Greetings! Could anyone enlighten me about the validity of this equation? I'm trying to prove it without success. $K$ is the number of clusters, $C_i$ is the $i$-th cluster, $m_i$ is the number of objects of $i$-th cluster and m is the total number of objects. Thanks in advance!

  • $\begingroup$ It's the sum of all objects (x) that belong to i-th cluster! $\endgroup$ Commented Dec 25, 2019 at 10:22
  • $\begingroup$ @IoannisTsirovasilis $m_i$ should be means, not counts; otherwise the equality doesn't hold. Let $x_1=1,x_2=-1$, and let they belong to different clusters, i.e. $m_1=1,m_2=1$. Apparently, the equation doesn't hold. $\endgroup$
    – gunes
    Commented Dec 25, 2019 at 10:43
  • $\begingroup$ Thank you! You are absolutely right! $\endgroup$ Commented Dec 25, 2019 at 11:51
  • $\begingroup$ Please choose a more descriptive title, e.g., by including the meaning of the equation, so that others can more easily find this. $\endgroup$ Commented Dec 26, 2019 at 8:43
  • $\begingroup$ @Anony-Mousse-ReinstateMonica Thanks for the tip I will keep that in mind! Happy new year's eve! $\endgroup$ Commented Dec 28, 2019 at 13:27

1 Answer 1


Via a couple of algebraical tricks, you'll have your zero only if $\mu_i$ and $\mu$ are the means, not the counts: $$\begin{align}S&=\sum_{i=1}^K\sum_{x\in C_i}(x-\mu_i)\cdot(\mu-\mu_i)\\&=\sum_{i=1}^K\overbrace{\left(\sum_{x\in C_i} (x-\mu_i)\right)}^0\cdot (\mu-\mu_i)\\&=0\end{align}$$

If it were the counts, letting $x_1=1\in C_1,x_2=-1\in C_2$ will cause invalidity of the equation shown. The equation is also shift-variant if $m$ are counts.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.