I think the median average temperature is around 2,200°C.

The Earth has a radius of 6,371km, giving a volume of 1.08e13km^3.

A sphere of half this volume would have a radius of 5,057km. Within the Earth, this sphere would sit at a depth of 6,371 - 5,057 = 1,314km.

From this chart, the temperature at that depth is around 2,200°C, so half the volume of the Earth has a temperature above that, and half a temperature below it.

Higher than you think. The inner core of earth is ~5K degrees C and the outer core is ~3K degrees C

The crust is minuscule compared to the core and mantle.

The mantle makes up about 84% of Earth’s total volume. The temperature varies from about 1 300 K (1 000°C, 1 832°F) near its boundary with the crust, to 4 000 K (3 700°C, 6 692°F) near its boundary with the core. https://education.nationalgeographic.org/resource/mantle/

The temperature in the Earth's core is uncertain: estimates at the inner core boundary range from 4 000 K to 8 000 K and at the core–mantle boundary from 3 000 to 4 500 K. https://www.homepages.ucl.ac.uk/~ucfbdxa/pubblicazioni/nat.pdf

By mass or by volume?

Let me get out my thermometer, brb

When you say "all molecules that comprise earth," are you including every molecule in the atmosphere out to the Karman line? Are you looking for an average of every molecule, or an average by volume? There are more molecules in solid matter than gaseous, obv

You will have a very difficult time finding this. The average temperature of all molecules on earth is absurdly difficult to calculate, nearly impossible to gather data on, and not something that's very useful for any practical calculations so no one has bothered to do it.

Black body radiation is probably more what you're looking for, I would suggest starting there.

I don't think that's an answer that really exists in any meaningful sense since temperature is a macroscopic phenomenon. When you get down to the scale of the microscopic, i.e. of molecules, then atoms, then particles, you really only have amounts of kinetic energy of said particles, typically measured in the unit electronvolt, or eV.

When said particles interact, they impart kinetic energy to one another, which directly constitutes the thermodynamic fluctuations we see in macroscopic systems.

Put simply, microscopic energy levels create macroscopic temperature readings.

In other words, "temperature" is just a macroscopic reading of collective microscopic energy levels.

tl;dr: Molecules don't have temperature; they have energy.

I imagine that if you look up the estimated temp for the Earth's Mantle, you'll be pretty close to what the average temp is.

After cruising the comments and reading your post twice I propose the following because I'm that asshole today:

Go by layer.

Okay, so, assload insta-death ballpark Celsius in the center, okay fine. Maybe median this one. Then, consider some other identifiable layer(s) between there & surface, with some more eyeballing. THEN, for good measure, as many surface temps you can get for sea/land/air if you have chosen the range to include atmosphere.

Report back. I'm half asleep and haven't checked my work. Also. I only ever need half a reason to suggest implementing excel to assist in your calculations 😂 I'm so sorry

Why can't you take an average of global average ocean, surface, and air temperatures? That seems like it'd be...an okay...estimate...

Hotter than it was last year, and the year before that, and...

Average?

The hottest place on earth is the core, about 4400°C to 6000°C(average around 5200°C)

https://education.nationalgeographic.org/resource/core/

A short google point out that the coldest place on earth is Eastern Antarctic Plateau, Antarctica (-94°C)

https://www.newscientist.com/question/coldest-places-earth/#:~:text=1)%20Eastern%20Antarctic%20Plateau%2C%20Antarctica,of%20coldest%20place%20on%20Earth.

And since no living thing is hotter than earth core and no living thing is colder than antarctica(other than my ex), then we calculate the average of this two((5200 + -94)÷2) and we get 2553°C. That's the average temperature of earth.