Arithmetic–geometric mean Puns

Hello r/learnmath, I was thinking about this: If the arithmetic mean is the number which the dataset accumulates around (i.e. is the least squared sum of errors), then what is the geometric mean?

I've read that it's the point of least distance from any of the numbers within a dataset.

Imagine a number increasing by 50%, 60%, and 70%. So, the geometric mean is the number the "increases" accumulate around. However, how can one think of this intuitively? I can't imagine "increases" accumulating, but imagining numbers in a dataset accumulating around a certain number is easy.

agm( a, b ) = c.

a, b and c are three different elements of the natural numbers. we are excluding trivial cases like agm( 1, 1 ) = 1. can the result c ever be a natural number? I have tried non-systematically and have not found a case for a < b < 10^4. A more systematic method of inquiry is needed.

Recently there has been a discussion about whether to use arithmetic mean or geometric mean to calculate the averages when comparing cpu/gpu frame averages against each other. I think it may be good to put the numbers out in the open so everyone can see the impact of using either:

Using this video showing 16 game average data by Harbor Hardware Unboxed, I have drawn up this table.

The differences are... minor. 1.7% is the highest difference in this data set between using geo or arith mean. Not a huge difference...

NOW, the interesting part is I think there might be cases where the differences are bigger and data could be misinterpreted:

Let's say in Game 7 the 10900k only scores 300 frames because Intel, using the arithmetic mean now shows an almost 11 frame difference compared to the 5600x but the geo mean shows 3.3 frame difference (3% difference compared to 0.3%)

So ye... just putting it out there so everyone has a clearer idea what the numbers look like. Please let me know if you see anything weird or this does not belong here, I lack caffeine to operate at 100%.

Cheers mates.

Edit: I am a big fan of using geo means, but I understand why the industry standard is to use the 'simple' arithmetic mean of adding everything up and dividing by sample size; it is the method everyone is most familiar with. Imagine trying to explain the geometric mean to all your followers and receiving comments in every video such as 'YOU DOIN IT WRONG!!'. Also in case someone states that i am trying to defend HU; I am no diehard fan of HU, i watch their videos from time to time and you can search my reddit history to show that i frequently criticise their views and opinions.

TL:DR

• The difference is generally very minor

• 'Simple' arithmetic mean is easy to undertand for all people hence why it is commonly used

• If you care so much about geomean than do your own calculations like I did

• There can be cases where data can be skewed/misinterpreted

• Everyone stay safe and take care

Show that the polynomial p(x, y, z) = x^(4)y^(2) + x^(2)y^(4) + z^(6) - 3x^(2)y^(2)z^(2) is non-negative using the arithmetic-geometric mean inequality. (if you have any ideas without using the arithmetic geometric mean inequality im all ears as well). Thanks!

The geometric mean is always less than or equal to the arithmetic mean. The only time the two means will be equal is when there is no variability in the observations.

I get this statement in the curiculum.. but why is this the case?

I've been looking into the inequality of arithmetic and geometric means, trying to prove it in general for N terms.

Proof for 2 terms

Proving it to be true for 2 terms is easy, a little harder with 3 terms, and I quickly run out of patience and paper and sanity for 4 terms. Is there a way to prove it in general for N terms?

General form for N terms

Two numbers differs by 40 and their arithmetic mean exceeds their geometric mean by 2.

What is the smaller number? the choices are 45,81,64 and 100, I chose the 100 because its the nearest but Im not sure about my answer :(

Hi! I'm learning the differences of all the means and this question just came to my mind as I learned that geometric mean is a good alternative to compares things from different scales. Most of the tutorials I've read said Geometric mean usually gives the same trend as the normalized arithmetic mean. So are there ever any exceptions where you can't use GM as a substitution for normalized AM? Thanks!

