# Fitts's law

Fitts' Law: Draft of target size W and distance to target D

Fitts's law (often cited as Fitts' law) is a predictive model of human movement primarily used in human–computer interaction and ergonomics. This scientific law predicts that the time required to rapidly move to a target area is a function of the ratio between the distance to the target and the width of the target.[1] 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.

Fitts's law has been shown to apply under a variety of conditions; with many different limbs (hands, feet,[2] the lower lip,[3] head-mounted sights,[4] eye gaze[5]), manipulanda (input devices[6]), physical environments (including underwater), and user populations (young, old, special educational needs, and drugged participants).

## Original model formulation

The original 1954 paper by Paul Morris Fitts proposed a metric to quantify the difficulty of a target selection task; the metric was based on an information analogy, where the distance to the center of the target (D) is like a signal and the tolerance or width of the target (W) is like noise. The metric is Fitts's index of difficulty (ID, in bits):

${\displaystyle {\text{ID}}=\log _{2}{\Bigg (}{\frac {2D}{W}}{\Bigg )}}$

Fitts also proposed an index of performance (IP, in bits per second) as a measure of human performance; the metric combines a task's index of difficulty (ID) with the movement time (MT, in seconds) in selecting the target. In Fitts's words, "The average rate of information generated by a series of movements is the average information per movement divided by the time per movement" (1954, p. 390). Thus,

${\displaystyle {\text{IP}}={\Bigg (}{\frac {\text{ID}}{\text{MT}}}{\Bigg )}}$

Today, IP is more commonly called throughput (TP), it is also common to include an adjustment for accuracy in the calculation.

Researchers after Fitts began the practice of building linear regression equations and examining the correlation (r) for goodness of fit; the equation expresses the relationship between MT and the D and W task parameters:

${\displaystyle {\text{MT}}=a+b\cdot {\text{ID}}=a+b\cdot \log _{2}{\Bigg (}{\frac {2D}{W}}{\Bigg )}}$
where:

• MT is the average time to complete the movement.
• a and b are constants that depend on the choice of input device and are usually determined empirically by regression analysis.
• ID is the index of difficulty.
• D is the distance from the starting point to the center of the target.
• 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 ±​W2 of the target's center.

Since shorter movement times are desirable for a given task, the value of the b parameter can be used as a metric when comparing computer pointing devices against one another; the first Human-Computer Interface application of Fitts's law was by Card, English, and Burr (1978), who used the index of performance (IP), interpreted as ​1b, to compare performance of different input devices, with the mouse coming out on top compared to the joystick or directional movement keys.[7] This early work, according to Stuart Card's biography, "was a major factor leading to the mouse's commercial introduction by Xerox".[8]

Many experiments testing Fitts's law apply the model to a dataset in which either distance or width, but not both, are varied; the model's predictive power deteriorates when both are varied over a significant range.[9] Notice that because the ID term depends only on the ratio of distance to width, the model implies that a target distance and width combination can be re-scaled arbitrarily without affecting movement time, which is impossible. Despite its flaws, this form of the model does possess remarkable predictive power across a range of computer interface modalities and motor tasks, and has provided many insights into user interface design principles.

## Bits per second: model innovations driven by information theory

The formulation of Fitts's index of difficulty most frequently used in the Human-Computer Interaction community is called the Shannon formulation:

${\displaystyle {\text{ID}}=\log _{2}{\Bigg (}{\frac {D}{W}}+1{\Bigg )}}$

This form was proposed by Scott MacKenzie,[10] professor at York University, and named for its resemblance to the Shannon–Hartley theorem.[11]

Using this form of the model, the difficulty of a pointing task was equated to a quantity of information transmitted (in units of bits) by performing the task; this was justified by the assertion that pointing reduces to an information processing task. Although no formal mathematical connection was established between Fitts's law and the Shannon-Hartley theorem it was inspired by, the Shannon form of the law has been used extensively, likely due to the appeal of quantifying motor actions using information theory. In 2002 the ISO 9241 was published, providing standards for human-computer interface testing, including the use of the Shannon form of Fitts's law, it has been shown that the information transmitted via serial keystrokes on a keyboard and the information implied by the ID for such a task are not consistent.[12]

## Adjustment for accuracy: Use of the effective target width

An important improvement to Fitts's law was proposed by Crossman in 1956 (see Welford, 1968, pp. 147–148)[13] and used by Fitts in his 1964 paper with Peterson.[14] With the adjustment, target width (W) is replaced by an effective target width (We). We is computed from the standard deviation in the selection coordinates gathered over a sequence of trials for a particular D-W condition. If the selections are logged as x coordinates along the axis of approach to the target, then

${\displaystyle W_{e}=4.133\times SD_{x}}$

This yields

${\displaystyle {\text{ID}}_{e}=\log _{2}{\Bigg (}{\frac {D}{W_{e}}}+1{\Bigg )}}$

and hence

${\displaystyle {\text{IP}}={\Bigg (}{\frac {ID_{e}}{MT}}{\Bigg )}}$

If the selection coordinates are normally distributed, We spans 96% of the distribution. If the observed error rate was 4% in the sequence of trials, then We = W. If the error rate was greater than 4%, We > W, and if the error rate was less than 4%, We < W. By using We, a Fitts' law model more closely reflects what users actually did, rather than what they were asked to do.

