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How to find the actual cluster sizes in a k-means algorithm?

I’m new to clustering and I’m currently working with the k-means algorithm. When I started implementing this algorithm, I read the notion of cluster quality like this:

The quality of the clusters is approximately equal to the ratio of the
within-cluster sum of squares to the total sum of squares.

This was just an intuitive definition which I still don’t really understand.
So my question is: Why is the sum of squares in a cluster equal to the sum of the squares of that cluster (we would call these the mean squares or the expected cluster centers)? In other words, how can we calculate this quantity?

A:

When you use k-means, you are trying to minimize the following objective function:
$J_{k-means} = \sum_{i=1}^{n} \sum_{j=1}^{k} (x_{i} – \mu_{j})^2$
Where $x_i$ represents the centroid of the $i$-th cluster (the cluster means) and $\mu_j$ represents the clustering center of the $j$-th cluster (the cluster centers).
The actual cluster sizes are unknown to you, so you need to guess the cluster means and the cluster centers (i.e. the centers of the clusters in your data). Note that these are just estimates, so they may not be what you intend.
If you apply the k-means algorithm to this data $x$, where $x$ is a real data matrix, then you will find that clusters will be formed where each cluster center will be near the group of data which belong to that group of points.
In order to calculate the actual cluster sizes, you will need the sum of the squares of the actual cluster centers (the cluster means) or you will need to estimate the actual cluster centers (using the actual cluster sizes).

A:

Your question (with my emphasis):

How can we calculate this quantity?

This is nothing

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