I am doing a clustering analysis using K-means and I have around 6 categorical variables that I want to consider in the model. When I transform these variables as dummy variables (binary values 1 - 0) I got around 20 new variables. Since two assumptions of K-means are Symmetric distribution (Skewed) and same variance and mean (scale). Do I need to transform these variables?
If you have exclusively binary variable you can use KModes, if you have both real and binary variables I would consider the KPrototypes algorithm. KModes use by default the hamming distance and prototype computation use the mod instead of the mean. KPrototypes mix both KMeans and KModes for each kind of features using euclidean and hamming for distance computation and mean and mod for getting both part of the prototypes.
You may be interested on DenseClus
DenseClus uses the uniform manifold approximation and projection (UMAP) and hierarchical density based clustering (HDBSCAN) algorithms to arrive at a clustering solution for both categorical and numerical data. With DenseClus, you provide a dataframe, and it will then generate homogeneous clusters with no need for extensive preprocessing or worrying about how to treat categorical features. This capability opens a wide range of use cases, from customer segmentation in marketing to mapping cells in biomedicine.