Fitts's law: a log-linear law and the search for the theory under it

An interface review produces the sentence "that close button is too small, and too far from where the hand already is". That is a claim about acquisition difficulty. Since 1954 there has been an equation for it: the time to reach a target grows linearly with an index of difficulty in bits.

This unit argues that the law is two claims inside one equation. The first is an ordering of tasks by difficulty, which replicates widely. The second is an account of that ordering as information transmitted through a channel of fixed rate, which does not. The derivation is an approximation valid where the signal-to-noise ratio is large, and pointing at an easy target is where it is not. Use the law to rank designs, not to explain why the ordering holds, and not to quote an absolute time as a person's measurement. Unit 7 is the other information-theoretic law, and it does not survive the same test.

The law is two claims wearing one equation

Fitts (1954) had participants tap between targets in reciprocal tapping and transfer tasks, and found response duration proportional to the minimum information needed for it, an index of difficulty built from the logarithm of the ratio of amplitude to target width. The mean index of performance was 10.10 bits per second, over indices of difficulty from 1 to 7 bits and movement times from 180 to 731 milliseconds.

Card, English and Burr (1978) evaluated four devices for selecting text on a screen: the mouse was fastest with the lowest error rate, and the law accounted for positioning time for the mouse and joystick, capturing more than 80 per cent of movement time variation (MacKenzie, 1992).

Only the first of the two claims is robust.

The derivation is an approximation, valid only where the signal-to-noise ratio is high

Fitts derived the relation by analogy. MacKenzie (1992) records that the model descends from Shannon's Theorem 17, which gives a channel capacity as bandwidth times the logarithm of signal plus noise over noise, with movement amplitude standing in for signal and target width for noise. MacKenzie (1989) showed that Fitts cited an approximation of that theorem, introduced with the caution that it is useful only if the signal-to-noise ratio is large. The exact adaptation gives the logarithm of amplitude over width plus one. The factor of two inside Fitts's logarithm kept the index above zero.

So the fidelity of the analogy varies with the task. Where the target is small and distant, the ratio of amplitude to width is large and the approximation is sound; where the target is large and close, that ratio sits near the bottom of its range and the law extrapolates into a region its derivation excluded, with Fitts's experiments running it as low as 1 to 11 (MacKenzie, 1992). The channel reading does not transfer to people either, since human coding is ill-defined and often unpredictable.

A design review does the arithmetic on a real widget. A 40-pixel button 400 pixels from where the hand rests gives an index of the base-2 logarithm of twice the amplitude over the width, the logarithm of 800 over 40, or 4.32 bits, which at the survey rate of 10.1 bits per second predicts about 428 milliseconds. Halving the width raises the index to 5.32 bits, and doubling the distance raises it by exactly one bit. The same equation on a 60-pixel target 60 pixels away gives one bit, the condition where the fitted rate falls to 5.56 bits per second, 3.4 standard deviations below Fitts's mean. The prediction is least trustworthy precisely where the arithmetic is easiest.

Where the fit fails, and the repair that keeps the law

MacKenzie (1992) notes that plots of movement time against difficulty curve upward away from the fitted line for the easy conditions. A law accurate except where the target is large and near still ranks designs, but its constant is not a capacity.

The repair redefines the task inputs rather than the law. Soukoreff and MacKenzie (2004) recommend the Shannon formulation of the index and the effective target width, 4.133 times the standard deviation of the observed endpoints, which gives 96 per cent of hits inside the target. Recomputing Fitts's first experiment with both changes raises the correlation with movement time from 0.9831 to 0.9937 (MacKenzie, 1992). The improvement comes from moving the inputs rather than from gaining a mechanism, so a better fit is not evidence for the channel.

The rate is not one number

MacKenzie's (1992) survey of six device studies shows the cost of treating the rate as a capacity. Across them the indices of performance range from 1.1 to 13.7 bits per second and the intercepts from minus 880 to 1030 milliseconds. The mouse is reported at 10.4 bits per second in one study and 2.6 in another, and one joystick appears at 4.5, 1.2 and 2.2 bits per second across three studies. The rate is not one number across people either: the coefficients were fitted on participants who were mostly young and able-bodied, and Xiao and colleagues (2024) record that studies of children, older and visually impaired users rest on insufficient samples. A target comfortable for a young adult can sit beyond the precision of a hand tremor, so the repair is to change the interface rather than the user.

