Data standardization vs. normalization for clustering analysis I'm performing clustering analysis and visualization (hierarchal, PCA, T-SNE etc.) on a dataset, and a bit confused about the method for data preparation. I understand that the typical options are to standardize, normalize, or log transform, but it seems like there are no hard and fast rules regarding when you apply one over the other? 
With standardization and log-transformation - my dataset splits into two clusters with a number of different algorithms. One cluster is large and heterogeneous (which is actually interesting as this is a biological problem and makes logical sense). However, if I normalize the data, I get three clusters out of it - splits the heterogeneous cluster into two. This could make sense as well, but it would be a stretch, and the clusters are not as clean. What could be causing this? The non-heterogeneous cluster remains the same, which is reassuring. Is it reasonable to conclude that the "instability" of the second cluster is further evidence of the heterogeneity in the dataset?
 A: There cannot be a general rule on what to do.
Any automatic normalization is usually "wrong". They only happen to usually work better than not weighting features at all, so people commony use them - in particular on data they don't understand.
But the right way is to weight and scale features such they have the right balanced amount of influence on the results. As there is no mathematical way to capture this "right balance" (it's not uniform!) there cannot be an automatic solution. You have to understand your data and scale each feature to give it he desired amount of influence.
A: I think standard scaling mostly depends on the model being used, and normalizing depend on how the data is originated 
Most of distance based models e.g. k-means need standard scaling so that large-scaled features don't dominate the variation. Same goes to PCA.
About the normalization, it mostly depends on the data. For example, if you have sensor data (each time step being a variable) with different scaling, you need to L2 normalize the data to bring them into the same scale. Or if you are working on customer recommendation and your entry are the number of times they bought each item (items being variables), you might need to L2 normalize the items if you don't want people who buy a lot to skew the feature. 
Personally, I think if the variables are well-defined, their log might result in losing interpretaility. So if you get good looking clusters without the log transform, I'd stick to it.
