# Variable standardization / scaling for PCA when all dimensions already have same scale [duplicate]

Often when PCA is performed on exam results where all variables (dimensions) have the same $0$ to $100$ scale, scaling is none the less applied. For different scales I can see the purpose of it, but not in this case. Why is it done?

• This is covered in various answers to our very popular thread on this topic: stats.stackexchange.com/questions/53. Look through all well-upvoted answers to get a feeling of different opinions and use cases. Apr 26, 2016 at 15:12
• @amoeba This question may be subtly different from the others: given that exam scores have a natural, common range, why would anyone ever standardize the scores? The accepted (and highly upvoted) answer to the proposed duplicate specifically suggests they would not. Other answers also suggest as much.
– whuber
Apr 26, 2016 at 16:59
• @whuber I agree that this question has its own spin (+1, by the way). I think the answers to the proposed duplicate all together (not only the accepted one) do provide some guidance and discussion of both options (standardizing or not). I am happy for this thread to stay open and alone, but I am afraid it will not get as thorough a discussion as already exists elsewhere... Apr 26, 2016 at 19:38
• @whuber but what if there are topics that really 'discriminate' students so that the variance for these particular topics are much greater than the other topics ? It is the same potential range but woudn't you end up with a really distorted space ? Wouldn't scaling be advisable in that situation ?
– Riff
Apr 27, 2016 at 6:58
• @Nicolas I'm not sure what you mean by "topic" or "really distorted space." If your point is that different exam variables can have substantially different variances, then that would imply the fact they have natural limits of $0$ and $100$ is irrelevant--and likewise this question becomes irrelevant (or trivial). Many tests, though, are designed so that (at least in some reference population) the variances of each variable are all equal.
– whuber
Apr 27, 2016 at 11:33

When you have variables on different scales, for exemple $m$ and $km$, the difference of scale often leads to difference in variance that may not be relevant (just an artificial product of scale) but scaling in a PCA is much more than just getting rid of that problem.