How can I interpret what I get out of PCA? As part of a University assignment, I have to conduct data pre-processing on a fairly huge, multivariate (>10) raw data set. I'm not a statistician in any sense of the word, so I'm a little confused as to what's going on.  Apologies in advance for what is probably a laughably simple question - my head's spinning after looking at various answers and trying to wade through the stats-speak.
I've read that:


*

*PCA allows me to reduce the dimensionality of my data

*It does so by merging / removing attributes / dimensions that correlate a lot (and thus are a little unnecessary)

*It does so by finding eigenvectors on covariance data (thanks to a nice tutorial I followed through to learn this)


Which is great.
However, I'm really struggling to see how I can apply this practically to my data. For instance (this is not the data set I'll be using, but an attempt at a decent example people can work with), if I were to have a data set with something like...
PersonID     Sex     Age Range    Hours Studied     Hours Spent on TV      Test Score     Coursework Score 
1            1       2            5                 7                      60             75
2            1       3            8                 2                      70             85 
3            2       2            6                 6                      50             77
...          ...     ...          ...               ...                    ...            ...

I'm not quite sure how I would interpret any results.
Most of the tutorials I've seen online seem to give me a very mathematical view of PCA. I've done some research into it and followed them through - but I'm still not entirely sure what this means for me, who's just trying to extract some form of meaning from this pile of data I have in front of me.
Simply performing PCA on my data (using a stats package) spits out an NxN matrix of numbers (where N is the number of original dimensions), which is entirely greek to me.
How can I do PCA and take what I get in a way I can then put into plain english in terms of the original dimensions?
 A: I would say your question is a qualified question not only in cross validated but also in stack overflow, where you will be told how to implement dimension reduction in R(..etc.) to effectively help you identify which column/variable contribute the better to the variance of the whole dataset. 
The PCA(Principal Component Analysis) has the same functionality as SVD(Singular Value Decomposition), and they are actually the exact same process after applying scale/the z-transformation to the dataset. 
Here are some resources that you can go through in half an hour to get much better understanding. 
I am not capable to give a vivid coding solution to help you understand how to implement svd and what each component does, but people are awesome, here are some very informative posts that I used to catch up with the application side of SVD even if I know how to hand calculate a 3by3 SVD problem.. :)


*

*Coursera Data Analysis Class by Jeff Leek: Video Lecture / Class Notes

*A Very Informative student post 

*A post from American Mathematical Society.
A: In PCA you want to describe the data in fewer variables. You can get the same information in fewer variables than with all the variables. For example, hours studied and test score might be correlated and we do not have to include both. 
In your example, let's say your objective is to measure how "good" a student/person is. Looking at all these variables, it can be confusing to see how to do this. PCA allows us to clearly see which students are good/bad. 
If the first principal component explains most of the variation of the data, then this is all we need. You would find the correlation between this component and all the variables. "Large" correlations signify important variables. For example, the first component might be strongly correlated with hours studied and test score. So high values of the first component indicate high values of study time and test score.