The main advantage in computing IP as above is that spatial variability, or accuracy, is included in the measurement. With the adjustment for accuracy, Fitts's law more truly encompasses the speed-accuracy tradeoff; the equations above appear in ISO 9241-9 as the recommended method of computing throughput.

## Welford's model: innovations driven by predictive power

Not long after the original model was proposed, a 2-factor variation was proposed under the intuition that target distance and width have separate effects on movement time. Welford's model, proposed in 1968, separated the influence of target distance and width into separate terms, and provided improved predictive power:[13]

${\displaystyle T=a+b_{1}\log _{2}(D)+b_{2}\log _{2}(W)}$

This model has an additional parameter, so its predictive accuracy cannot be directly compared with 1-factor forms of Fitts's law. However, a variation on Welford's model inspired by the Shannon formulation,

${\displaystyle T=a+b_{1}\log _{2}(D+W)+b_{2}\log _{2}(W)=a+b\log _{2}\left({\frac {D+W}{W^{k}}}\right)}$

reduces to the Shannon form when k = 1. Therefore, this model can be directly compared against the Shannon form of Fitts's law using the F-test of nested models;[15] this comparison reveals that not only does the Shannon form of Welford's model better predict movement times, but it is also more robust when control-display gain (the ratio between e.g. hand movement and cursor movement) is varied. Consequently, although the Shannon model is slightly more complex and less intuitive, it is empirically the best model to use for virtual pointing tasks.

## Extending the model from 1D to 2D and other nuances

### Extensions to two or more dimensions

In its original form, Fitts's law is meant to apply only to one-dimensional tasks. However, the original experiments required subjects to move a stylus (in three dimensions) between two metal plates on a table, termed the reciprocal tapping task; the target width perpendicular to the direction of movement was very wide to avoid it having a significant influence on performance. A major application for Fitts's law is 2D virtual pointing tasks on computer screens, in which targets have bounded sizes in both dimensions.

Fitts's law has been extended to two-dimensional tasks in two different ways. For navigating e.g. hierarchical pull-down menus, the user must generate a trajectory with the pointing device that is constrained by the menu geometry; for this application the Accot-Zhai steering law was derived.

For simply pointing to targets in a two-dimensional space, the model generally holds as-is but requires adjustments to capture target geometry and quantify targeting errors in a logically consistent way.[16][17]

### Characterizing performance

Since the a and b parameters should capture movement times over a potentially wide range of task geometries, they can serve as a performance metric for a given interface. In doing so, it is necessary to separate variation between users from variation between interfaces; the a parameter is typically positive and close to zero, and sometimes ignored in characterizing average performance. Multiple methods exist for identifying parameters from experimental data, and the choice of method is the subject of heated debate, since method variation can result in parameter differences that overwhelm underlying performance differences.[18][19]

An additional issue in characterizing performance is incorporating success rate: an aggressive user can achieve shorter movement times at the cost of experimental trials in which the target is missed. If the latter are not incorporated into the model, then average movement times can be artificially decreased.

## Temporal targets

Fitts's law deals only with targets defined in space. However, a target can be defined purely on the time axis, which is called a temporal target. A blinking target or a target moving toward a selection area are examples of temporal targets. Similar to space, the distance to the target (i.e., temporal distance Dt) and the width of the target (i.e., temporal width Wt) can be defined for temporal targets as well. The temporal distance is the amount of time a person must wait for a target to appear; the temporal width is a short duration from the moment the target appears until it disappears. For example, for a blinking target, Dt can be thought of as the period of blinking and Wt as the duration of the blinking. As with targets in space, the larger the Dt or the smaller the Wt, the more difficult it becomes to select the target.

The task of selecting the temporal target is called temporal pointing; the model for temporal pointing was first presented to Human-computer Interaction field in 2016 [20]. The model predicts the error rate, the human performance in temporal pointing, as a function of temporal index of difficulty (IDt):

${\displaystyle {\text{ID}}_{t}=\log _{2}{\Bigg (}{\frac {D_{t}}{W_{t}}}{\Bigg )}}$

