Fitts' law
Encyclopedia
Fitts's law is a model of human movement primarily used in human–computer interaction
and ergonomics
that predicts that the time required to rapidly move to a target area is a function of the distance to the target 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 monitor using a pointing device
. It was proposed by Paul Fitts
in 1954.
, and named for its resemblance to the Shannon–Hartley theorem
) for movement along a single dimension:
where:
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.
of .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 , to compare performance of different input device
s, 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 interface
s, 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. Fitts's law has also been shown to model target-directed hand and head motions in a virtual environment.
In its original and strictest form:
If, as generally claimed, the law does hold true for pointing with the mouse, some consequences for user interface
design include:
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.
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
where
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 = IDavg/MTavg (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:
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.
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 . 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:
Human–computer interaction
Human–computer Interaction is the study, planning, and design of the interaction between people and computers. It is often regarded as the intersection of computer science, behavioral sciences, design and several other fields of study...
and ergonomics
Ergonomics
Ergonomics is the study of designing equipment and devices that fit the human body, its movements, and its cognitive abilities.The International Ergonomics Association defines ergonomics as follows:...
that predicts that the time required to rapidly move to a target area is a function of the distance to the target and the size of the target. Fitts's law is used to model
Conceptual model
In the most general sense, a model is anything used in any way to represent anything else. Some models are physical objects, for instance, a toy model which may be assembled, and may even be made to work like the object it represents. They are used to help us know and understand the subject matter...
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 monitor using a pointing device
Pointing device
A pointing device is an input interface that allows a user to input spatial data to a computer...
. It was proposed by Paul Fitts
Paul Fitts
Paul M. Fitts was a psychologist at Ohio State University . He developed a model of human movement, Fitts's law, based on rapid, aimed movement, which went on to become one of the most highly successful and well studied mathematical models of human motion...
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 UniversityYork University
York University is a public research university in Toronto, Ontario, Canada. It is Canada's third-largest university, Ontario's second-largest graduate school, and Canada's leading interdisciplinary university....
, and named for its resemblance to the Shannon–Hartley theorem
Shannon–Hartley theorem
In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy channel coding theorem to the archetypal case of a continuous-time...
) for movement along a single dimension:
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 (intercept) and
- b stands for the inherent speed of the device (slope). 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 ± 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 coefficientPearson product-moment correlation coefficient
In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
of .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
Human–computer interaction
Human–computer Interaction is the study, planning, and design of the interaction between people and computers. It is often regarded as the intersection of computer science, behavioral sciences, design and several other fields of study...
(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 , to compare performance of different input device
Input device
In computing, an input device is any peripheral used to provide data and control signals to an information processing system such as a computer or other information appliance...
s, with the mouse
Mouse (computing)
In computing, a mouse is a pointing device that functions by detecting two-dimensional motion relative to its supporting surface. Physically, a mouse consists of an object held under one of the user's hands, with one or more buttons...
coming out on top. This early work, according to Stuart Card
Stuart Card
Stuart K. Card is an American researcher and Senior Research Fellow at Xerox PARC. He is considered to be one of the pioneers of applying human factors in human–computer interaction.- Biography :...
's biography, "was a major factor leading to the mouse's commercial introduction by Xerox
Xerox
Xerox Corporation is an American multinational document management corporation that produced and sells a range of color and black-and-white printers, multifunction systems, photo copiers, digital production printing presses, and related consulting services and supplies...
".
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 interface
Graphical user interface
In computing, a graphical user interface is a type of user interface that allows users to interact with electronic devices with images rather than text commands. GUIs can be used in computers, hand-held devices such as MP3 players, portable media players or gaming devices, household appliances and...
s, Fitts's law has been applied to tasks where the user must position a mouse cursor
Cursor (computers)
In computing, a cursor is an indicator used to show the position on a computer monitor or other display device that will respond to input from a text input or pointing device. The flashing text cursor may be referred to as a caret in some cases...
over an on-screen target, such as a button or other widget. Fitts's law models both point-and-click
Point-and-click
Point-and-click is the action of a computer user moving a cursor to a certain location on a screen and then pressing a mouse button, usually the left button , or other pointing device...
and drag-and-drop
Drag-and-drop
In computer graphical user interfaces, drag-and-drop is the action of selecting a virtual object by "grabbing" it and dragging it to a different location or onto another virtual object...
actions. Dragging has a lower IP associated with it, because the increased muscle tension makes pointing more difficult. Fitts's law has also been shown to model target-directed hand and head motions in a virtual environment.
