I believe it's traditionally been held to be more on the synthetic side of the spectrum, but why? Are there any quantitative analyses to back this up?
According to The Welsh Language Welsh is an analytic language and that the relation of one word to another is conveyed by the use of prepositions or by the placing of the word in the sentence. It does imply, however, that there was a shift from a synthetic Brythonic language to the modern analytic Welsh language some time around 600 AD.
Language is not monolithic: some of its subsystems can be regarded as ‘(more) analytic’ and some ‘(more) synthetic’. Welsh is in this regard not different.
There cannot be a linear quantitative scale or spectrum partially because of the this; language isn't made of one piece. If one looks for a simple number (‘Welsh is 0.65 synthetic and 0.35 analytic’) xe is bound to oversimplify and erase important, non-negligible details. Oversimplification, caused by mixing flattened descriptions of many language en masse, is unfortunately too common in typology, resulting in contradictory and unusable conclusions.
Specifically, considering the language in question, let's take the Welsh verbal system — at least in Modern Literary Welsh, with which I'm more familiar — as an example. It is twofold: it has a basically synthetic sub-system (canai, for example) and a basically analytic one (oedd hi yn canu, for example). Even if we could quantify the analytic/synthetic-ness of each sub-system, how can one mix them in order to obtain a single, simple quantitative result? Should we calculate an average…? ;-)
Moreover, different linguists might mean different things when using these terms. For introductory reading, I'd recommend chapters V and VI of Sapir's Language.