Can averaging all the variables be seen as a crude form of PCA? This question just occurred to me out of the blue. PCA is a way to reduce dimensions. Another way that is often (perhaps too often) used is to take mean values of two or more variables. This is done a great deal to create "scales" or "ratings" that get published in the popular press about various political issues. Dimensions are naively "averaged" and this cumulative "average" is portrayed as a magical one-number-sums-all figure.
Is this, nevertheless, a very crude relative of PCA? If so, what assumptions does it have? I can think of the following:


*

*Sphericity is perfect or near-perfect.

*All variables are perfectly orthogonal from all other variables.

*Variables are all perfectly scaled against each other.


What else would there be?
 A: PCA forms linear combinations of the variables, and averaging all the variables is also taking a linear combination -- namely one where all the weights are equal to $1/d$, where $d$ is the number of variables. So one can view these approaches as conceptually related.
Moreover, under certain conditions averaging can indeed be called "a very crude relative of PCA", in a sense that PCA will result in the first principal component being proportional to the average of all variables (or close to it). What are these conditions?

Sphericity is perfect or near-perfect. All variables are perfectly orthogonal from all other variables. Variables are all perfectly scaled against each other.



*

*If all variables are "perfectly scaled", let's assume that they are centread and standardized to have variances equal to $1$. This means that covariance matrix and correlation matrix coincide. 

*Note that if all the variables are indeed "perfectly orthogonal" to each other as you suggest, then the covariance/correlation matrix becomes identity matrix, and any vector can be chose to represent its first principal component; all eigenvalues are equal to $1$ and PCA would be useless (as would be the averaging). So let's rather consider small but non-zero pairwise covariances/correlations.
Now if all pairwise correlations are equal to the same number $c$, i.e. the covariance matrix looks like that: $$\left(\begin{array}{}1&c&c&c\\c&1&c&c\\c&c&1&c\\c&c&c&1\end{array} \right),$$ then the first eigenvector will be proportional to $$\left(\begin{array}{}1\\1\\1\\1 \end{array}\right),$$ i.e. the first PC will be proportional to the average over all variables. This should be obvious from the permutation-invariance of this covariance matrix.
This is true with any number of variables, not necessarily four. This also remains true for any value of $c \in (0,1)$, whether the variables are nearly orthogonal ($c\approx 0$) or not.
Moreover, this often remains approximately true (as noted by @whuber in the comments) if the off-diagonal elements are not exactly equal, but are of similar magnitude. Then the first PC will often be close to the average as well. For a nice real-life example of such a situation, see this answer illustrating the dataset of crab body measurements.
