Two Laws Designers Quote and Misuse

The apparatus was ten pea lamps arranged in an irregular circle, deliberately irregular so that the subject would not need to move his eyes, and ten Morse keys with the fingers resting on them. A punched tape fed stimuli in irregular sequences at roughly five second intervals. In Experiment I of On the Rate of Gain of Information, the subject was the experimenter himself, W. E. Hick, who had accumulated over 8,000 practice reactions before he started collecting data. Experiment II had two subjects.

That is the empirical base of the law a design system deck will cite this week to argue for removing two items from a navigation bar. It is worth knowing what the experiment did and did not do before borrowing its authority.

Hick 1952: what was actually measured

The paper appeared in the Quarterly Journal of Experimental Psychology, volume 4, issue 1, 1952, pages 11 to 26. The number of alternatives ran from two to ten, in sequences of 100 to 200 stimuli. In Experiment I the instruction was to perform without errors. In Experiment II the instruction was inverted: the subject was "asked to try reacting quickly, with as many errors as he liked", which is where the speed and accuracy trade was deliberately opened up.

Hick reported reaction time as a function of the logarithm of the number of alternatives plus one, and estimated a rate of gain of information of roughly five bits per second. The reason for the plus one is a detail almost nobody repeats, and it is the most interesting thing in the paper: Hick reasoned that during the waiting interval the subject treats "no stimulus yet" as one more equiprobable possibility. In his words, "the missing portion is due to the possibility of 'no stimulus' at any instant."

Three conditions are built into that design. The alternatives are known in advance. They are equiprobable. Each one maps to a single dedicated response the finger is already resting on. Remove any of the three and you are no longer running Hick's experiment.

Hyman 1953: why the law is not about counting items

Ray Hyman's "Stimulus information as a determinant of reaction time", Journal of Experimental Psychology volume 45, number 3, 1953, generalised the finding, and in doing so proved that the number of alternatives was never the causal variable. As Robert Proctor and Darryl Schneider summarise in their review, Hyman manipulated information three different ways: "(a) varying set size, (b) altering the amount of information conveyed by a stimulus through varying the probability of occurrence of particular S-R pairs and (c) instituting sequential dependencies between successive trials." The result was that "when the data from these three experiments were plotted as a function of bits of information transmitted...each subject's results could be fit with a single function."

So the quantity that predicts reaction time is transmitted information, not item count. Eight equiprobable options and eight options where one is chosen 90 per cent of the time are the same count and very different tasks. Every real interface is the second case.

Fitts 1954: what was actually measured

Paul M. Fitts published "The information capacity of the human motor system in controlling the amplitude of movement" in the Journal of Experimental Psychology, volume 47, number 6, 1954, pages 381 to 391. It reports four experiments: reciprocal tapping with a one ounce stylus, reciprocal tapping with a one pound stylus, a disc transfer task and a pin transfer task. In the tapping experiments, subjects moved a stylus back and forth between two metal plates and tapped them at their centres.

In I. Scott MacKenzie's reanalysis, the one ounce condition used amplitudes of 2, 4, 8 and 16 inches and target widths of 0.25, 0.5, 1 and 2 inches, giving sixteen conditions with indices of difficulty from 1 to 7 bits. Mean movement times ranged from 180 to 731 milliseconds, each score derived from over 600 observations, and the correlation between movement time and index of difficulty was r =.9831. Mean index of performance for the one ounce stylus was 10.10 bits per second, with a standard deviation of 1.33.

Note what the task is. A back-and-forth, self-paced, one dimensional aimed movement with a physical stylus against a physical plate, performed until the subject is practised, with an error rate held near a constant. MacKenzie's summary of the scope is unambiguous: "Fitts' law is a model for rapid, aimed movements, and the presence of a nominal yet consistent error rate in subjects' behaviour is assumed and arguably vital."