## Notes

1. ^ Fitts, Paul M. (June 1954). "The information capacity of the human motor system in controlling the amplitude of movement". Journal of Experimental Psychology. 47 (6): 381–391. doi:10.1037/h0055392. PMID 13174710.
2. ^ Hoffmann, Errol R. (1991). "A comparison of hand and foot movement times". Ergonomics. 34 (4): 397–406. doi:10.1080/00140139108967324. PMID 1860460.
3. ^ Jose, Marcelo Archajo; Lopes, Roleli (2015). "Human-computer interface controlled by the lip". IEEE Journal of Biomedical and Health Informatics. 19 (1): 302–308. doi:10.1109/JBHI.2014.2305103. PMID 25561451.
4. ^ So, R. H. Y.; Griffin, M. J. (2000). "Effects of target movement direction cue on head-tracking performance". Ergonomics. 43 (3): 360–376. doi:10.1080/001401300184468. PMID 10755659.
5. ^ Zhang, Xuan; MacKenzie, I. Scott (2007). Evaluating eye tracking with ISO9241 - Part 9. Proceedings of HCII 2007. Lecture Notes in Computer Science. 4552. pp. 779–788. CiteSeerX 10.1.1.72.8113. doi:10.1007/978-3-540-73110-8_85. ISBN 978-3-540-73108-5.
6. ^ MacKenzie, I. Scott; Sellen, A.; Buxton, W. A. S. (1991). A comparison of input devices in elemental pointing and dragging tasks. Proceedings of the ACM CHI 1991 Conference on Human Factors in Computing Systems. pp. 161–166. doi:10.1145/108844.108868. ISBN 978-0897913836.
7. ^ Card, Stuart K.; English, William K.; Burr, Betty J. (1978). "Evaluation of mouse, rate-controlled isometric joystick, step keys, and text keys for text selection on a CRT" (PDF). Ergonomics. 21 (8): 601–613. CiteSeerX 10.1.1.606.2223. doi:10.1080/00140137808931762.
8. ^ "Stuart Card". PARC. Archived from the original on 2012-07-11.
9. ^ Graham, Evan (1996). Pointing on a Computer Display (Ph.D.). Simon Fraser University.
11. ^ MacKenzie, I. Scott (1992). "Fitts' law as a research and design tool in human–computer interaction" (PDF). Human–Computer Interaction. 7: 91–139. doi:10.1207/s15327051hci0701_3.
12. ^ Soukoreff, R. William; Zhao, Jian; Ren, Xiangshi (2011). "The Entropy of a Rapid Aimed Movement: Fitts' Index of Difficulty versus Shannon's Entropy". Human Computer Interaction: 222–239.
13. ^ a b Welford, A. T. (1968). Fundamentals of Skill. Methuen.
14. ^ Fitts, Paul M.; Peterson, J. R. (1964). "Information capacity of discrete motor responses". Journal of Experimental Psychology. 67 (2): 103–112. doi:10.1037/h0045689.
15. ^ Shoemaker, Garth; Tsukitani, Takayuki; Kitamura, Yoshifumi; Booth, Kellogg (December 2012). "Two-Part Models Capture the Impact of Gain on Pointing Performance". ACM Transactions on Computer-Human Interaction. 19 (4): 1–34. doi:10.1145/2395131.2395135.
16. ^ Wobbrock, J.; Shinohara, K (2011). The effects of task dimensionality, endpoint deviation, throughput calculation, and experiment design on pointing measures and models. Proceedings of the ACM Conference on Human Factors in Computing Systems. Vancouver, British Columbia. pp. 1639–1648. CiteSeerX 10.1.1.409.2785. doi:10.1145/1978942.1979181. ISBN 9781450302289.
17. ^ MacKenzie, I. Scott; Buxton, William A. S. (1992). Extending Fitts' law to two-dimensional tasks. Proceedings of the ACM CHI 1992 Conference on Human Factors in Computing Systems. pp. 219–226. doi:10.1145/142750.142794. ISBN 978-0897915137. (Subscription required (help)). Cite uses deprecated parameter |subscription= (help)
18. ^ Soukoreff, R. William; MacKenzie, I. Scott (2004). "Towards a standard for pointing device evaluation, perspectives on 27 years of Fitts' law research in HCI". International Journal of Human-Computer Studies. 61 (6): 751–789. doi:10.1016/j.ijhcs.2004.09.001.
19. ^ Zhai, Shumin (2002). On the Validity of Throughput as a Characteristic of Computer Input (pdf) (Technical report). San Jose, California: Almaden Research Center. RJ 10253.
20. ^ Lee, Byungjoo; Oulasvirta, Antti (2016). Modelling Error Rates in Temporal Pointing. Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems. CHI '16. New York, NY, USA: ACM. pp. 1857–1868. doi:10.1145/2858036.2858143. ISBN 9781450333627.

## References

• Accot, Johnny; Zhai, Shumin (2002). More than dotting the i's—foundations for crossing-based interfaces. Proceedings of ACM CHI 2002 Conference on Human Factors in Computing Systems. pp. 73–80. doi:10.1145/503376.503390. ISBN 978-1581134537.
• Accot, Johnny; Zhai, Shumin (2003). Refining Fitts' law models for bivariate pointing. Proceedings of ACM CHI 2003 Conference on Human Factors in Computing Systems. pp. 193–200. doi:10.1145/642611.642646. ISBN 978-1581136302.
• Card, Stuart K.; Moran, Thomas P.; Newell, Allen (1983). The Psychology of Human–Computer Interaction. Hillsdale, NJ: L. Erlbaum Associates. ISBN 978-0898592436.
• Fitts, Paul M.; Peterson, James R. (February 1964). "Information capacity of discrete motor responses". Journal of Experimental Psychology. 67 (2): 103–112. doi:10.1037/h0045689.