Can the Arithmetic Mean of Complex Numbers Be Calculated?

[lee.lenton]

Can the arithmetic mean of a data set of complex numbers be calculated?
if so, can the method be demonstrated?

 

[Simon Bridge]

To figure that out, start from the definitions of your terms ... write them down.

 

[robphy]

Simon gave a great response.

For a visualization, think "centroid" or "center of mass" of n identical point-particles in a plane.

 

[mathman]

The most direct way would be to get the arithmetic means of the real and imaginary parts separately to get the resultant complex mean.

 

[lee.lenton]

Thank you for your assistance in trying to understand this slippery concept.

The difficulty I have in grasping the idea of an arithmetic mean of a data set of complex numbers is that complex numbers form an unordered number field.
That is; there is no greater than or less than order of magnitude of any group of complex numbers.
Therefore how can they be arranged in any order of magnitude to be able to find the arithmetic mean.

 

[robphy]

The median would require an ordering... but not the mean.
A sort is not needed to compute the mean.

If you regard the complex number as a vector in the plane... then your mean is essentially the centroid I mentioned earlier.

 

[Simon Bridge]

@lee.lenton:

The median would require an ordering... but not the mean.

Exactly - that's why I wanted to see you write down the definitions, which you haven't done yet.

Obtaining a median requires making a decision about how to order the numbers.
Consider, $\mathbb{R}^n$ are not intrinsically ordered either - but there are ways of making sense of the concepts of the mean and median over sets of them.

But you must start from defining your terms.
If you do not define your terms, you will get confused.

 

[lee.lenton]

I appreciate your comments on the need for definition.

I agree totally, as my impresion is that when this issue is discussed the terms arithmetic mean and geometric mean are interchanged.

I agree the centroid will give a measure of the dispersion around a central point in the plane comparable to the geometric mean. Therefore substitutiing a geometric concept for an algebraic concept doesn't solve the issue

However I can't quite understand why arithmetic mean and I suspect the harmonic mean of a data set of complex numbers cannot be calculated.

 

[Simon Bridge]

You still have not written out the definition that you want to use - is there some reason you are reluctant to do this? I mean it - actually physically type out how you would find the arithmetic mean of a bunch of numbers - write out what it means for those numbers to be complex, then apply the definition to the numbers. Do it right now.

i.e. you have N numbers $\{z_1, z_2, z_3, \cdots , z_{N-1}, z_N\}$
Then the arithmetic mean of them is given by ...

 

[lee.lenton]

The definition is exactly as for positive real numbers i.e. the arithmetic mean lies within the minimum and maximum range of the data set and calculated as per real positive numbers .

Therefore for complex numbers this would be: min z to max z of the data set and the arithmetic mean would be calculated by summing the complex numbers in the data set and dividing by the sample number and the resultant mean value should lie within the data set's minimum and maximum values.

What I am not grasping is this: can this method be performed on a data set of complex numbers?

 

[robphy]

This is the definition...

(snip)
the arithmetic mean would be calculated by summing the complex numbers in the data set and dividing by the sample number
(snip)

Everything else may (or may not) be true... but those would be results (to be proven) of the definition above and the properties of elements being averaged.

 

[mathman]

The definition is exactly as for positive real numbers i.e. the arithmetic mean lies within the minimum and maximum range of the data set and calculated as per real positive numbers .

Therefore for complex numbers this would be: min z to max z of the data set and the arithmetic mean would be calculated by summing the complex numbers in the data set and dividing by the sample number and the resultant mean value should lie within the data set's minimum and maximum values.

What I am not grasping is this: can this method be performed on a data set of complex numbers?

There is no min z or max z. However, the calculation of the mean does not require either. Your description of the calculation (summing and dividing by the number of entires) gives the mean.

In passing: the geometric mean a completely different animal (multiply all the numbers and take the nth root).

 

[Simon Bridge]

...for complex numbers this would be: min z to max z of the data set and the arithmetic mean would be calculated by summing the complex numbers in the data set and dividing by the sample number and the resultant mean value should lie within the data set's minimum and maximum values.

That's how you do the calculation all right ... but you need to write that down in math terms...

$$\bar{z} = \frac{1}{N}\sum_{i=1}^N z_i$$

Notice how the bit where the resultant mean lies between maximum and minimum values does not appear in the math-form of the definition? That not part of the definition: it's a consequence.

What I am not grasping is this: can this method be performed on a data set of complex numbers?

You are having trouble grasping it because you steadfastly refuse to follow the suggestions. You have to actually do the math. It is OK to start the math without knowing where you will end up - just do it.