The formulae

HICK 1952, as reported

RT = 0.518 * log10(n + 1)       seconds, Experiment I
n = number of equiprobable alternatives
the +1 accounts for the "no stimulus yet" state during waiting

HICK-HYMAN, modern statement (Proctor & Schneider 2018)

RT = a + b * log2(n)
and more generally
RT = a + b * H
H = - SUM( p_i * log2(p_i) )    bits of transmitted information
Hyman showed set size, probability and sequential dependency
all collapse onto one function when plotted against H

FITTS 1954, original formulation

ID = log2(2A / W)               bits, index of difficulty
MT = a + b * ID                 movement time
A = amplitude (distance to target centre)
W = target width along the axis of motion

WELFORD 1960 variant

MT = a + b * log2(A / W + 0.5)

SHANNON FORMULATION (MacKenzie 1989, 1992)

ID = log2(A / W + 1)            cannot go negative
IP = ID / MT                    bits per second, throughput

derived from Shannon's Theorem 17:
C = B * log2((P + N) / N)
Fitts had used Goldman's approximation:
C = B * log2(P / N)

EFFECTIVE WIDTH CORRECTION (required for honest comparison)

We = 4.133 * SDx                SDx = s.d. of endpoint
                                coordinates
We = W * (2.066 / z)            when the error rate is not 4%
the model assumes a normal spread of hits with about 96 per
cent inside the target and 4 per cent outside

FFITTS, FOR FINGER TOUCH (Bi, Li & Zhai 2013)

T = a + b * log2( A / sqrt(2*pi*e*(sigma^2 - sigma_a^2)) + 1 )
sigma_a = 0.94 mm               one-dimensional tasks
sigma_a = 1.50 mm               two-dimensional tasks
sigma_a is the finger's absolute precision limit and does not
shrink when the user slows down

HICK 1952, as reported

RT = 0.518 * log10(n + 1)       seconds, Experiment I
n = number of equiprobable alternatives
the +1 accounts for the "no stimulus yet" state during waiting

HICK-HYMAN, modern statement (Proctor & Schneider 2018)

RT = a + b * log2(n)
and more generally
RT = a + b * H
H = - SUM( p_i * log2(p_i) )    bits of transmitted information
Hyman showed set size, probability and sequential dependency
all collapse onto one function when plotted against H

FITTS 1954, original formulation

ID = log2(2A / W)               bits, index of difficulty
MT = a + b * ID                 movement time
A = amplitude (distance to target centre)
W = target width along the axis of motion

WELFORD 1960 variant

MT = a + b * log2(A / W + 0.5)

SHANNON FORMULATION (MacKenzie 1989, 1992)

ID = log2(A / W + 1)            cannot go negative
IP = ID / MT                    bits per second, throughput

derived from Shannon's Theorem 17:
C = B * log2((P + N) / N)
Fitts had used Goldman's approximation:
C = B * log2(P / N)

EFFECTIVE WIDTH CORRECTION (required for honest comparison)

We = 4.133 * SDx                SDx = s.d. of endpoint
                                coordinates
We = W * (2.066 / z)            when the error rate is not 4%
the model assumes a normal spread of hits with about 96 per
cent inside the target and 4 per cent outside

FFITTS, FOR FINGER TOUCH (Bi, Li & Zhai 2013)

T = a + b * log2( A / sqrt(2*pi*e*(sigma^2 - sigma_a^2)) + 1 )
sigma_a = 0.94 mm               one-dimensional tasks
sigma_a = 1.50 mm               two-dimensional tasks
sigma_a is the finger's absolute precision limit and does not
shrink when the user slows down

HICK 1952, as reported

RT = 0.518 * log10(n + 1)       seconds, Experiment I
n = number of equiprobable alternatives
the +1 accounts for the "no stimulus yet" state during waiting

HICK-HYMAN, modern statement (Proctor & Schneider 2018)

RT = a + b * log2(n)
and more generally
RT = a + b * H
H = - SUM( p_i * log2(p_i) )    bits of transmitted information
Hyman showed set size, probability and sequential dependency
all collapse onto one function when plotted against H

