The disagreement between you and the user is probably due to different definitions of clutter. Characterizing and quantifying clutter first depends on the meaning you choose.
Information density is simply the amount of information items per square unit of area on the screen (e.g., controls, words, sentences, icons, pictures, or combinations thereof). By this definition and measure, clearly Yahoo is more cluttered than Google.
There is a line of research that attempted understand the effects of clutter as measured this way on user performance, going back to the days of green-screen character-cell UIs. Not surprisingly, the more stuff you cram on the screen, the harder it is to find a specific item of information. Researchers have also held the total amount of information per page constant but varied the space between the items, thus reducing density alone. This does not seem to consistently affect search time. Here’re some recent examples:
Lingberg T (2003). Effects of Information Density and Size on the Perception of Graphics in User Interfaces. Masters thesis at Helsinki University of Technology.
Weller, D (2004) The Effects of Contrast and Density on Visual Web Search. Wichita State University.
Applying information density (or simply total information quantity) to design decisions is tricky because the alternative to high information per page is to spread the information over more pages, so now you have to weigh search-time-per-page against navigation time. It’s also complicated be the fact that search time per page depends on how well organized the information is, which is our second definition of clutter.
Information organization is the degree related information items are next to each other. This can mitigate the effects of high information density on search time because a user only has to look at one item in each region of the page. If that item isn’t close to what the user is looking for, then neither are any of the other items around it, and the user doesn’t even have to look at them.
The importance of organization has been shown in various studies comparing randomly organized menus with semantically organized menus (you can think of a web page as an enormous menu of links). For example:
McDonald JE, Stone JD, Leibelt LS, & Karat J (1982). Evaluating a Method for Structuring the User-System Interface, Proceedings of the 26th Annual Meeting of the Human Factors Society, p551-555
Again, no shocker, semantically organized menus outperform random menus.
From this perspective, both Google and Yahoo have low clutter, being well organized. I don’t know of any quantitative measure of information organization, but I suppose one measure would be to run cluster analysis or multi-dimensional scaling on card-sort results for all the information, then correlate the statistical distances between each pair of items with the physical distance between each pair on the page. That would make a good master’s thesis, if it hasn’t been done already. However, it fails to account for the effects of visual organization, the next definition of clutter.
Visual organization is the degree that related information appears to go together. This is primarily accomplished by applying the rules of Gestalt. Those includes separating related information from other information with white space (which confounds visual organization with information organization), but it also can be accomplished by such graphic techniques as enclosing the information in a box or using the same font or color for the related information. The Weller study cited above showed the importance of visual organization.
Glancing casually at the Google and Yahoo pages, they both appear to have decent visual organization, and therefore low clutter by that definition. Unfortunately, I don’t know of any quantitative measures of the degree visual information presentation forms “good” gestalts.
Perhaps a more promising approach to measuring clutter is to define it as visual variability, which is the degree colors and shades change irregularly over the page, regardless of the amount of information in those colors and shades. Measuring clutter this way involves applying models of human visual perception, which analyzes the spatial frequencies in the page. For details, see:
Rosenholtz R, Li Y, Mansfield J, & Jin Z (2005), Feature congestion: A measure of display clutter. SIGCHI 2005, p761-770
This definition of clutter seems to accurately capture the perception of clutter one gets when first looking at a page. The Yahoo page, for example, has more graphic variability (e.g., changes in colors per square inch) than Google, and thus looks more cluttered. However, I expect graphic variability is only partially related to visual search. For example, if we re-made the Yahoo page to use all the same type and size of font, all colored gray, and all evenly distributed over the page, it would look, at first glance, uncluttered (the way a page in a book looks uncluttered). However, by removing all the visual organization, I expect you made visual search harder.
Personally, I decided to concern myself with the impact of clutter, rather than clutter per se. If the problem with clutter is that it increases visual search, then what makes visual search difficult? Among the factors is the degree the target information is visually similar from its context. The target will be hard to find if it’s surrounded by a lot of similar-looking imagery –that’s the clutter effect, or, more generally, the poor contrast effect. A means of measuring this are in:
Zuschlag MK (2004). Quantification of Visual Clutter Using a Computational Model of Human Perception: an Application for Head-Up Displays. Human Performance Situation Awareness and Automation II Conference, Daytona Beach, FL (Mar. 22-25).
This measure does not take into account other factors that influence visual search time, such as visual variability and organization, so it won’t tell you the whole story. For example, see:
Lohenz MC, Trafton JG, Beck MR, Gendron ML (2009). A Model of Clutter for Complex, Multivariate Geospatial Displays. Human Factors, 51(1), p90-101.
The bottom line is clutter is complicated, and we’ve yet to develop a single measure for it. But even if we do put a number on it, what does it mean? “Clutter” sounds negative, implying the real definition of clutter is “stuff on the screen I don’t care about.” Perhaps the disagreement between you and your user over the relative clutter of VS has more to do with what you each find useful than how much is in the window.
However, when using any of the above definitions and measures, is more clutter necessarily worse that less clutter? Sometimes the user’s task requires you display a lot of information in a small space. Sometimes reality is disorganized and it’s important that users understand that (e.g., in some data visualizations).
The best we have now are guidelines for balancing breadth with depth, following the rules for good layout and Gestalt, and balancing the graphic distinctiveness of your information with their importance.