Calculate the rank correlation coefficient of the given problem

[chwala]

Homework Statement

See attached.

Relevant Equations

Spearman's rank correlation coefficient.

Find the problem and solution here; I am refreshing on this topic of Correlation.

The steps are pretty much clear..my question is on the given formula $\textbf{R}$. Is it a generally and widely accepted formula or is it some form of improvised formula approach for repeated entries/data? How did they arrive at... $m^3-m?$ ... Any proofs? Supposing $9$ entries are repeated and $1$ entry is different would the formula still hold?
Are there other different ways of solving this particular problem?

Cheers...

 

[WWGD]

Homework Statement:: See attached.
Relevant Equations:: Spearman's rank correlation coefficient.

Find the problem and solution here; I am refreshing on this topic of Correlation.

View attachment 305574

The steps are pretty much clear..my question is on the given formula $\textbf{R}$. Is it a generally and widely accepted formula or is it some form of improvised formula approach for repeated entries/data? How did they arrive at... $m^3-m?$ ... Any proofs? Supposing $9$ entries are repeated and $1$ entry is different would the formula still hold?
Are there other different ways of solving this particular problem?

Cheers...

This just measures the degree to which the two rankings agree. There are other such measures:

https://en.wikipedia.org/wiki/Rank_correlation

https://en.wikipedia.org/wiki/Rank_correlation/Spearman%27sSpearman's $\rho$ as a particular case


Edit: I think the expression $m^3 -m$ comes from properties of the Uniform Distribution

Edit 2 : Per the article linked, this is the case, i.e., the expression $m^3-m$ ad others are derived from the Uniform Distribution:
Ranks are just elements of permutations
of $S_n$ , the group of permutations of the elements in $\{ 1,2,3,..n\}$

 

[chwala]

Thanks...let me look at the links and make some short notes. Cheers.