FITTS 1954, original formulation

ID = log2(2A / W)               bits, index of difficulty
MT = a + b * ID                 movement time
A = amplitude (distance to target centre)
W = target width along the axis of motion

WELFORD 1960 variant

MT = a + b * log2(A / W + 0.5)

SHANNON FORMULATION (MacKenzie 1989, 1992)

ID = log2(A / W + 1)            cannot go negative
IP = ID / MT                    bits per second, throughput

derived from Shannon's Theorem 17:
C = B * log2((P + N) / N)
Fitts had used Goldman's approximation:
C = B * log2(P / N)

EFFECTIVE WIDTH CORRECTION (required for honest comparison)

We = 4.133 * SDx                SDx = s.d. of endpoint
                                coordinates
We = W * (2.066 / z)            when the error rate is not 4%
the model assumes a normal spread of hits with about 96 per
cent inside the target and 4 per cent outside

FFITTS, FOR FINGER TOUCH (Bi, Li & Zhai 2013)

T = a + b * log2( A / sqrt(2*pi*e*(sigma^2 - sigma_a^2)) + 1 )
sigma_a = 0.94 mm               one-dimensional tasks
sigma_a = 1.50 mm               two-dimensional tasks
sigma_a is the finger's absolute precision limit and does not
shrink when the user slows down

The Shannon correction is not pedantry. MacKenzie's 1989 note in the Journal of Motor Behavior shows that Fitts derived his index from an approximation of Shannon's theorem "originally introduced with the caution that it is useful only if the signal-to-noise ratio is large". With the original formulation, a target wider than twice the distance to it yields a negative index of difficulty, which is meaningless. The Shannon form cannot go negative and fits slightly better: reanalysing Fitts's own data, MacKenzie reports the correlation rising from.9831 to.9904 with the effective width adjustment, and to.9937 with the Shannon form as well.

Four misapplications

1. Hick's law applied to a menu that is being read

This is the common one, and it has been formally disproved. Andy Cockburn, Carl Gutwin and Saul Greenberg's "A Predictive Model of Menu Performance", Proceedings of CHI 2007, pages 472 to 481, models menu selection as two different tasks depending on experience. A first-time user must visually search. An experienced user can go straight to a remembered location and only has to decide. The authors are explicit that earlier modelling attempts "failed because it used the Hick-Hyman Law to predict visual search time, which is not modelled by the law".

"Hick and Hyman's original experiments timed decision time in response to clear visual stimuli."

Their calibrated components make the difference concrete:

NOVICE, VISUAL SEARCH: linear in the number of items

    Tvs = 0.08 * n + 0.30 seconds

EXPERT, HICK-HYMAN DECISION: logarithmic in item probability

    Thh = 0.08 * H + 0.24 seconds,  H = log2(1 / p)

POINTING, IN BOTH CASES

    Tp = 0.37 + 0.13 * ID seconds,  ID = log2(A / W + 1)

NOVICE, VISUAL SEARCH: linear in the number of items

    Tvs = 0.08 * n + 0.30 seconds

EXPERT, HICK-HYMAN DECISION: logarithmic in item probability

    Thh = 0.08 * H + 0.24 seconds,  H = log2(1 / p)

POINTING, IN BOTH CASES

    Tp = 0.37 + 0.13 * ID seconds,  ID = log2(A / W + 1)

NOVICE, VISUAL SEARCH: linear in the number of items

    Tvs = 0.08 * n + 0.30 seconds

EXPERT, HICK-HYMAN DECISION: logarithmic in item probability

    Thh = 0.08 * H + 0.24 seconds,  H = log2(1 / p)

POINTING, IN BOTH CASES

    Tp = 0.37 + 0.13 * ID seconds,  ID = log2(A / W + 1)

Linear against logarithmic is not a rounding difference. Going from five items to ten adds about 0.4 seconds of search for the novice and about one bit, roughly 0.08 seconds, of decision for the expert. Cite Hick's law for a nav bar a visitor has never seen, and you are using a logarithmic model for a linear process, which systematically understates the cost of the items you added.

