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Bill Shillito @solidangles.bsky.social · 6h

Cool thing I just realized about f-means!

If deg(f) < deg(g), then the f-mean of a data set is less than the g-mean.

So since f(x)=log x has "degree" 0 and g(x)=x has degree 1, the geometric mean is less than the arithmetic mean.

(I always forget which way it goes — now I'll always remember!)

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Bill Shillito @solidangles.bsky.social · 21h

I tend to remember it as:

∫ u dv = uv - ∫ v du

So for me, it's not "what's u and what's v" but "what's u and what's dv?"

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Bill Shillito @solidangles.bsky.social · 1d

I was worried that my musical example would fall flat. 🙃

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Bill Shillito @solidangles.bsky.social · 1d

UGH, more mistakes that of course I only catch after posting ... here's the corrected graphic.

One of these days I'll manage to post something completely right the first time. 😅

A diagram on a dark green background showing how the quadratic mean relates to variance and standard deviation.

Top left: {−6, 1, 5}, with a rightward arrow labeled “QM” leading to ≈ 4.546 (standard deviation*).

A downward arrow labeled “( )²” points to {36, 1, 25}.

From {36, 1, 25}, a rightward arrow labeled “AM” points to 20⅔ (variance*).

An upward arrow labeled “√( )” connects 20⅔ back to 4.546.
At the bottom, the label reads “Quadratic Mean.”

A footnote adds: “*Technically this is for a population variance and standard deviation. If this were for a sample, we’d actually divide by n - 1 instead of n to account for underestimating the variability, so the sample standard deviation would be √31 ≈ 5.568.”

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Bill Shillito @solidangles.bsky.social · 1d

*Oops, this should have said "logarithms of the values" — plural.

And by the way, we could use any base log (and corresponding exponential) we want!

I chose log base 10 because scientific notation (and the English language) generally uses powers of 10.

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Bill Shillito @solidangles.bsky.social · 1d

What other relationships between these means have you come across? I’m sure there are more unexpected links I've left out — if you’ve got a favorite, please go ahead and share it! 🙂

[20/end]

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Bill Shillito @solidangles.bsky.social · 1d

And I think this topic shows an excellent interplay between the kinds of things we want students to do in mathematics:

* Understanding the concepts behind the procedures and the relationships between them

* Seeing how those ideas can be applied to real-world problems

BOTH are important.

[19]

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Bill Shillito @solidangles.bsky.social · 1d

So while the geometric, harmonic, and quadratic means seem like very different formulas, they're really just variations on a theme: "transform, average, undo."

It's just a matter of figuring out what shift in perspective helps you strike the right balance for the problem at hand.

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Bill Shillito @solidangles.bsky.social · 1d

Now, technically the squaring function isn't invertible because it's not one-to-one — for example, 3² = (-3)² = 9.

But if we're okay with only getting positive numbers as a result, we can just restrict the domain of f(x) = x² to x ≥ 0, and everything turns out just fine.

[17]

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Bill Shillito @solidangles.bsky.social · 1d

The root mean square may also sound familiar from physics — it's used in the kinetic molecular theory of gases for the same reason. (Remember, kinetic energy varies with velocity squared!)

It's also used for RMS voltage for alternating current, which also fluctuates about an equilibrium.

[16]

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Bill Shillito @solidangles.bsky.social · 1d

It turns out we can instead use the quadratic mean, aka the root mean square (RMS).

Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units.

(BTW, here's a video on why squaring really is the best choice:
www.youtube.com/watch?v=q7se...)

[15]

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Bill Shillito @solidangles.bsky.social · 1d

Let's look at one more application.

In statistics, we often want to describe not just the center but the spread of a data set.

The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do?

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A slide on a dark red background showing a data set and its mean and deviations.

Data: 10, 17, 21.

Arithmetic mean: AM{10, 17, 21} = (10 + 17 + 21) / 3 = 16.

Deviations listed:
10 − 16 = −6
17 − 16 = 1
21 − 16 = 5

On the right, under “Average deviation,” it shows AM{−6, 1, 5} = 0…?!, showing that the deviations cancel out.

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Bill Shillito @solidangles.bsky.social · 1d

It turns out these are both examples of a more general idea called an "f-mean." Here's the basic idea:

1. Apply some function f to your data.
2. Find the arithmetic mean.
3. Undo f with the inverse function f⁻¹.

The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x.

[13]

$A diagram on a brown background illustrating the general process for finding an f-mean. Top left: {a, b}, with a rightward arrow labeled “M_f” pointing to M_f{a, b} on the right. A downward arrow labeled “f( )” leads from {a, b} to {f(a), f(b)}. From {f(a), f(b)}, a rightward arrow labeled “AM” points to AM{f(a), f(b)}. An upward arrow labeled “f⁻¹( )” connects AM{f(a), f(b)} back to M_f{a, b}. At the bottom, the label reads “f-Mean.”$

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Bill Shillito @solidangles.bsky.social · 1d

The harmonic mean is also useful for problems involving rates!

Say you travel 60 miles going 30 mph, and another 60 miles going 20 mph. What's your average speed?

See if you can convince yourself why the harmonic mean makes sense here.

Hint: average speed = total distance / total time.

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Bill Shillito @solidangles.bsky.social · 1d

One quick side "note" — that E isn't QUITE 660 Hz.