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
User interface
The user interface, in the industrial design field of human–machine interaction, is the space where interaction between humans and machines occurs. The goal of interaction between a human and a machine at the user interface is effective operation and control of the machine, and feedback from the...
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 monitor (e.g., the location of the Start buttonStart menuThe Start Menu and Start Button are user interface elements used in the later versions of the Microsoft Windows operating systems and in some X window managers...
in Microsoft WindowsMicrosoft WindowsMicrosoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...
and the menus and DockDock (computing)The Dock is a prominent feature of the graphical user interface of the Mac OS X operating system. It is used to launch applications and switch between running applications...
of Mac OS X) are particularly easy to acquire with a mouse, touchpad or trackball 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. This doesn't apply to touchscreens, though. - Similarly, top-of-screen menus (e.g., Mac OS) are sometimes easier to acquire than top-of-window menus (e.g., Windows OS).
- Pop-up menuMenu (computing)In computing and telecommunications, a menu is a list of commands presented to an operator by a computer or communications system. A menu is used in contrast to a command-line interface, where instructions to the computer are given in the form of commands .Choices given from a menu may be selected...
s can usually be opened faster than pull-down menus, since the user avoids travel: the pop-up appears at the current cursor position. - Pie menuPie menuIn computer interface design, a pie menu is a circular context menu where selection depends on direction. A pie menu is made of several "pie slices" around an inactive center and works best with stylus input, and well with a mouse...
items typically are selected faster and have a lower error rate than linear menuMenu (computing)In computing and telecommunications, a menu is a list of commands presented to an operator by a computer or communications system. A menu is used in contrast to a command-line interface, where instructions to the computer are given in the form of commands .Choices given from a menu may be selected...
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
Accot-Zhai steering law
The steering law in human–computer interaction and ergonomics is a predictive model of human movement that describes the time required to navigate, or steer, through a 2-dimensional tunnel. The tunnel can be thought of as a path or trajectory on a plane that has an associated thickness or width,...
, which is derived from Fitts's law.
Mathematical details
The logarithmLogarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
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
where
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
Stylus
A stylus is a writing utensil, or a small tool for some other form of marking or shaping, for example in pottery. The word is also used for a computer accessory . It usually refers to a narrow elongated staff, similar to a modern ballpoint pen. Many styli are heavily curved to be held more easily...
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 = IDavg/MTavg (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:
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 . 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:
-
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) .
See also
- Hick's lawHick's lawHick's Law, named after British psychologist William Edmund Hick, or the Hick–Hyman Law , describes the time it takes for a person to make a decision as a result of the possible choices he or she has. The Hick-Hyman Law assesses cognitive information capacity in choice reaction experiments...
- Point-and-clickPoint-and-clickPoint-and-click is the action of a computer user moving a cursor to a certain location on a screen and then pressing a mouse button, usually the left button , or other pointing device...
- Accot-Zhai steering lawAccot-Zhai steering lawThe steering law in human–computer interaction and ergonomics is a predictive model of human movement that describes the time required to navigate, or steer, through a 2-dimensional tunnel. The tunnel can be thought of as a path or trajectory on a plane that has an associated thickness or width,...
- Crossing-based interface
External links
- Peer-reviewed Encyclopedia entry on Fitts' Law by Mehmet Göktürk
- Fitts' Law at AskTog
- A Quiz Designed to Give You Fitts at AskTog
- Fitts' Law at CS Dept. NSF-Supported Education Infrastructure Project
- Fitts’ Law: Modeling Movement Time in HCI
- Bibliography of Fitts’ Law Research compiled by I. Scott MacKenzie
- Fitts' Law in Microsoft Office User Interface by Jensen Harris
- Visualizing Fitts's Law by Kevin Hale
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