2. Hick's law applied where the slope has already collapsed

Proctor and Schneider's review, Quarterly Journal of Experimental Psychology volume 71, issue 6, 2018, pages 1281 to 1299, catalogues the boundary conditions. Three of them matter for interface work.

Compatibility: "It is possible for set size to have little or no effect when compatibility is high." A row of controls in a fixed spatial order, each mapped to one obvious action, is a high-compatibility task.

Practice: "The slope reduces with practice", and "with sufficient practice, RT might become independent of set size". In one illustration they cite, after extensive practice on an eight-choice task the difference in mean reaction time between eight and two alternatives was only 149 milliseconds.

Range: they cite Seibel's 1963 suggestion that the linear relation between reaction time and log set size "held in the range of two to eight alternatives, but not beyond that range".

So invoking Hick's law to cut a daily-use toolbar from nine items to six, for a population that has used it for two years, is applying a model outside all three conditions at once. There may be excellent reasons to cut the toolbar. Decision time is unlikely to be one of them, and the saving, if any, is measured in tens of milliseconds against a two-second task.

The popular statements of the law hide this. Laws of UX renders it as "The time it takes to make a decision increases with the number and complexity of choices" and recommends that designers "Minimize choices when response times are critical to decrease decision time." The phrase "and complexity" has no counterpart in Hick or Hyman. Their variable is transmitted information among known, equiprobable alternatives, not how complicated the options are.

3. Fitts's law applied to a finger as if it were a pointer

Xiaojun Bi, Yang Li and Shumin Zhai's "FFitts Law: Modeling Finger Touch with Fitts' Law", Proceedings of CHI 2013, pages 1363 to 1372, shows precisely where the standard model breaks. Their argument is that endpoint spread on a touchscreen has two independent components: a relative one governed by the usual speed and accuracy trade, and an absolute one produced by the finger's contact area and the device's centroid estimate, which does not shrink however carefully the user aims.

They measure that absolute component as a standard deviation of 0.94 mm in one dimensional tasks and 1.5 mm in two dimensional tasks. Conventional Fitts's law degrades substantially below roughly 5 mm of target width. At 2.4 mm targets, observed error rates were 29 to 38 per cent in one dimension and 65 to 66 per cent in two, against the 4 per cent the model's information-theoretic reading assumes. Their revised model recovers the fit: R squared of.96 for two dimensional tasks against.85 for the nominal index of difficulty and.79 for the effective-width version.

The practical consequence is that below about 5 mm, Fitts's law stops telling you how long a tap takes and starts telling you nothing, because the user is no longer trading speed for accuracy, they are simply missing. A target twice as far away but comfortably above the finger's precision floor will often be faster and far more accurate than a nearer target below it. Raw Fitts arithmetic says the opposite.

4. The "infinite" screen edge, generalised past its premise

Bruce Tognazzini's A Quiz Designed to Give You Fitts, published on AskTog in February 1999, contains the claim every designer half-remembers: "the screen edge is, for all practical purposes, infinitely deep. It doesn't matter how fast that mouse is going when it hits the screen edge, that pointer absolutely will not overshoot." From this he concludes that "The Apple menu bar is a lot faster than menu bars in windows. Why? Because, since the menu bar lies on a screen edge, it has an infinite height."

The reasoning is sound and the premise is narrow. The effective width becomes unbounded because software clamps the pointer at the display boundary. That clamp is the whole mechanism. It is not a property of edges, it is a property of a pointer that cannot leave the coordinate space.

Carry the conclusion to a touchscreen and the premise is gone: a finger is not clamped by anything, and a control flush against the bezel is if anything harder to hit because half the finger lands off the glass. Carry it to a multi-monitor desktop and the shared edge between two displays stops clamping at all, which is exactly why a menu bar on an internal edge behaves worse than the same bar on a single screen. The corner claim survives the move to touch even less well.

