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In my textbook it says the data is assumed to be zero-centered. I am not sure what this means. I read it stand for substracting the mean from the individual values, however I would like to know do you divide the values by the standard deviation as well.

For example if one dimension had values 8 10 12 and another 80 100 120, the latter would have much more variation and consequently PCA would assign it a greater weight. My question is whether we divide the data by standard deviation as well.

Borut Flis
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You're right, each attribute should always be normalised before PCA occurs. Check out this answer: Why do we need to normalize data before principal component analysis (PCA)?