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I hase described these here:

http://en.wikipedia.org/wiki/Talk:Average#Relationship_between_different_types_of_mean

I would be very happy if someone could check them and add them to either article.

thanks.

Plugging in some numbers, it seems to me that none of these hold. Why should they? There is the exception that ${\displaystyle A(x,y)H(x,y)=G(x,y)^{2}}$, but this only holds for ${\displaystyle n=2}$. In this case you can simplify the harmonic mean as ${\displaystyle H(x,y)={\frac {2xy}{x+y}}}$. xo Ergur (talk) 11:15, 5 August 2025 (UTC)Reply

The Rational Mean is the unifying concept which embraces all known Means and the most simple principle for generating high-order root-solving methods, as shown at: The Rational Mean

So, it is disturbing to realize so many people seem unaware of this basic arithmetical principles and methods. — Preceding unsigned comment added by Arithmonic (talk • contribs) 20 October 2006 (UTC)

i think it would be very useful to visualize the concept with such a graphic: [[1]]

These are about how certain things should be pronounced: first, ${\displaystyle {\bar {x}}}$ and second, ${\displaystyle \mu }$. Both of these have their own articles that are linked next to where the note is located, so the notes feel redundant. Additionally, in regards to note 1, the precise pronunciation of mathematics notation is not standardized and this types of descriptions come of as amateurish. I suggest removing both notes and clarifying "the greek letter ${\displaystyle \mu }$...". I will do this if no one objects. — xo Ergur (talk) 12:41, 5 August 2025 (UTC)Reply

There are actually only two entries in the table: the power mean and the geometric mean. The other two are just special cases of the power mean. I suggest removing the table and adding text mentioning these two cases, un less there are other things to add to the table. If another example, or two, are added to the table, then I suggest removing the harmonic and arithmetic means, and referencing the previous subsection instead. I will do this if no one objects. This will also have the nice side effect of removing three notes, as they will either be unneeded or inlined. — xo Ergur (talk) 12:50, 5 August 2025 (UTC)Reply

There are relationships between quantities.

• ⁠(u₁ + u₂)(v₁ + v₂)/u₁ + u₂ + v₁ + v₂⁠ = ⁠u₁u₂v₁ + u₁u₂v₂ + u₁v₁v₂ + u₂v₁v₂/(u₁ + v₁)(u₂ + v₂)⁠ means either ⁠u₂/u₁⁠ = ⁠v₂/v₁⁠ or ⁠v₁/u₁⁠ = ⁠v₂/u₂⁠.
• ⁠(u₁ + v₁)(u₂ + v₂)/u₁ + u₂ + v₁ + v₂⁠ = ⁠u₁u₂v₁ + u₁u₂v₂ + u₁v₁v₂ + u₂v₁v₂/(u₁ + u₂)(v₁ + v₂)⁠ means either ⁠u₂/u₁⁠ = ⁠v₂/v₁⁠ or ⁠v₁/u₁⁠ = ⁠v₂/u₂⁠.
• (u₁ + u₂)(v₁ + v₂)(u₁ + v₁)(u₂ + v₂) = (u₁ + u₂ + v₁ + v₂)(u₁u₂v₁ + u₁u₂v₂ + u₁v₁v₂ + u₂v₁v₂) means u₁v₂ = u₂v₁.
• ⁠(E₁ + E₂)(R₁ + R₂)/E₁ + E₂ + R₁ + R₂⁠ = ⁠E₁E₂R₁ + E₁E₂R₂ + E₁R₁R₂ + E₂R₁R₂/(E₁ + R₁)(E₂ + R₂)⁠ means either ⁠E₂/E₁⁠ = ⁠R₂/R₁⁠ or ⁠R₁/E₁⁠ = ⁠R₂/E₂⁠.
• ⁠(E₁ + R₁)(E₂ + R₂)/E₁ + E₂ + R₁ + R₂⁠ = ⁠E₁E₂R₁ + E₁E₂R₂ + E₁R₁R₂ + E₂R₁R₂/(E₁ + E₂)(R₁ + R₂)⁠ means either ⁠E₂/E₁⁠ = ⁠R₂/R₁⁠ or ⁠R₁/E₁⁠ = ⁠R₂/E₂⁠.
• (E₁ + E₂)(R₁ + R₂)(E₁ + R₁)(E₂ + R₂) = (E₁ + E₂ + R₁ + R₂)(E₁E₂R₁ + E₁E₂R₂ + E₁R₁R₂ + E₂R₁R₂) means E₁R₂ = E₂R₁.
• ⁠(x₁ + x₂)(y₁ + y₂)/x₁ + x₂ + y₁ + y₂⁠ = ⁠x₁x₂y₁ + x₁x₂y₂ + x₁y₁y₂ + x₂y₁y₂/(x₁ + y₁)(x₂ + y₂)⁠ means either ⁠x₂/x₁⁠ = ⁠y₂/y₁⁠ or ⁠y₁/x₁⁠ = ⁠y₂/x₂⁠.
• ⁠(x₁ + y₁)(x₂ + y₂)/x₁ + x₂ + y₁ + y₂⁠ = ⁠x₁x₂y₁ + x₁x₂y₂ + x₁y₁y₂ + x₂y₁y₂/(x₁ + x₂)(y₁ + y₂)⁠ means either ⁠x₂/x₁⁠ = ⁠y₂/y₁⁠ or ⁠y₁/x₁⁠ = ⁠y₂/x₂⁠.
• (x₁ + x₂)(y₁ + y₂)(x₁ + y₁)(x₂ + y₂) = (x₁ + x₂ + y₁ + y₂)(x₁x₂y₁ + x₁x₂y₂ + x₁y₁y₂ + x₂y₁y₂) means x₁y₂ = x₂y₁.
• ⁠(m + n)(p + q)/m + n + p + q⁠ = ⁠mnp + mnq + mpq + npq/(m + p)(n + q)⁠ means either ⁠n/m⁠ = ⁠q/p⁠ or ⁠p/m⁠ = ⁠q/n⁠.
• ⁠(m + p)(n + q)/m + n + p + q⁠ = ⁠mnp + mnq + mpq + npq/(m + n)(p + q)⁠ means either ⁠n/m⁠ = ⁠q/p⁠ or ⁠p/m⁠ = ⁠q/n⁠.
• (m + n)(p + q)(m + p)(n + q) = (m + n + p + q)(mnp + mnq + mpq + npq) means mq = np.

172.56.35.162 (talk) 15:49, 16 September 2025 (UTC)Reply