|Adding Value to Your Interface by Using Simple Math and Matching|
|Fitts' law (Wikipedia)|
|Hick's law (Wikipedia)|
|Power Law of Practice in UsabilityFirst glossary|
By Gerd Waloszek, SAP User Experience, SAP AG – August 5, 2009
When a colleague of mine attended a workshop organized by the Nielsen Norman group in London, she took part in a tutorial held by Bruce Tognazzini, or Tog for short. Among others, Tog presented Fitts' law (also known as Fitts's law) and was eager to mention that he probes job applicants on this law (as you can imagine, if you fail the "Tog test" you can forget about working for the NN group...).
My colleague had not heard of Fitts' law before and was curious to know whether she was the only one with this gap in her knowledge. Back at work, she probed her colleagues for Fitts' law in a team meeting. None of them had heard of it either – even those with a background in psychology. Then she asked me. Yes, I had heard of the law, but what I recalled was something different, namely Hick's law. I assume that similar gaps abound outside of SAP. Therefore, I would like to offer a short refresher of Fitts' law and include two other "classics," Hick's law and the power law of practice.
According to Wikipedia, "Fitts' law still remains one of the few hard, reliable predictive models in human-computer interaction." In short it says:
Fitts published his law in two articles (Fitts, 1954; Fitts and Peterson, 1964). The formula is best known in the "Shannon" formulation proposed by Scott and McKenzie:
T = time to complete the movement; D = distance to center of target; W = width of target in the direction of the motion; a and b are constants.
One might interpret this formula as follows: In order to achieve the same movement speed, the target width has to be increased according to the distance of the target (see figure 1). More details are given in Wikipedia.
Figure 1: Demonstration of Fitts' law
In the 1980s, Fitts' law was extended to graphical user interfaces. Here, the user's task is to move the mouse cursor and position it over an on-screen target, such as a button (Fitts' law can model both point-and-click and drag-and-drop actions). In its original and strictest form, however, the law:
But what is the use of a law if we cannot apply it? Therefore, we had better assume that it is also valid for pointing with the mouse. According to Wikipedia, some consequences for user-interface design include:
Tog and other Apple Macintosh pioneers like to tell the story of why the Apple Macintosh menu bar solution is superior to the window-based menu bars in Windows. On the Mac, there is only one menu bar at the top edge of the screen, which reflects the active application. In Windows, however, each window has a menu bar of its own. Because the Mac menu bar is at the edge of the screen, menu items can be selected much faster than in Windows – you simply "slam" the cursor against the edge of the screen.
Popup menus, such as context menus, which are opened with the right mouse button, have the additional advantage that functionality can be offered not only "in-place" but also selectively. A long time ago, I called this the "locality" principle. However, popup menus, which are opened using small trigger icons, are not a good design choice, at least when seen from a Fitts' law perspective. Also note that Fitts' law tells us that the usual linear popup menus are not optimal – pie menus are better. However, the only pie menu that I know of is offered by the Logitech mouse driver (see figure 2; note that the menu items do not extend to the edge of the screen). I must admit that I have never used it.
Figure 2: Pie menu
Hick's law deals with the time needed to choose between alternatives of equal probability. Originally, the law was targeted at simple motor decisions, such as hitting a number key on a numerical keypad or hitting a "yes" or "no" button in a psychological experiment. In short, Hick's law says:
That sounds rather hard to understand. Somewhat oversimplified, one might say:
The formula is:
T = average reaction time required to choose between n equally probable alternatives; b is a constant.
By the way, the "plus one" is due to the fact that a decision about "no key" is also a decision. More details are given in Wikipedia.
Hick's law is similar in form to Fitts' law because both laws have their roots in information theory. An intuitive argument for the logarithmic form of Hick's law is that people subdivide the total set of choices into binary categories: At each step, they eliminate about half of the remaining choices, rather than consider the choices one by one, which would require a linear amount of time (from Wikipedia, adapted).
Table 1: Relationship between time and number of choices
As already mentioned, Hick's law can be applied to hitting keys on a keyboard, where each key has the same probability of being a target.
Hick's law is sometimes also used to justify menu designs. However, applying the model to menus must be done with care: The order of the commands plays an important role as to which strategy people use. For example, to find a given word in a randomly ordered word list – in this case, the name of a command in a menu – requires each word in the list to be scanned. This strategy consumes linear time; thus, Hick's law does not apply here. Whereas, if the list is in alphabetical order, users may use a subdividing strategy and only need logarithmic time (from Wikipedia, adapted).
The power law of practice states that
*) I found this formulation in the UsabilityFirst glossary. Personally, however, I would not characterize such a decay as an exponential one. For me, "exponential" means a decay by the same factor for each time step (for example, 1/2, 1/4, 1/8, etc.), that is, not "n to the power of a constant" but "a constant to the power of n."
Regrettably, Wikipedia's article on the power law of practice is incomplete, but there are other sources on the Web, such as the UsabilityFirst glossary, where you can find the formula and some details:
T(n) = the time to perform a task after n trials, T(1) = ditto in the first trial, n = the number of trials, a ~ 0.4
Note that according to Wikipedia, Heathcote, Brown, and Mewhort (2000) have suggested that at the individual-level a three-parameter exponential function tends to fit observed data better than a three-parameter power function (so the UsabilityFirst glossary might be right, but it still shows a power law).
The power law of practice can be visualized as a learning curve or "the learning curve effect on performance" (see figure 3):
Figure 3: The power law of practice shown as a so-called learning curve
The quantitative statement of the power law of practice has been applied to a wide variety of different human behaviors: immediate response tasks, motor perceptual tasks, recall tests, text editing, and more high-level, deliberate tasks such as game playing (from University of Michigan, Artificial Intelligence Laboratory: Power law of Practice, adapted).
Because of the decay according to a power function, we can make two observations:
Please note that typical reaction times at the computer are the result of a complex interplay of different, often competing, influences – Fitts' law, Hick's law, and the power law of practice are just some of them.
Further references can be found in the Wikipedia articles.