There is a quieter version of the same error in measurement. If two designs are compared on movement time alone, a design that is faster because users are missing more often will win. That is what the effective width correction exists to prevent: normalising to a constant error rate, or adjusting width by the observed spread of endpoints, before any comparison is made. Reporting a speed improvement without an error rate is not a Fitts's law result.

What I could not verify, and what I had wrong

I did not obtain the 1954 Fitts paper itself. Everything attributed to it here, including the amplitudes, the widths, the index of performance of 10.10 bits per second and the correlation of.9831, comes from MacKenzie's reanalysis in Human-Computer Interaction volume 7, 1992, pages 91 to 139, and from his doctoral chapter. That is a careful and widely used secondary reading, but it is secondary, and I am flagging it rather than implying I read the original. Hyman's 1953 paper I have only through Proctor and Schneider's review; I have confirmed its citation but not read its method at first hand.

The thing I had wrong for a long time was simpler. I treated Fitts's law as a claim about distance. It is a claim about a ratio. Doubling the distance and doubling the target width leaves the index of difficulty almost unchanged, which means the familiar advice to "move things closer together" can be worth nothing if the targets shrink to fit. The variable is log of A over W, and the only reliable way to make a pointing task easier is to move the ratio, not either number on its own.

Sources

• W. E. Hick, "On the Rate of Gain of Information", Quarterly Journal of Experimental Psychology, volume 4, issue 1, 1952, pages 11 to 26. doi 10.1080/17470215208416600. 1952.pdf

• Ray Hyman, "Stimulus information as a determinant of reaction time", Journal of Experimental Psychology, volume 45, number 3, 1953, pages 188 to 196. Citation verified via PhilPapers. philpapers.org/rec/HYMSIA

• Robert W. Proctor and Darryl W. Schneider, "Hick's law for choice reaction time: A review", Quarterly Journal of Experimental Psychology, volume 71, issue 6, 2018, pages 1281 to 1299. doi 10.1080/17470218.2017.1322622. web.ics.purdue.edu/~dws/pubs/ProctorSchneider_2018_QJEP.pdf

• Paul M. Fitts, "The information capacity of the human motor system in controlling the amplitude of movement", Journal of Experimental Psychology, volume 47, number 6, 1954, pages 381 to 391. Citation verified via PhilPapers. philpapers.org/rec/FITTIC

• I. Scott MacKenzie, "Fitts' law as a research and design tool in human-computer interaction", Human-Computer Interaction, volume 7, 1992, pages 91 to 139. yorku.ca/mack/hci1992.html

• I. Scott MacKenzie, doctoral thesis chapter 2, on Fitts's original experiments, the Welford variant and the assumptions of the model. yorku.ca/mack/phd-ch2.html

• I. Scott MacKenzie, "A note on the information-theoretic basis for Fitts' law", Journal of Motor Behavior, volume 21, 1989, pages 323 to 330. doi 10.1080/00222895.1989.10735486. yorku.ca/mack/JMB89.html

• Xiaojun Bi, Yang Li and Shumin Zhai, "FFitts Law: Modeling Finger Touch with Fitts' Law", Proceedings of CHI 2013, Paris, pages 1363 to 1372. doi 10.1145/2470654.2466180. www3.cs.stonybrook.edu/~xiaojun/pdf/FFitts.pdf

• Andy Cockburn, Carl Gutwin and Saul Greenberg, "A Predictive Model of Menu Performance", Proceedings of CHI 2007, pages 472 to 481. csse.canterbury.ac.nz/andrew.cockburn/papers/paper191-cockburn.pdf

• Bruce Tognazzini, "A Quiz Designed to Give You Fitts", AskTog, February 1999. asktog.com/columns/022DesignedToGiveFitts.html

• Laws of UX, "Hick's Law" and "Fitts's Law", consulted as examples of how the two laws are currently stated to designers. lawsofux.com/hicks-law/ and lawsofux.com/fittss-law/

Related reading

• Gestalt: What Replicated and What Did Not

• Norman Corrected Himself, and Almost Nobody Followed

• Muriel Cooper and Typography That Moves

• Where the Eight Point Grid Came From

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