Rule of thumb on the best k in k-means clustering The rule of thumb on choosing the best k for a k-means clustering suggests choosing $k$
$$
k \sim \sqrt{n/2}
$$
$n$ being the number of points to cluster. I'd like to know where this comes from and what's the (heuristic) justification. I cannot find good sources around. 
The only references I can find about this are a comment on reserchgate and this review, which does not explain it anyway.
 A: For the purpose of data approximation, this value can give you desired properties. Computing the pairwise distances of $\sqrt{n/2}$ takes approximately linear time, so if you want to reduce the size of your data set, this can be a value of interest.
For actual clustering, the value usually is unreasonably large.
A: I'm not sure if there is a "best" answer to this-I could only find a few references to your rule and no underlying theory.  I went through some of the Springer texts (ISLR and ELSL) here on my laptop and the chapters mention K means reference there are ways to choose k-but there is no consensus on the matter. 
There is just a single reference to additional material on the subject (Hastie et al. (2009)) in ISLR.  It appears that this method might begin with assigning p values to your clusters, but the details are a bit thin and I have yet to open that part up...  However that might be a place to start!
A: The references ("good sources") for the "rule of thumb":


*

*Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate Analysis. Academic Press, London. ISBN 0‐12‐471252‐5. See Multivariate Analysis, p.365  

*Cluster Validity in Clustering Methods (PhD dissertation)

*Clustering Approach in Wireless Sensor Networks Based on K-means: limitations and Recommendations

*Ahmed, M. & Mahmood, A.N. Ann. Data. Sci. (2015) 2: 111. https://doi.org/10.1007/s40745-015-0035-y
