How to calculate gear range mean step?

Question

On this wikipedia article, all the mean steps in the gearing range chart appear to be wrong, but my guess is that I am misinterpreting something:
https://en.wikipedia.org/wiki/Bicycle_gearing
180%/3 != 34.2%
250%/5 != 25.7%
300%/7 != 20.1%

etc
How are the authors arriving at these mean step figures?
Edit:
Looking at the source of the page, I see this:
{{#expr:(1.8^(1/2)-1)*100 round 1}}%

So it appears that the formula they are using is:
mean_gear_step = ((gear_range/10)^1/(number of gears - 1) - 1) * 100

No idea why this is though.

Wilskt · Answer

100 x 1.34 x 1.34 = 180.
With three gears you've got a starting gear (100%) and then two change-ups of 34%, leaving you at 180% of where you started.
You'd probably find similar sums work for the other two examples you gave.

juhist · Answer

And a formula how to calculate the gear step in the other direction:
1.8 ^ (1/(3-1)) = 1.3416 so 34.16% mean step
2.5 ^ (1/(5-1)) = 1.2574 so 25.74% mean step
3.0 ^ (1/(7-1)) = 1.2009 so 20.09% mean step

RLH · Answer

The formula they're using is the geometric mean. It:

Starts with the ratio between the biggest and smallest gears
Takes the nth root of this ratio (where n is the number of steps between the smallest and largest gears) to find the ratio R for which R^n takes you from the smallest to biggest gear
Subtracts 1 from this value to get a gear difference rather than a gear ratio

This formula is preferred over the formula you suggested because gear comparisons are fundamentally ratios and not differences.
For example, if you have gears that are radius 1, 2, and 4, then the total ratio is 400%, the step ratio is 200%, and the middle gear is exactly at the geometric mean -- going from 1:2 will feel like the same change as going from 2:4.

How to calculate gear range mean step?

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