Tyler Schnoebelen, "The Weirdest Languages", Idibon blog 6/21/2013:
The World Atlas of Language Structures evaluates 2,676 different languages in terms of a bunch of different language features. These features include word order, types of sounds, ways of doing negation, and a lot of other things—192 different language features in total. […]
The data in WALS is fairly sparse, so we restrict ourselves to the 165 features that have at least 100 languages in them (at this stage we also knock out languages that have fewer than 10 of these—dropping us down to 1,693 languages).
Now, one problem is that if you just stop there you have a huge amount of collinearity. Part of this is just the nature of the features listed in WALS—there’s one for overall subject/object/verb order and then separate ones for object/verb and subject/verb. Ideally, we’d like to judge weirdness based on unrelated features. We can focus in on features that aren’t strongly correlated with each other (between two correlated features, we pick the one that has more languages coded for it). We end up with 21 features in total.
For each value that a language has, we calculate the relative frequency of that value for all the other languages that are coded for it. So if we had included subject-object-verb order then English would’ve gotten a value of 0.355 (we actually normalized these values according to the overal entropy for each feature, so it wasn’t exactly 0.355, but you get the idea). The Weirdness Index is then an average across the 21 unique structural features. But because different features have different numbers of values and we want to reduce skewing, we actually take the harmonic mean (and because we want bigger numbers = more weird, we actually subtract the mean from one).
Tyler offers a spreadsheet with the full set of languages and scaled feature-values behind the ordering.
There's a fairly large prior literature on how to quantify similarities or differences among languages, though usually the goal is to look for relationships rather than to identify conformity vs. freakiness.
An older information-theoretic measure can be found in Patrick Juola, "Cross-entropy and linguistic typology", CoNLL 1998. A recent effort, pointed out to me by Pieter Muysken, is Harald Hammarström & Loretta O'Connor, "Dependency-Sensitive Typological Distance" (those are slides from a Workshop on Comparing Approaches to Measuring Linguistic Differences, 24-25 October 2011, University of Gothenburg).
The problem here, in my opinion, is that there's no clearly-good way to decide how to choose and weight "typological" features of this type. And since Schoebelen used only 21 WALS features in that blog post, a few more or less would make a pretty big impact.
It's easy to list dozens of plausible features that happen not to be covered in WALS, and seem roughly as important as (say) whether comitative and instrumental are the same or not. For example, if we counted auxiliary-like uses of a verb meaning do or make, English would (according to a presentation by Aaron Ecay at a historical syntax workshop this past weekend) join Korean (Hagstrom 1995), a Northern Italian dialect (Benincà and Poletto 2004), and perhaps the earliest attested Icelandic (Viðarsson 2009) as one of a minority of languages with this feature, thereby boosting our freakiness coefficient.