While one could compute the average aspect-ratio or, even better, the most frequent aspect-ratio of your displays for people of visit your site, it will be of little consequence.
As you have noted, the browser bar takes some space. It takes a different amount of space on each browser, at different resolutions and with different toolbars installed. Scroll bars will also add to this.
The biggest problem though, I strongly suspect, is that most people other than tablet users do not browse with their window full-screen. Even if they browse maximized, the browser will not always cover the entire screen, at least on Windows, depending on the start bar position, size and auto-hide state.
The best you can do is decide the ratio of image you have and do some scaling and cropping to get the balance you want. If you want to avoid space outside of the image, start with something like 4:3 and crop to it which mean most people will get to see most of the width of the image. Remember some people use their screens in portrait orientation, so you should be ready for that too.