I've tried to find one, but haven't come into any luck doing so.

Edit: I have already ascertained my regions Unitarian Universalist ministries.

CONTEXT:

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By some estimates, as many as 46% of UUs are atheists.

here

In 1997, the Unitarian Universalist Association surveyed more than 8,000 active UUs and found that humanists — a category that includes agnostics and atheists — constituted 46% of the membership.

The most recent nationwide survey to ask that question was conducted in 1987 by the UUA Commission on Appraisal, which found that 7% of Unitarian Universalists picked "atheist" over other options, including "humanist." Going back even further, 21% of Unitarian Universalists in 1979 said that the concept of God is irrelevant or harmful, down from 30% in 1967.

According to a 2001 report in The Christian Century, a study by Ohio University professor James Casebolt found that 18% of Unitarian Universalists whom he polled in Ohio, West Virginia, and western Pennsylvania considered themselves atheists. By comparison, a 2002 City University of New York study found that just 0.4% of all Americans say they're atheists.

Kennedy, D. (2003) 'Are You with the Atheists?'. Unitarian Universalist Association. Available at: here.

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EDIT: This question encompasses all types of irreligion, including those who approach UU as a secular religion, from a social scientific standpoint, syncretists, etc.

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EDIT: More studies.

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According to a 2001 report in The Christian Century, a study by Ohio University professor James Casebolt found that 18% of Unitarian Universalists whom he polled in Ohio, West Virginia, and western Pennsylvania considered themselves atheists. By comparison, a 2002 City University of New York study found that just 0.4% of all Americans say they're atheists.

Kennedy, D. (2003) 'Are You with the Atheists?'. Unitarian Universalist Association. Available at: here.

Moreover, this study (page nine, figure seven) exhibits further contrastive trends in "religiosity" between the 'general' American population versus the UU population:

As Figure 7 shows Unitarians are less religious and by far more secular in their outlook compared with the general American population. Nationally, 37% of American adults regard themselves as “religious” and 38% as “somewhat religious,” namely 75% of the adult population. This is in contrast with only 37% of religious outlook Unitarians. Furthermore, the ARIS sub-sample indicates that 39% of Unitarian regard themselves as “secular” and 19% as “somewhat secular” in outlook whereas only 10% of adult Americans describe their outlook as “secular” and 6% as “somewhat secular.”

Interestingly, the Unitarian pattern tends to closely resemble that of American Nones, those who profess no religion among whom 40% regarded themselves “secular” and 13% as “somewhat secular” in 2001. However, the Nones perhaps not unexpectedly have a smaller proportion with a “religious” outlook (only 8%).

Even more interesting is the finding that the UUA pattern (58% secular) also somewhat resembles that of America‟s Jews (i.e. persons who identify as Jewish-by religion). Among these “religious” Jews, 44% describe their outlook as “secular” or “somewhat secular.”

Kosmin, B. A., and Keysar, A. (n.d.) Unitarian-Universalists in the United States 1990-2008: Socio-demographic Trends and Religious Patterns, A Report Based on the American Religious Identification Survey. Trinity College, Hartford Connecticut. Unitarian Universalist Association. Available at: here.

I'll try to keep this short: a friend and I have plans to attend a UU service this weekend. I saw pictures on Facebook of the congregation with their hands on each other's shoulders, which makes me really nervous because neither my friend nor I tolerate physical touch well. If we sit and the end of a row, would it be unforgivably rude for us to say something like "no thank you?" Is there any chance of us passing on this without making a scene?