The 12-note chromatic scale many of us are accustomed to uses a geometric progression to evenly space the notes out.

So since a perfect fifth is 7 semitones, the real frequency is 440 · 2^(7/12) ≈ 659.255 Hz.

But hey, still pretty close!

[11]

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Bill Shillito @solidangles.bsky.social · 1d

Again, why did this work?

Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows:

1. Take the reciprocals of the data.
2. Find the arithmetic mean.
3. Undo the reciprocal with another reciprocal.

That's the harmonic mean.

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A diagram on a dark purple background showing how the harmonic mean relates to reciprocals and the arithmetic mean.

Top left: {1/5, 1/7}, connected by a rightward arrow labeled “HM” to 1/6 on the right.

A downward arrow labeled “1/( )” leads from {1/5, 1/7} to {5, 7}.

From {5, 7}, a rightward arrow labeled “AM” points to 6.

An upward arrow labeled “1/( )” connects 6 back to 1/6.

At the bottom, the label reads “Harmonic Mean.”

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Bill Shillito @solidangles.bsky.social · 1d

It turns out we need to use the harmonic mean: flip the string fractions upside-down, add, and divide 2 (the number of values) by the result.

So I have to place my finger to make the vibrating string 2/3 as long.

(BTW, if you lightly touch here, you get what's called a "string harmonic!")

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A formula for the harmonic mean on a dark purple background. It shows “HM{1, ½} = 2 / (1/1 + 1/(½)) = 2/3,” with the label “Harmonic Mean” in bold.

Below is a violin with four labeled points along one string:

“A440 (open)” near the scroll for the full-length string,

A dotted line with an X about 1/4 of the way long the string, showing the incorrect location for E660.

“E660!” about 1/3 of the way along the string, that is, making the string 2/3 of its usual length.

“A880” halfway between the scroll and the bridge, indicating the octave point where the string length is halved.

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Bill Shillito @solidangles.bsky.social · 1d

When I used to play viola in high school (alto clef represent!) I tuned to A440. An octave up the A string (making it half as long) is A880, and a perfect fifth is E660. (Almost — more on that later.)

Where should I put my finger to play that E?

It's NOT halfway to the octave mark!

[8, lol]

A viola on a dark purple background with three labeled points along the A string:

Near the scroll, “A440 (open)” marks the full string length.

Along the neck, “E660?” indicates the conjectured location for the note E.

“A880” marks the halfway point between the scroll and the bridge, one octave above the open A.

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Bill Shillito @solidangles.bsky.social · 1d

By the way, I'm using "trick" and "procedure" here because I don't think they're dirty words in math.

They highlight how useful — and fun — it is to recognize structure and turn it into a reusable technique.

(As long as we understand WHY they work.)

So how else can we use this trick?

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Bill Shillito @solidangles.bsky.social · 1d

It turns out that this "do A, do B, undo A" trick is one of the most useful algebraic techniques out there!

* Linear algebra → change of basis
* Group theory → conjugation (Rubik's cubes!)
* Differential equations → Laplace/Fourier transform

It's all about shifting your perspective.

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Bill Shillito @solidangles.bsky.social · 1d

The trick is to use logarithms to extract the exponents!

We do a three-step procedure:

1. Take logarithms of the value.
2. Find the arithmetic mean.
3. Undo the logarithm with an exponential function.

What we get is the geometric mean.

(Fun exercise: check this algebraically!)

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A diagram on a dark blue background showing how the geometric mean relates to logarithms and the arithmetic mean.

Top left: {10⁵, 10⁷}, connected by a rightward arrow labeled “GM” to 10⁶ on the right.

A downward arrow labeled “log( )” leads from {10⁵, 10⁷} to {5, 7}.

From {5, 7}, a rightward arrow labeled “AM” points to 6.

An upward arrow labeled “10( )” connects 6 back to 10⁶.

At the bottom, the label reads “Geometric Mean.”

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Bill Shillito @solidangles.bsky.social · 1d

When we humans make low and high guesses, our over- and under-estimates are often off by an order of magnitude, NOT by a fixed linear width.

If, say, 10⁵ is too small, and 10⁷ is too large, then 10⁶ is often "just right."

So in what sense can we say 10⁶ in the "center" of 10⁵ and 10⁷?

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Bill Shillito @solidangles.bsky.social · 1d

We can get a more reasonable estimate with what's called the geometric mean: multiply the values and take a root (here a square root, since there were 2 numbers).

This gives a much better estimate — as you can check on Wikipedia.

But why did this work? How would anyone think to try this?

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A formula for the geometric mean on a dark blue background. It shows “GM{250, 4000} = √(250 × 4000) = 1000,” with the label “Geometric Mean” in bold below.

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Bill Shillito @solidangles.bsky.social · 1d

Let's say you're trying to estimate the weight of a moose.

250 lb (a bit more than me) is too low.
4000 lb (roughly a car) is too high.

So what might be a good "mid-range" guess?

The arithmetic mean would be 2125. But that seems a bit high somehow.

Can we do better?

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Bill Shillito @solidangles.bsky.social · 1d

Let's start with ol' reliable: the arithmetic mean. The average.

What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values.

But as anyone who's studied triangles knows, there may be multiple kinds of "center!"

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A formula for the arithmetic mean on a dark red background. It shows “AM{5, 7} = (5 + 7)/2 = 6,” with the label “Arithmetic Mean” in bold below.

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