Zhai (2004) presses the point, arguing that throughput mixes the informational and non-informational aspects of pointing; Soukoreff and MacKenzie (2004) codify it as the mean of the per-condition means and record that Zhai proposed a different calculation in the same issue. When the headline statistic's definition is contested within one issue, it is a protocol, not a constant.

The logarithm can be derived without a channel

MacKenzie (1992) states the position this unit takes: the law remains an analogy waiting for a theory. Meyer, Abrams, Kornblum, Wright and Smith (1988) built a stochastic optimised-submovement model in which the logarithmic trade-off follows from minimising expected movement time under noisy force production, not from information theory.

But the candidate is one of several: Plamondon and Alimi (1997) argue that the number of interpretations proposed calls into question the ability of many models to explain the observations consistently.

The submovement account makes a prediction the channel reading does not. Where the target is wide relative to the distance, the model expects a single ballistic movement; where it is narrow, an initial movement that undershoots, followed by corrections, with the logarithm emerging from the correction count rather than from a channel. The number of submovements is visible in the velocity profile, so the account is testable in a way the information-theoretic reading is not: whether the count rises as the target narrows is the experiment that would choose between them.

Contested question: is the touch repair an improvement, and is throughput information?

The first disagreement is whether the touchscreen repair is an improvement or a local patch. Bi, Li and Zhai (2013) propose FFitts's law, in which finger-touch endpoint variability combines a component governed by the speed-accuracy trade-off with an absolute precision of the finger, reporting an R-squared of 0.91 or better. The evidence is small-scale and unevenly sourced: each result rests on one dataset or one laboratory session, with the positive results from the model's own authors and the negative ones from independent groups. Woodward, Cato, Smith, Wang and Benda (2020) put the same question to 54 children aged 5 to 10 and found the standard law with nominal target widths more accurate. Ko, Zhao, Ramakrishnan, Zhai and Bi (2021) used two touch datasets, asking whether the model should use nominal or effective width and whether its ambiguity term should be fixed or free: the better width choice varies between cases, while the free parameter always fitted better.

Xiao and colleagues (2024) recommend the touch model for small targets on small touchscreens, reporting that the conventional law becomes less accurate exactly there. Their review also contains an anomaly worth recording: it describes Woodward's result as the Shannon formulation outperforming the conventional law, whereas Woodward's own abstract reports nominal-width Fitts's law against FFitts's law. What would settle the question is a preregistered comparison across datasets that fixes the model-comparison criterion in advance, fits nominal and effective widths and fixed and free ambiguity terms on the same data, and reports which model was rejected. Meanwhile, use the law to rank designs within your own device and population, and add the finger-ambiguity term only when your data prefer it.

The second disagreement is whether throughput measures information at all. Zhai (2004) argues that it does not, while Soukoreff and MacKenzie (2004) treat it as the field's standard summary and specify how to compute it. One is a definitional argument about what the statistic measures, the other a protocol for computing it, and neither is an experiment. Roig-Maimo, Mas-Sanso and MacKenzie (2025) show that the protocol is not even applied consistently. The targets of the ISO 9241-411 multi-directional tapping task can be equally spaced only when their number is odd, and about a third of 200 studies using the task chose an even number and so computed throughput from the wrong movement distance. What would settle it is a demonstration that throughput predicts performance outside the conditions it was measured under.

Where the argument is heading

The direction of travel is to look for the missing mechanism rather than another index. Trendafilov and Polani (2019) propose empowerment, a task-independent utility capturing uncertainty in the perception-action loop including noise and delay, simulated in a Fitts-type task. Xiao and colleagues (2024) map the other gap: two-dimensional formulations have reached consensus, three-dimensional work diverges, and no accepted way exists to compare the competing indices.

