Question: ### Context:

I'm a Psychology PhD student. As with many psychology PhD students, I know how to perform various statistical analyses using statistical software, up to techniques such as PCA, classification trees, and cluster analysis.
But it's not really satisfying because though I can explain why I did an analysis and what the indicators mean, I can't explain how the technique works.

The real problem is that mastering statistical software is easy, but it is limited. To learn new techniques in articles requires that I understand how to read mathematical equations. At present I couldn't calculate eigenvalues or K-means. Equations are like a foreign language to me.

### Question:

* Is there a comprehensive guide that helps with understanding equations in journal articles?

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### Edit:

I thought the question would be more self explanatory: above a certain complexity, statistical notation becomes gibberish for me; let's say I would like to code my own functions in R or C++ to understand a technique but there's a barrier. I can't transform an equation into a program.
And really: I don't know the situation in US doctoral schools, but in mine (France), the only courses I can follow is about some 16th century litterary movement...

Answer: 
### Overview:

* My impression is that your experience is common to a lot of students in the social sciences.
* The starting point is a motivation to learn.
* You can go down either **self-taught** or **formal instruction** routes.

### Formal instruction:

There are many options in this regard.
You might consider a masters in statistics or just taking a few subjects in a statistics department.
However, you'd probably want to check that you have the necessary mathematical background. Depending on the course, you may find that you need to revisit pre-calculus mathematics, and perhaps some material such as calculus and linear algebra before tackling university-level mathematically rigorous statistics subjects.

### Self-taught

Alternatively, you could go down the self-taught route.
There are heaps of good resources on the internet.
In particular, reading and doing exercises in mathematics text books is important, but probably not sufficient. It's important to listen to instructors talking about mathematics and watch them solve problems.

It's also important to think about your mathematical goals and the mathematical prerequisites required to achieve those goals. If equations are like a foreign language to you, then you may find that you need to first study elementary mathematics. 

I've prepared a few resources aimed at assisting people who are making the transition from using statistical software to understanding the underlying mathematics.

* **Videos**: [List of Free Online Mathematics Videos](http://jeromyanglim.blogspot.com/2009/05/online-mathematics-video-courses-for.html) - This post also provides some guidance regarding what would be an appropriate mathematical sequence starting from pre-calculus and working through calculus, linear algebra, probability, and mathematical statistics.
Also see this question on [mathematical statistics videos](https://stats.stackexchange.com/questions/485/mathematical-statistics-videos).
* **Reading and pronunciation** - One of the first challenges is learning how to pronounce and read mathematical equations. I wrote two posts, one on [pronunciation](http://jeromyanglim.blogspot.com/2009/05/pronunciation-guides-for-mathematical.html) and another on [tips for reading mathematics for the non-mathematician](http://jeromyanglim.blogspot.com/2010/01/tips-on-reading-mathematics-for-non.html).
* **Writing** - Learning to write mathematics can help with reading mathematics. Try learning LaTeX and check out some of the guides on [mathematics in LaTeX](http://en.wikibooks.org/wiki/LaTeX/Mathematics)
* **Books**: When it comes to learning mathematics, I think it is worth investing in a few good textbooks. However, there are a lot of [free online options](http://jeromyanglim.blogspot.com/2010/01/free-online-mathematics-books.html) these days
I understand your difficulty as I have a similar problem when I try to do something new in statistics (I'm also a grad student, but in a different field). I have found examining the R code quite useful to get an idea how something is calculated. For example, I have been recently learning how to use `kmeans` clustering and have many basic questions, both conceptual and how it is implemented. Using an `R` installation (I recommend `R Studio`, <http://www.rstudio.org/>, but any installation works), just type `kmeans` in to the command line. Here is an example of *part* of the output:

```
x <- as.matrix(x)
    m <- nrow(x)
    if (missing(centers)) 
        stop("'centers' must be a number or a matrix")
    nmeth <- switch(match.arg(algorithm), `Hartigan-Wong` = 1, 
        Lloyd = 2, Forgy = 2, MacQueen = 3)
    if (length(centers) == 1L) {
        if (centers == 1) 
            nmeth <- 3
        k <- centers
        if (nstart == 1) 
            centers <- x[sample.int(m, k), , drop = FALSE]
        if (nstart >= 2 || any(duplicated(centers))) {
            cn <- unique(x)
            mm <- nrow(cn)
            if (mm < k) 
                stop("more cluster centers than distinct data points.")
            centers <- cn[sample.int(mm, k), , drop = FALSE]
        }
    } 

```

I'm not sure how practical it is to examine the source every time, but it really helps me get an idea what is going on, assuming you have some familiarity with the syntax. 

A previous question I asked on [stackoverflow](https://stackoverflow.com/questions/5297889/how-do-i-find-the-parameters-used-in-kmeans-to-create-clusters) pointed me in this direction, but also helpfully told me that the comments about the code are sometimes included [here](http://svn.r-project.org/R/trunk/src/library/stats/R/kmeans.R).

---

More generally, the [Journal of Statistical Software](http://www.jstatsoft.org/) illustrates this link between theory and implementation, but it is frequently about advanced topics (that I personally have difficulty understanding), but is useful as an example.