Fitts' law

In ergonomics, Fitts' law (or Fitts's law) is a principle of human movement published in 1954 by Paul Fitts which predicts the time required to move from a starting position to a final target area. The kind of motion it describes is aimed and rapid. The time needed to acquire a target is a function of the distance to the target, and the size of the target. Fitt's law is used to model the act of pointing, both in the real world, e.g. with a hand or finger, and on a computer, e.g. with a mouse.

Table of contents
1 The model
2 Some mathematical details
3 Success and implications of Fitts' law
4 A derivation of Fitts' law
5 References
6 External links

The model

Mathematically, Fitts' law has been formulated a few different ways. One common form is the Shannon formulation (due to its resemblance to Shannon's theorem), which, for movement along a single dimension, states

where

  • T is the average time taken to complete the movement
  • a and b are empirical constants, and can be determined by fitting a straight line to measured data
  • D is the distance from the starting point to the center of the target. Some authors instead use A, for 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 centre.

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

Some mathematical details

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

Thus the units for b are time/bit, e.g. milliseconds/bit. The reciprocal of b is the index of performance IP = 1/b, with units bits/time.

The values for a, b, and IP 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, b, IP associated with them. Since IP is the reciprocal of the slope of the line T = a+b ID, it follows that IP is a measure of how quickly pointing tasks can be completed (in bits/time), independent of the particular targets involved.

Slightly different from Shannon's formulation is the original formulation by Fitts

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' original form. The Shannon form has the advantage that the ID is always non-negative, and has been shown to better fit measured data.

Success and implications of Fitts' law

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

Although Fitts himself only published two articles on his law (Fitts 1954, Fitts and Peterson 1964), there have been 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 application of Fitts' law to HCI was by Card et al. (1978), who used the index of performance IP to compare the performance of different input devices. Fitts' law has been shown to apply under a variety of conditions, with many different limbs (hands, feet, head-mounted sights, eye gaze), manipulanda (i.e. input devices), physical environments (including underwater!), and user populations (young, old, retarded, and drugged subjects). Note that the constants a, b, IP have different values under each of these conditions.

Since the advent of graphical user interfaces, Fitts' law has been applied to tasks where the user must position the mouse pointer over an on-screen target, such as a button or other widget. Fitts' law can be used to model both point-and-click and drag-and-drop actions (note that dragging has a lower IP associated with it, because the increased muscle tension makes pointing more difficult). Despite the appeal of the model, it should be remembered that in its original and strictest form:

  • it applies only to movement in a single dimension and not to movement in two dimensions (though it has been extended, more or less successfully, to two dimensions)
  • it describes the simple motor response of, say, the human hand, and does not account for the software acceleration usually implemented for a mouse pointer
  • it describes untrained movements, not movements that are executed after months or years of practice (though some argue that Fitts' 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 are
  • buttons and other widgetss to be selected in GUIs should not be made too small, as this would make them very difficult to click on
  • the edges (e.g. menubar in Mac OS) and corners of the computer display are particularly easy for a user to acquire: because the pointer remains at the edge of the screen no matter how much further the mouse is moved, objects at these positions can be considered as having infinite width
  • popup menuss can usually be opened faster than pull-down menuss, since the user does not have to travel at all before popping up a popup menu
  • items in a pie menu can often be selected faster than items in a linear menu, because in the former, the items are all the same, small distance from the centre of the menu

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

See also Hick's law

A derivation of Fitts' law

Fitts' 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 simplistic, it provides some intuition for Fitts' 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,

Solving for N,

N
(because logxy = (logzy)/(logzx))
(because logxy = - logx1/y)

The time required for all submovements is

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

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).

References

External links


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