https://preview.redd.it/7pnde78s99d51.jpg?width=362&format=pjpg&auto=webp&s=a3e21fbccb5d032888aa34e9f02c948d548d652b

http://edspi31415.blogspot.com/2020/07/hp-41c-hp-42s-ti-60-arithmetic.html

Arithmetic-Geometric Mean

The program AGM calculates the arithmetic-geometric mean of two positive integers x and y. As the graphic above suggests, an iterative process is used to find the AGM, computing both the arithmetic mean and geometric mean until the two means converge.

a0 = x

g0 = y

Repeat:

Arithmetic Mean: a1 = (a0 + g0)/2

Geometric Mean: g1 = √(a0 * g0)

Transfer new to old: a0 = a1, g0 = g1

Until |a1 - g1| < tolerance

You can set the tolerance as low as you want. The programs presented on this blog set tolerance at 10^(-10) (1E-10), to fit the calculator's display.

Click on the link above to get the program listings.

The question actually asks to 'explain' but I can't think of an intuitive explanation.

So I try and simplify sqrt(xy) <= (x+y)/2

(Note x,y bigger 0)

I tried rearranging but didn't get anywhere.

Then I tried using y=cx (c>0):

Sqrt(xcx) <= (x+cx)/2

x sqrt(c) <= x (1+c)/2

sqrt(c) < (1+c)/2

That looks better! But I'm not sure where to go from there, rearranging I get to a quadratic:

0 <= c^2 - 2c + 1

solution c=1, which makes sense, but I don't see how that helps showing the inequality holds? I feel like I'm missing some final step to conclude the proof.

Hi! I need help with a proof. Sorry about the poor format. I tried to post a photo of the question, but couldn't figure out how.

If A1 and b1 are given positive numbers, with a1<b1, and two sequences of positive numbers are defined by:

A n+1=Sqrt(AnBn) [<- Geometric Mean]

and

B n+1=1/2(An+Bn) [<-Arithmetic mean]

Prove that 0<An<A n+1<Bn+1<Bn

The answer says "needs Geometric mean < Arithmetic Mean, and induction." The induction is where I get lost. I understand the concept of mathematical induction, but I don't see how to apply it here. All I can get is:

A n+1= G

B n+1=A

G<A

and the 2 sequences are positive, so

0<An+1<Bn+1

How do I get the An and Bn in there?

So, I have the recurring formula...

Xn = 1/2 (Xn-1 + (a/Xn-1)) with a being any positive real number.

This finds the square root of a. using the arithmetic mean - geometric mean inequality...

((x+y)/2 > or equal to sqrt(xy))

i must show that

Xn >(or equal to) sqrt(a).

this question isn't worth very much so I feel like I am missing something fairly obvious but I cannot see any answer through manipulation.

formatting: no subscripts so I compromised with Xn and Xn-1. Hope all is clear

edit: sign error

Given a set of positive numbers A, when is the geometric mean of the harmonic mean of A and the arithmetic mean of A equal to the geometric mean of A? This doesn't always happen, as one can see with the example A={1,2,3}. But one interesting case where it does happen is when the set A is the set of positive divisors of an integer. In this case, the harmonic mean is exactly (n tau(n))/sigma(n), the geometric mean is exactly n^{1/2} and the arithmetic mean is exactly sigma(n)/tau(n). Here tau(n) is the number of positive divisors of n and sigma(n) is the sum of the positive divisors. One can construct other examples from these by multiplying everything in the set A by a scaling factor. Do all such examples arise from scaling a set of a divisors this way? If not, is there a nice characterization of these sets?

The limit as x approaches infinity of 2*x*AGM(1, 2^(-x))*Ln(2) = Pi , and also that the limit as x goes to infinity of 2*Ln(x)*AGM(1, 4/x) = Pi.

It was the clean connection I was looking for between AGM's and Pi, and is a good introduction to lemniscates.

I've found students understand logarithms easier than lemniscates... thought I'd share.

For the plots on the bottom of the page:

http://www.procalcitonin.com/clinical-utilities/sepsis/reference-values.html

I read on wikipedia the rationale for when you'd want to use geometric mean but I'm not seeing that apply here. So why was it used instead of the arithmetic mean?