The sources this unit rests on

• Fitts, P. M. (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 Ran reciprocal tapping and transfer tasks and found response duration proportional to minimum information. The source of both the ordering kept and the channel questioned.
• Card, S. K., English, W. K. and Burr, B. J. (1978). Evaluation of mouse, rate-controlled isometric joystick, step keys and text keys for text selection on a CRT. Ergonomics, 21(8), 601-613. doi:10.1080/00140137808931762 Evaluated four devices and found the mouse fastest with the lowest error rate. Origin of the law as a design instrument.
• Meyer, D. E., Abrams, R. A., Kornblum, S., Wright, C. E. and Smith, J. E. K. (1988). Optimality in human motor performance: ideal control of rapid aimed movements. Psychological Review, 95(3), 340-370. doi:10.1037/0033-295X.95.3.340 Derives the logarithmic trade-off from minimising expected movement time under noisy force production. The clearest alternative derivation.
• MacKenzie, I. S. (1989). A note on the information-theoretic basis for Fitts' law. Journal of Motor Behaviour, 21(3), 323-330. doi:10.1080/00222895.1989.10735486 Shows that Fitts's index departs from the theorem he invoked and that an exact adaptation fits better. The repair is to the derivation, where this unit's thesis bites.
• MacKenzie, I. S. (1992). Fitts' law as a research and design tool in human-computer interaction. Human-Computer Interaction, 7(1), 91-139. doi:10.1207/s15327051hci0701_3 Reanalyses Fitts's data, reports the deviation at low difficulty, and reviews six device studies whose rates disagree widely. It carries the unit's numbers, the index repair, and the judgement that the law waits for a theory.
• Plamondon, R. and Alimi, A. M. (1997). Speed/accuracy trade-offs in target-directed movements. Behavioural and Brain Sciences, 20(2), 279-303. doi:10.1017/S0140525X97001441 Argues that the many interpretations proposed call into question the ability of models to explain the observations consistently. The strongest statement that the competitors are numerous.
• Soukoreff, R. W. and MacKenzie, I. S. (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 Recommends the Shannon index, the effective width of 4.133 standard deviations and stated trial counts, converting the law into a protocol.
• Zhai, S. (2004). Characterising computer input with Fitts' law parameters: the information and non-information aspects of pointing. International Journal of Human-Computer Studies, 61(6), 791-809. doi:10.1016/j.ijhcs.2004.09.006 Argues that throughput conflates the information rate with non-informational properties of device and task, which bears on whether the field's headline statistic is information at all.
• Bi, X., Li, Y. and Zhai, S. (2013). FFitts law: modelling finger touch with Fitts' law. Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). doi:10.1145/2470654.2466180 Derives a dual-distribution model for finger touch that predicts better than Fitts's law at an R-squared of 0.91 or above: the best-documented repair of the law by adding a term.
• Woodward, J., Cato, J., Smith, J., Wang, I. and Benda, B. (2020). Examining Fitts' and FFitts' law models for children's pointing tasks on touchscreens. Proceedings of the International Conference on Advanced Visual Interfaces (AVI). doi:10.1145/3399715.3399844 Had 54 children aged 5 to 10 do a two-dimensional touchscreen task, where the standard law with nominal widths fit better, challenging the repair.
• Ko, Y.-J., Zhao, H., Ramakrishnan, I., Zhai, S. and Bi, X. (2021). Issues related to using Finger-Fitts law to model one-dimensional touch pointing tasks. Proceedings of the International Symposium of Chinese CHI. doi:10.1145/3490355.3490360 Tests two touch datasets for the width choice and the ambiguity term, finding the better choice varies by case, so the repair is not stable across datasets.
• Trendafilov, D. and Polani, D. (2019). Information-theoretic sensorimotor foundations of Fitts' law. Extended Abstracts of the CHI Conference on Human Factors in Computing Systems. doi:10.1145/3290607.3313053 Proposes empowerment, a utility capturing noise and delay in the perception-action loop, simulated in a Fitts-type task; that it remains a candidate shows the question is open.
• Xiao, H., Sun, Y., Duan, Z., Huo, Y., Liu, J., Luo, M., Li, Y. and Zhang, Y. (2024). A study of model iterations of Fitts' law and its application to human-computer interactions. Applied Sciences, 14(16), 7386. doi:10.3390/app14167386 Reviews the law across touchscreens, virtual reality and vulnerable populations, reporting consensus in two dimensions, divergence in three, and insufficient samples for children, older and visually impaired users.
• Roig-Maimo, M. F., Mas-Sanso, R. and MacKenzie, I. S. (2025). Design of 2D Fitts' law experiments: an odd thing about targets. Applied Ergonomics, 131, 104680. doi:10.1016/j.apergo.2025.104680 Shows that the ISO 9241-411 multi-directional tapping task can space its targets equally only when their number is odd, and that about a third of 200 studies using the task chose an even number and so computed throughput from the wrong distance.