The question asks for stats comparing sites' mobile versions with not having such a thing.
Be wary of any such stats... a "Mobile optimised website" covers the span from, at one end, brilliant pieces of design that present the key essence of a site in a small space requiring only simple interactions - to, at the other end, poorly conceived, barely managed fad-following beached whales that don't accommodate all of a site's key content or functions, were conceived based on wild unsupported assumptions talking about "Mobile Users" as if they're some different species, were 'tested' on one guy's iPhone on less than 10% of the site's content, aren't even thought about by the site's content creators when creating content, and end up turning customers away because it makes them think a site doesn't do important things the real site actually does do.
So be careful of any studies aggregating all such things together and calling them a group - they would be like studies investigating the question "Does music aid studying" where an unknown number of people in the "With music" group got Brahms and an unknown number got Beiber. Some mobile sites are brilliant, some are terrible.
For example, a Nielsen study from 2011 (their latest at time of writing) discusses a small but significant increase in task success rates in mobile sites (64%) compared to desktop sites on mobile (58% - desktop sites on desktop get 84%). This seemingly small difference on average doesn't show the important extremes - the potential for an individual great mobile site to reach parity with desktop-on-desktop, versus potential real harm from a spurious, poorly conceived one.
Instead of deciding based on general stats that aggregate wildly different cases, focus on your specific case and the exact nature of the problem you would be trying to solve with a mobile site.
To quote the article Stop Designing Mobile Websites (arguing essentially to not just unthinkingly churn out mobile sites because the big boys do it), "don’t think mobile… think universal". What's the design/UX problem, weakness or missed opportunity you hope a mobile version may improve?
Most often, it will be that much of the key content is difficult to navigate on small devices. Is that definitely just a problem with small devices, or is there a problem on all devices that is bad-but-tolerable on larger screens and pushed beyond breaking point on mobile? Is there an underlying problem that can be fixed for all devices?
If it's definitely a good desktop design, and there's an unavoidable reason why it won't work on small devices - e.g. something that works exceptionally well on desktop that can't work on mobile - can the problem be solved with responsive design? Is it okay if you take the same site, and same content, but re-arrange it and re-style it below a certain width?
If no, for example if there are features that just won't work on a smartphone or that require processing or download time that definitely just isn't worth it on a smartphone, the defining feature of your mobile site is that elements are removed. What are the costs and benefits of removing or adapting these features? How could someone accessing your site on a mobile get these features if they needed them?
One thing that is definite - never assume that mobile users are 'on the go' - a growing number (40% according to one Feb 2013 study) of people browse the web on mobile as their primary way of browsing the web - presumably even at home 10 feet from a 'real' computer, because it's more convenient. Those people aren't "Mobile users", they're users, who happen to be on a mobile (or other small device) right now, for reasons you can't guess. Since it's an area that is changing fast, it's unwise to make assumptions about them, and be aware that evidence in this area has a short shelf life.
(and a personal plea: please please please never ever ever take a user away from the specific page they followed a link to and dump them lost confused and surprised onto your homepage just because they're on the 'wrong' type of device or switched from mobile to desktop view or vica versa. Not that you would have, I'm sure, but some sites still do...)