Hello,

I'm using betas against the S&P 500 and Nasdaq 100 to calculate my exposure to the two indices. I'm trying to decide whether to use geometric and arithmetic mean. Searching with Google give me mixed answers, but the number of pages point to geometric mean. Which do you recommend I use?

For any who don't happen to know, the geometric mean of two numbers is the square root of their product, as opposed to the usual--or "arithmetic"--mean which is, of course, half their sum.

A few years ago when I was in theory-based calculus, we were introduced to the geometric mean and I was very curious about what would happen if you started with two numbers a and b, took their arithmetic and geometric means to get two new numbers, and then took the arithmetic and geometric means of the two new numbers, and so on to the limit of infinitely many reiterations. Unfortunately, I gave up trying to figure it out after a few months of no success.

I vaguely remember finding out that this concept is called the arithmetic-geometric mean, and that Gauss found a result involving "elliptic integrals." (not sure what those are, just path integral over ellipse or something?)

Does anyone know a way to evaluate the limiting behavior? Is there a closed-form solution for attaining the limit from a and b? Also, is anyone aware of any applications of the arithmetic-geometric mean?

EDIT:

Thanks very much for the sources and matlab experiment guys! So that clears up that half of the question. Does anyone know what this mean is used for though? Or any other interesting facts about it?

How do I prove that the arithmetic mean is greater than the geometric mean, and equal, if and only if x1=x2=...=x_n ? We have already proven Bernoulli's inequality, the binomial theorem and: for all n it holds that for n real numbers whose product is equal to 1, their sum is greater than n, in case this is relevant.

http://i.imgur.com/kyyPxYq.jpg

I've just started working through a book on some analysis problems, but I'm having trouble with the first proof - specifically why the author can replace a_k with a_knroot(a1a2...an) . It doesn't seem to make any sense to me. Any help would be greatly appreciated.

Thankyou

I never formally studied statistics, so sorry if I'm being dense.

Say you have an optimization that improves a computing system's performance (time or memory usage). Let x1 and x2 be the old and new systems' performances respectively, so that the improvement factor is x2/x1 (e.g. x2/x1 = .5 if it's twice as fast).

You want to determine this factor empirically. You take multiple measurements of x1, and average these datapoints together; and likewise for x2. Should this averaging use an arithmetic mean or a geometric mean?

I have two ways of thinking about this that lead me to opposite conclusions. One is that you are trying to determine an unknown but fixed improvement factor, and you are gathering datapoints with random error, so the arithmetic mean should be used to get as close to the true x1 and x2 as possible.

The second is that the improvement factor varies over different test runs, but has an average value. Therefore we need the geometric mean of the x2 datapoints to get a meaningful average of these different factors.

Is there a decisively "best" way, or does it come down to a choice? Thanks.

eom

I was just analysing some (really nice) data and a colleague told me to use pLog IC50 data to average several compound curves rather than molar. I've managed to deduce that one is an arithmetic mean and the other is geometric but without calculus knowledge I’m struggling to understand why the two give different values.

If anyone could take a minute to explain this difference I would really appreciate it.

Please can someone help me with these equations, I'm sure they're easy but I can't get my head around them. What formulas do I use and how do I use them?

https://preview.redd.it/eti7gadbmlz71.png?width=407&format=png&auto=webp&s=a69ca729c3fe3a521bcb97c3f125299380ee5625

https://preview.redd.it/i3w7zgdbmlz71.png?width=374&format=png&auto=webp&s=dd343d3227ee4d7eac9f23a11a425b7d50fb8238

Hi /r/learnmath,

I'm trying to understand how Maxwell's 2nd formula for the mutual inductance of filaments can be transformed from the form seen in equation 11 to the form seen in equation 12.

I somewhat understand that the expression K(k) - E(k) = K(k)*C_s, and that K(k) can be approximated by AGM to be pi/2a_n. However, I don't fully get how K(k) is transformed to its final form, nor how these equations are manipulated.

The main goal of this is to use these concepts to calculate the inductance of multi-layered solenoids.

Link