You have the definition of the arithmetic mean in math symbols, now define a complex number in math terms ... then apply the definition above to the definition of the complex number.

Perhaps you are happier with concrete numbers ... say you have a set of 3 real numbers {1,2,3} ... what is the mean of those numbers (write it out one step at a time).

Now you have a set of three complex numbers {1+2i, 2+3i, 3+4i} ... now do the exact same calculation on those and see what happens.

Also see:
https://en.wikipedia.org/wiki/Arithmetic_mean

 

[lee.lenton]

I totally agree with the definition of arithmetic mean for positive real numbers, however it would not be a mean of the data set if it fell outside the minimum and maximum range of the data.

However the requirement that the arithmetic mean fall within the minimum and maximum range of the data set is what reveals the flaw with applying the concept of arithmetic mean to complex numbers.
In the example you give; the mean of the sample is 2 +3i, and it follows the expectation that it should lie within the range set by the minimum and maximum values of the data set.

However this appears to be the result of the fact that the data set chosen for your example, is approx. Collinear in the plane.

When a similar calculation is performed on non collinear widely dispersed data points the calculated mean falls outside the range of the data set.

For example the arithmetic mean of: 1 + 3i, 3 + i is 2 + 2i which in fact lies outside of the minimum value of the data set.

It appears that an arithmetic mean of a data set of complex numbers is not a valid mathematical concept because of this inconsistency. That is if your data set is collinear in the plane you can calculate a valid meaningful arithmetic mean, however if your data is widely dispersed over the plane the arithmetic mean may fall outside of the minimum value range of your data completely invalidating the calculation.

 

[Simon Bridge]

The reason you keep hitting a brck wall is because you keep leaving information out.
i.e. how did you determine whether one complex number was bigger or smaller than another?

How do you have to modify the formula for the arithmetic mean to stay true to the concepts and still get a mean for complex numbers?

You are correct that you have to figure out what a "mean" actually is when complex numbers are involved.
I cannot tell if you don't understand the terms you are using in your question or don't understand how to investigate a property mathematically or what is going on for you. The concept of a mean for complex numbers is well defined. You'll just have to read around a bit to see if it fits your ideas for what an arithmetic mean is but whatever you decide, you'll come away with a better understanding of your question.

Have a look at:
http://faculty.sfasu.edu/robersonpamel/txcmj/vol1/MeansOfComplexNumbers.PDF
http://planetmath.org/complexarithmeticgeometricmean

Good luck.

 

[lee.lenton]

Thanks for the links, I have already read the articles in depth; what I trying to get a simple answer to a question that remains elusive.
My original question was can anyone demonstrate how to calculate the arithmetic mean of a data set of complex numbers given that the complex numbers form an unordered field.
If you follow the early responses and search widely on the net the responses usually talk about component addition then divide by the sample number or talk about centroids.
When clearly the answer is an unequivocal no: the arithmetic mean of a data set of complex numbers cannot be calculated, as per the article you quoted.
However the example you gave as quoted in your post, shows that an arithmetic mean as define for real positive numbers can be calculated if the data are collinear in the plane.
If the data are widely dispersed and non collinear the arithmetic mean cannot be calculated as per the article you quoted in your post.
In other words there appears to be an implied structure in the complex plane that hasn't been taken into account in assessing the mathematical validity of calculating the arithmetical mean of complex numbers.

I appreciate your help and input into achieving clarity into what I believe is an important topic.

 

[D H]

I totally agree with the definition of arithmetic mean for positive real numbers, however it would not be a mean of the data set if it fell outside the minimum and maximum range of the data.

However the requirement that the arithmetic mean fall within the minimum and maximum range of the data set is what reveals the flaw with applying the concept of arithmetic mean to complex numbers.

Nonsense. This is a straw man argument.

The complex numbers cannot be ordered in a way that is consistent with the mathematical operations on the complex numbers (+ and *). The same applies to any vector space ℝ^N with any N>1. For example, defining z₁<z₂ on the complex numbers as meaning that |z₁|<|z₂| (i.e., comparison is by magnitude) is incompatible with addition. A lexicographic comparison is incompatible with multiplication. There is no comparison scheme on the complex numbers that is compatible with complex arithmetic.

The solution is to generalize the concept of "betweenness". That generalization is the convex hull. The arithmetic mean of a set of complex numbers always lies on or inside the convex hull of that set. This generalization also works on the vector spaces ℝ^N, and even on ℂ^N.

