of your measurment by averaging.

Yeah, that's kind of how I saw it as well. I just got thrown for a loop when I read the other guy's paper and he used the geometric mean instead. Though I should have known, since the guy is a quack (based on his other publications). 😛 Thanks! :beer:

 

[bobsmith1492]

The only place I've heard of the geometric mean being used (assuming we're talking the same thing, sqrt(A*B)), was in Circuit Analysis I when designing a bandpass filter where the center frequency is the geometric mean of the corner frequencies...

What else can it be used for?

 

[jmcoreymv]

Geometric is also used in certain types of filters in digital image processing. It has slightly different effects over arithmetic (I forget the exact differences).

 

[CycloWizard]

Originally posted by: bobsmith1492
The only place I've heard of the geometric mean being used (assuming we're talking the same thing, sqrt(A*B)), was in Circuit Analysis I when designing a bandpass filter where the center frequency is the geometric mean of the corner frequencies...

What else can it be used for?

Computing average interest rates and other things that are averages of products. I'm not exactly sure why it's called the 'geometric' mean, as I can't really find any references to its utility in geometry other than the example above. It's probably just a lingering term from centuries ago.

 

[djhuber82]

I'm fairly confident that when refering to means, the terms 'aithmetic' and 'geometric' are used in the same sense as they are when refering to sequences.

 

[BrownTown]

geometric mean is used in all the CPU benchmarks and stuff. Basically If you have things on differnt scales (like test A is between 1-10, and test B is between 1-100) then the geometric means weigts everything equally. IF you used an arithmetic mean then some things would be weighted too much and others to little.

 

[thesurge]

Arithmetic mean=
The value to acheive the same sum.

Geometric mean=
The value to acheive the same product.

I remember reading a good explanation somewhere and uses, but I forgot... for example, one uses geometric mean for average percent change.

 

[Mday]

for your application, arithmetic mean is what you should be using. Also, for this case, there is no difference between your arithmetic mean and the geometric mean.