The basic idea of ruggedness is to calculate the average spire measure of every point in a region. However there are some subtleties to this process. (Click the link above for explanations, data, and acknowledgements regarding spire measure.)
First, let's assume that we have a fixed region K for which we want to find the ruggedness. We can think of any point in the region as a reference point for spire measure; intuitively, think of standing at that point and evaluating the impressiveness of the view based on steepness and local relief of the surrounding terrain. The simplest thing to do would be to take a simple average of all of the resulting spire measures. Instead it turns out to be better to take a root-mean-square (RMS) average of the spire measures at each point. The result is called the ruggedness R(K) of the region.
The resulting measure rewards both relief and steepness. In particular it is different from a commonly quoted measure of ruggedness, namely average slope. Average slope would reward a very low-relief but steep washboard landscape the same as it rewards a landscape with the same slope but much more relief. Ruggedness, as defined by (RMS) average spire measure, will reward the landscape with higher relief as compared to the one with less relief. I find this a very desirable feature.
So far so good, if we have a given region K. However if we just start with wanting to find "the ruggedness of the Alps" or "the ruggedness of the North Cascades" we have problem. Mountain ranges do not have precisely delineated borders or a well-defined size. So what exactly should we use for the region K, say, in the case of the Alps?
There are two separate issues we can identify. First, how big a region should we take, i.e. what area A should K have? We will postpone dealing with this until below. Second, once we choose a fixed A, exactly what are the right boundaries for K? For example, should it include a lot of the foothills or not?
Here is my strategy: always let a range show its best face for comparison. So what I do is the following, in the case of the Alps, to be specific: first, find the largest possible region anyone could think of as "the Alps." Then, given a choice of area A, find the most rugged region K with the specified area A which fits inside that huge starting region. That will be my version of "the Alps" at size A.
Now back to the first question, what size A to choose? This is where I have decided not to decide. I don't see a good way to nail down exactly how big a region to take, what outlying areas to include or exclude, or how down into the foothills to go. So I have calculated the ruggedness of various "versions" of each range. Three versions are displayed on the main ruggedness page; a large version (hopefully too large for most; it is intended to encompass most reasonable definitions of the range), a small version (hopefully too small for most), and (an attempt at) a "just right" version.
Here is how this affects the ruggedness calculation: as the area A gets smaller, we are selecting a more and more rugged core region of the range, so the ruggedness R increases. So we do not get a well-defined single value for "the ruggedness of the Alps" or "of the North Cascades." But we do get an interval of ruggedness values. When that interval is not too large, we can still make meaningful statements and comparisons, as I do on the main page.
For example, consider the Sierra Nevada and the Colorado mountains. Even the largest version of the Sierra Nevada (48700 km^2, R = 95m) beats the smallest version of the Colorado mountains (36200 km^2, R=79m). Also, the sizes are not completely different. Hence it makes sense to say that the Sierra Nevada are more rugged than the Colorado mountains.
In the case of the North Cascades and the West Chugach, it is harder to make a clear-cut comparison. The North Cascades come out a little ahead on a head-to-head basis, but the differences are within the margin of error for the calculation.
A word about units and accuracy: the units of spire measure are the same as the units of height; I use meters. To get a feel for the meaning of the number, note that the spire measure of a point on top of a conical peak of height H and slope 1 (45 degrees) is equal to H/2. So for example, when I say that the ruggedness of the Alps is 177m, that means roughly that the "impressiveness" of the average point in the Alps is comparable to standing atop a conical peak of height 354 meters and slope 1. That's pretty rugged!
The accuracy of most of these ruggedness values is plus or minus 5 percent, sometimes up to 10 percent. Since the main use of the numbers is for rough comparison, this seems sufficient.