You can evaluate the performance of k-means by convergence rate and by the sum of squared error(SSE), making the comparison among SSE.

read about it more at url

https://towardsdatascience.com/k-means-clustering-algorithm-applications-evaluation-methods-and-drawbacks-aa03e644b48a

K-means has usually stop conditions. The sum of squared error goes from quantization theory (there k-means is equivalent Lloyd algorythm):

1) point x is estimated by point y, the error is d(x,y) - i.e. the distance between them

2) area Y is estimated by single point y, the error is \int_Y d^2(x,y) dx - i.e. the integral of the distance square

3) All domain Z is broken to Yi, each of Yi is estimated by point yi, the squared error is sum_i \int_Y d^2(x,y) dx

It is similar to sums of inertia moments of clusters.

For further, see Vector Qiuantization Lloyd Algorythms.

You can compute performance metrics: True positive, False Positive, True Negative, False Negative for the well clustered or not based on labeled testing data.

As Abdelkader Dairi mentioned, if you have the labeled ground truth data, you can use external validation and compute metrics like purity, rand index. Both purity and rand index values should be close to 1 for good clustering.

In case you don't have the labeled ground truth, you can perform internal validation, i.e. check that the points inside each cluster have high similarity and the clusters have low similarity between them. In this case, try computing Silhouette and Dunn index values. The higher the better!

The link below presents different evaluation methods of clustering (in case you don't have a labelled database) : https://stats.stackexchange.com/questions/21807/evaluation-measures-of-goodness-or-validity-of-clustering-without-having-truth

Good Luck ;)

Like others have said, there are many ways (supervised and unsupervised), I have used:

Silhouette coefficient

Davies–Bouldin index and

Calinski Harabasz.

You can see those and more performance measures in scikit-learn documentation.

https://scikit-learn.org/stable/modules/clustering.html#clustering-performance-evaluation