Fitts's law

Fitts's law (often cited as Fitts' law) is a model of human movement in human-computer interaction and ergonomics which predicts that the time required to rapidly move to a target area is a function of the distance to and the size of the target. Fitts's law is used to model the act of pointing, either by physically touching an object with a hand or finger, or virtually, by pointing to an object on a computer display using a pointing device. It was proposed by Paul Fitts in 1954.

Model
Fitts's law has been formulated mathematically in several different ways. One common form is the Shannon formulation (proposed by Scott MacKenzie, professor of York University, and named for its resemblance to the Shannon-Hartley theorem) for movement along a single dimension:
 * $$T = a + b \log_2 \Bigg(1+\frac{D}{W}\Bigg)$$

where:
 * $$T$$ is the average time taken to complete the movement. (Traditionally, researchers have used the symbol MT for this, to mean movement time.)
 * $$a$$ represents the start/stop time of the device and $$b$$ stands for the inherent speed of the device. These constants can be determined experimentally by fitting a straight line to measured data.
 * $$D$$ is the distance from the starting point to the center of the target. (Traditionally, researchers have used the symbol $$A$$ for this, to mean the amplitude of the movement.)
 * $$W$$ is the width of the target measured along the axis of motion. $$W$$ can also be thought of as the allowed error tolerance in the final position, since the final point of the motion must fall within ±$W/2$ of the target's center.

From the equation, we see a speed-accuracy trade off associated with pointing, whereby targets that are smaller and/or further away require more time to acquire.

Success and implications
Fitts's law is an unusually successful and well-studied model. Experiments that reproduce Fitts's results and/or that demonstrate the applicability of Fitts's law in somewhat different situations are not difficult to perform. The measured data in such experiments often fit a straight line with a correlation coefficient of 0.95 or higher, a sign that the model is very accurate.

Although Fitts only published two articles on his law (Fitts 1954, Fitts and Peterson 1964), there are hundreds of subsequent studies related to it in the human-computer interaction (HCI) literature, and quite possibly thousands of studies published in the larger psychomovement literature. The first HCI application of Fitts's law was by Card, English, and Burr (1978), who used the index of performance (IP), defined as $1/b$, to compare performance of different input devices, with the mouse coming out on top. (This early work, according to Stuart Card's biography, "was a major factor leading to the mouse's commercial introduction by Xerox" .) Fitts's law has been shown to apply under a variety of conditions, with many different limbs (hands, feet, head-mounted sights, eye gaze), manipulanda (input devices), physical environments (including underwater), and user populations (young, old, special educational needs, and drugged participants). Note that the constants a, b, IP have different values under each of these conditions.

Since the advent of graphical user interfaces, Fitts's law has been applied to tasks where the user must position a mouse cursor over an on-screen target, such as a button or other widget. Fitts's law models both point-and-click and drag-and-drop actions. Dragging has a lower IP associated with it, because the increased muscle tension makes pointing more difficult.

In its original and strictest form:
 * It applies only to movement in a single dimension and not to movement in two dimensions (though it is successfully extended to two dimensions in the Accot-Zhai steering law);
 * It describes simple motor response of, say, the human hand, failing to account for software acceleration usually implemented for a mouse cursor;
 * It describes untrained movements, not movements that are executed after months or years of practice (though some argue that Fitts's law models behaviour that is so low level that extensive training doesn't make much difference).

If, as generally claimed, the law does hold true for pointing with the mouse, some consequences for user-interface design include:
 * Buttons and other GUI controls should be a reasonable size; it is relatively difficult to click on small ones.
 * Edges and corners of the computer display (e.g., Start button in Microsoft Windows and the menus and Dock of Mac OS X) are particularly easy to acquire because the pointer remains at the screen edge regardless of how much further the mouse is moved, thus can be considered as having infinite width.
 * Similarly, top-of-screen menus (e.g., Mac OS) are easier to acquire than top-of-window menus (e.g., Windows OS).
 * Pop-up menus can usually be opened faster than pull-down menus, since the user avoids travel.
 * Pie menu items typically are selected faster and have a lower error rate than linear menu items, for two reasons: because pie menu items are all the same, small distance from the centre of the menu; and because their wedge-shaped target areas (which usually extend to the edge of the screen) are very large.

Fitts's law remains one of the few hard, reliable human-computer interaction predictive models, joined more recently by the Accot-Zhai steering law, which is derived from Fitts's law.

Mathematical details
The logarithm in Fitts's law is called the index of difficulty ID for the target, and has units of bits. We can rewrite the law as


 * $$T = a + b ID,\,$$

where


 * $$ID = \log_2 \left(\frac{D}{W}+1\right).$$

Thus, the units for b are time/bit; e.g., ms/bit. The constant a can be thought of as incorporating reaction time and/or the time required to click a button.

The values for a and b change as the conditions under which pointing is done are changed. For example, a mouse and stylus may both be used for pointing, but have different constants a and b associated with them.

An index of performance IP (also called throughput TP), in bits/time, can be defined to characterize how quickly pointing can be done, independent of the particular targets involved. There are two conventions for defining IP: one is IP = 1/b (which has the disadvantage of ignoring the effect of a), the other is IP = IDaverage/MTaverage (which has the disadvantage of depending on an arbitrarily chosen "average" ID). For a discussion of these two conventions, see Zhai (2002). Whatever definition is used, measuring the IP of different input devices allows the devices to be compared with respect to their pointing capability.

Slightly different from the Shannon formulation is the original formulation by Fitts:


 * $$ID = \log_2 \left(\frac{2D}{W}\right).$$

The factor of 2 here is not particularly important; this form of the ID can be rewritten with the factor of 2 absorbed as changes in the constants a, b. The "+1" in the Shannon form, however, does make it different from Fitts's original form, especially for low values of the ratio D/W. The Shannon form has the advantage that the ID is always non-negative, and has been shown to better fit measured data.

Derivation
Fitts's law can be derived from various models of motion. A very simple model, involving discrete, deterministic responses, is considered here. Although this model is overly simple, it provides some intuition for Fitts's law.

Assume that the user moves toward the target in a sequence of submovements. Each submovement requires a constant time t to execute, and moves a constant fraction 1-r of the remaining distance to the centre of the target, where 0 < r < 1. Thus, if the user is initially at a distance D from the target, the remaining distance after the first submovement is rD, and the remaining distance after the nth submovement is rnD. (In other words, the distance left to the target's centre is a function that decays exponentially over time.) Let N be the (possibly fractional) number of submovements required to fall within the target. Then,


 * $$r^N D = \frac{W}{2}.$$

Solving for N:



\begin{align} N & = \log_r \frac{W}{2D} \\ & = \frac{1}{\log_2 r} \log_2 \frac{W}{2D}\quad(\text{since } \log_x y = (\log_z y)/(\log_z x)) \\ & = \frac{1}{\log_2 1/r} \log_2 \frac{2D}{W}\quad(\text{since } \log_x y = - \log_x(1/y)). \end{align} $$

The time required for all submovements is:



\begin{align} T = Nt & = \frac{t}{\log_2 1/r} \log_2 \frac{2D}{W} \\ & = \frac{t}{\log_2 1/r} + \frac{t}{\log_2 1/r} \log_2 \frac{D}{W}. \end{align} $$

By defining appropriate constants a and b, this can be rewritten as


 * $$T = a + b \log_2 \frac{D}{W}.$$

The above derivation is similar to one given in Card, Moran, and Newell (1983). For a critique of the deterministic iterative-corrections model, see Meyer et al. (1990).