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Refreshing Three HCI Laws: Fitts' Law, Hick's Law, and the Power Law of Practice

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.


Fitts' Law

According to Wikipedia, "Fitts' law still remains one of the few hard, reliable predictive models in human-computer interaction." In short it says:

  • The time to acquire a target is a function of the distance to and width of the target (formulation by Tog, adapted).

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 = a + b * log2 (1 + D/W)

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.

Demonstration of Fitts' law

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:

  • Applies only to movements in a single dimension
  • Describes simple motor responses – it does not account for software acceleration usually implemented for a mouse cursor
  • Describes untrained movements

The first restriction was dealt with in the Accot-Zhai steering law, which extends Fitts' law to two dimensions (see the Wikipedia article for details).

Applying Fitts' 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:

  • Buttons and other controls to be selected in GUIs should be of a reasonable size; it is very hard to click small targets.
  • Edges and corners of the computer display are easy to reach because the pointer is "caught" at the edges, irrespective of how much further the mouse is moved (they can be considered as having infinite width). There is no danger of "overshooting," which requires corrective movements that cost time.
  • Popup menus can usually be opened faster than pull-down menus, since the user avoids movement.
  • Pie menu items are typically selected faster and have a lower error rate than linear menu items, for two reasons:
    • Menu items are all the same, small distance from the center of the menu
    • Their wedge-shaped target areas (which usually extend to the edge of the screen) are very large

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.

Pie menu

Figure 2: Pie menu


Hick's Law

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:

  • Given n equally probable choices, the average time required to choose between them is approximately proportional to the logarithm to the basis 2 of the number of choices plus one.

That sounds rather hard to understand. Somewhat oversimplified, one might say:

  • The reaction time increases with the number of choices but at a considerably slower rate (namely, at a logarithmic one)

The formula is:

  • T = b * log2 (n + 1)

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).


# Choices 1 2 3 7 15
Relative Time 1 1.6 2 3 4

Table 1: Relationship between time and number of choices

Applying Hick's Law

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

The power law of practice states that

  • When performing a task based on practice trials, people improve in speed at a decaying exponential rate *)
  • Or: The time needed for a particular task decreases in proportion to the number of practice trials taken raised to a power of about -0.4
  • Or: The logarithm of the time needed for a particular task decreases linearly with the logarithm of the number of practice trials taken (this formulation is for the math geeks...)

*) 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) = T(1)* power (n, a)    [n to the power of a]

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):

Power Law of Practice

Figure 3: The power law of practice shown as a so-called learning curve

Applying The Power Law of Practice

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:

  • The largest improvements in speed are made during the very first trials. Therefore, we should be careful with generalizing timing results from first-time users.
  • The learning process lasts virtually endlessly. With workers who rolled cigars, small improvements were demonstrated even after tens of thousands of trials.


Final Word: A Word of Caution...

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.




Fitts' Law

  • Wikipedia: Fitts' law
  • Wikipedia: Accot-Zhai steering law
  • Visualizing Fitts's Law (Kevin Hale, Particletree):
  • Paul M. Fitts (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, volume 47, number 6, June 1954, pp. 381-391. (Reprinted in Journal of Experimental Psychology: General, 121(3):262-269, 1992).
  • Paul M. Fitts and James R. Peterson (1964). Information capacity of discrete motor responses. Journal of Experimental Psychology, 67(2):103-112, February 1964.
  • Johnny Accot and Shumin Zhai (1997). Beyond Fitts' law: models for trajectory-based HCI tasks. Proceedings of ACM CHI 1997 Conference on Human Factors in Computing Systems, pp. 295-302.

Hick's Law

  • Stuart K. Card, Thomas P. Moran, Allen Newell (1983). The Psychology of Human-Computer Interaction. CRC.

Power Law of Practice

  • Wikipedia: Power law of practice (incomplete article)
  • Power Law of Practice in UsabilityFirst glossary:
  • Power Law of Practice (University of Michigan, Artificial Intelligence Laboratory):
  • Newell, A., & Rosenbloom, P. S. (1981). Mechanisms of skill acquisition and the law of practice. In J. R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 1-55). Hillsdale, NJ: Erlbaum.
  • Heathcote, A., Brown, S., & Mewhort, D. J. K. (2000). The power law repealed: The case for an exponential law of practice. Psychonomic Bulletin & Review, 7(2), 185-207.

Further references can be found in the Wikipedia articles.


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