 

[lee.lenton]

It appears that the answers to the question: can the arithmetic mean of a data set of complex numbers be calculated if the set of complex numbers is an unordered number field? , is dependent on the underlying inherent structure of the complex plane. That is the complex plane has a preferred origin and preferred direction, i.ie. every direction away from the origin is in a direction of increasing magnitude of the modulus of the complex number.

This structure means that a mathematically valid arithmetic mean as per the definition of complex arithmetic mean can be calculated i.e. the mean lies between the min and max modulii value of the min and max data points. This only possible if the data set is precisely collinear and/ or precisely in phase.

If the data are not collinear and not in phase and widely dispersed over the complex plane than a mathematically valid arithmetic mean cannot be calculated.as per the link.

faculty.sfasu.edu/robersonpamel/txcmj/.../MeansOfComplexNumbers.PD...


To Demonstrate:

data set :

1+0i 0.924+0.383i 0.707+0.707i 0.383+0.924i 0+i

2+0i 1.848+0.766i 1.414+1.414i 0.766+1.848i 0+2i

3+0i 2.772+1.149i 2.121+2.121i 1.149+2.772i 0+3i

4+0i 3.696+1.532i 2.828+2.828i 1.532+3.696i 0+4i

5+0i 4.62+1.915i 3.535+3.535i 1.915+4.62i 0+5i

sum 15+0i 13.86 + 5.745i 10.605+10.605i 5.745+1386i 0+15i

divide
by 3+0i 2.772 +1.149 2.121+2.121 1.149+2.772 0+3i
sample
size

square 9+0 7.684+1.320 4.5+4.5 1.320+7.684 0+9
root = 3 = 3 = 3 = 3 =3
modulus

=MEAN 3 3 3 3 3

Therefore; if the data are collinear /in phase then a valid arithmetic mean can be calculated and its value lies between the min and max values of the data sample range.


However if the the data sample is collected from widely dispersed points in the complex plane; that is the data are not collinear or in phase then a mathematically valid arithmetic mean cannot be calculated .


To Demonstrate:


Data sample collected over the quadrant from 0 degrees to 90 degrees i.e widely dispersed/not collinear/not in phase:


1+0i
0.924+0.383i
0.707+0.707i
0.383+0.924i
0+1i

sum 3.014+3.014i

divide by sample size

= 0.6026+0.6028i

square root of sum of the squares

=0.85

which does not equal the moduli of the sample data which is equal to 1

It therefore falls below/outside of the data minimum value, but within the Hull Convexity of the data sample, but could never be called a arithmetic mean of sample.

In summary: it would appear that a mathematically valid arithmetic mean as defined for the positive real numbers, can be calculated for the complex numbers if the sample is collinear/directionally in phase, but not if the data are widely dispersed in an angular sense, not collinear/not phase.
This would be as a consequence of an intrinsic structure built into the complex plane, i.e. there is a fixed given origin and in every direction around the origin there is an inherent increasing magnitude of moduli from zero to positive infinity.

 

[robphy]

There seems to be a distinction between "[arithmetic] average" and "[arithmetic] mean".

Tracing back the very few references provided in that paper (from the Texas College Mathematics Journal) leads to "[arithmetic] mean" as treated in
Inequalities by Hardy, Littlewood, and Polya
https://www.amazon.com/dp/0521358809/?tag=pfamazon01-20

Thus, "[arithmetic] mean" in this case is associated with an ordering [which is an additional structure] beyond what is in an affine space (e.g. http://www.cut-the-knot.org/triangle/medians.shtml ).

It is the affine space structure [a vector space that forgot its origin] alone which allows the quantity
$\frac{ a_1+ a_2 +...a_n}{n}$
to be written down, which is taken to be the definition of an arithmetic average. (The set ${a_i}$ is just a set of elements, with no ordering implied.) So, this quantity can be calculated unambiguously for positive numbers, for real numbers, for complex numbers, for positions on a plane [giving the centroid], for m-by-n matrices, etc...

So, the OP's use of "[arithmetic] mean" is a very special case of the more general and more familiar "[arithmetic] average". (My characterization of "very special" is based on the above paper and its references (and papers similar to it) being relatively specialized [i.e. obscure, not typical].)

Simon's first response offers the best approach: first, start with clear definitions[/color]... [as opposed to results and desirable properties, which should be obtained from the hopefully clear definitions].

 

[D H]

In summary: it would appear that a mathematically valid arithmetic mean as defined for the positive real numbers, can be calculated for the complex numbers if the sample is collinear/directionally in phase, but not if the data are widely dispersed in an angular sense, not collinear/not phase.

This is utter nonsense, and with this, this thread is closed.