Open Journal of Applied Sciences, 2013, 3, 360-368 Published Online October 2013 (
A Note on the Validity of the Shannon Formulation for
Fitts’ Index of Difficulty
Ian Scott MacKenzie
Department of Computer Science and Engineering, York University, Toronto, Canada
Received August 13, 2013; revised September 20, 2013; accepted September 30, 2013
Copyright © 2013 Ian Scott MacKenzie. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The three most common variations of Fitts’ index of difficulty are the Fitts formulation, the Welford formulation, and
the Shannon formulation. A recent paper by Hoffmann [1] critiqued the three and concluded that the Fitts and Welford
formulations are valid and that the Shannon formulation is invalid. In this paper, we challenge Hoffmann’s position
regarding the Shannon formulation. It is argued that the issue of validity vs. invalidity is ill-conceived, given that Fitts’
law is a “model by analogy” with no basis in human motor control. The relevant questions are of utility: Does a model
work? How well? Is it useful? Where alternative formulations exist, they may be critiqued and compared for strengths
and weaknesses, but validity is an irrelevant construct. In a reanalysis of data from Fitts’ law experiments, models built
using the Shannon formulation are (re)affirmed to be as good as, and generally better than, those built using the Fitts or
Welford formulation.
Keywords: Fitts’ Law; Index of Difficulty; Shannon Formulation; Effective Target Width; Scientific Method
1. Introduction
Scientists pursue their research using a body of tech-
niques known as the scientific method. Ideas are framed
as hypotheses that challenge conventional wisdom about
the physical world. The goal is discovery. The method is
empirical: Observations are made, measurements are ta-
ken, evidence is gathered. Hypotheses are tested against
evidence and conclusions are drawn. While strong evi-
dence yields strong conclusions, hypotheses are never
proven. That’s the nature of science. Proof lies within the
realm of mathematics—the study of numbers, including
their relationships, operations, structure, and so on. If a
numeric relationship is proposed and subsequently de-
monstrated by analysis to violate the allowed and correct
operations, the relationship is deemed invalid. Validity or
invalidity is an inherent dichotomy, established through
analysis. There is no need for empirical evidence to sup-
port the case.1
In this paper, we examine Hoffmann’s claim that the
Shannon formulation for Fitts’ index of difficulty is inva-
lid, and that the Fitts and Welford formulations are valid.
Along the way, deficiencies in his analysis and a few
errors are noted. As inferred in the preceding paragraph,
we will touch on issues pertaining to the methodology in
scientific inquiry, such as the roles of analysis and em-
pirical evidence and the demand for rigor and due dili-
gence. We will also reach into a rather unique property of
Fitts’ law: Fitts’ law is a model by analogy, with no basis
in human motor control. Because of this, we argue that
validity is an ill-conceived and meaningless construct.
The only test is utility. On this point, there is ample evi-
dence—both old and new—that the Shannon formulation
works as wells as, and generally better than, the Fitts or
Welford formulation.
This paper is organized in the three parts: background,
analysis, evidence. In the first part, we provide back-
ground discussion on Fitts’ law, emphasising issues re-
levant to Hoffmann’s analysis and the contrary points
developed herein. The discussion is not a thorough re-
view of Fitts’ law. For that, the reader is directed to a few
published sources which are also available online [2-4].
In the second part, we present and critique the analysis
developed by Hoffmann on the validity and invalidity of
the formulations. Hoffmann’s position is built on a pre-
sumption that human movements are equivalent to elec-
trical signals. As no such equivalence exists, we demon-
strate that Hoffmann’s arguments, while perhaps interest-
1This final point is emphasised for a reason. Hoffmann declared the
Fitts and Welford formulations valid and the Shannon formulation
invalid, but also pursued an empirical analysis to determine which
formulation was “more valid” [1, p. 205].
opyright © 2013 SciRes. OJAppS
ing, are meaningless and futile to his purpose. The third
section examines the empirical evidence Hoffmann of-
fers in reanalysing published data. While such analyses
are common and often provide insight, they are irrelevant
on the question of validity vs. invalidity for alternative
formulations of Fitts’ index of difficulty. Nevertheless,
deficiencies in the analyses are noted. We conclude with
an analysis of a recently published data set. The analysis
reaffirms the utility of the Shannon formulation for Fitts’
index of difficulty.
2. Background
Like many psychologists in the 1950s, Fitts was moti-
vated to investigate whether human performance could
be quantified using a metaphor steeped in the new and
emerging language of information theory. Fitts’ particu-
lar interest was rapid-aimed movements, where a human
operator acquires or selects targets of a certain size over
a certain distance. Fitts proposed a model—now la w
that is widely used in fields such as ergonomics, engi-
neering, psychology, and human-computer interaction
[5,6]. The starting point for Fitts’ law is an equation
known as Shannon’s Theorem 17, which appears on the
first page of Fitts’ influential paper [6, p. 381].2 The
equation gives the information capacity C (in bits/s) of a
communications channel of bandwidth B (in s1 or Hz) as
log 1
where S is the signal power and N is the noise power [8,
pp. 100-103]. Fitts reasoned that a human operator that
performs a movement over a certain amplitude to acquire
a target of a certain width or tolerance is demonstrating a
“rate of information transfer” [6, p. 381]. In Fitts’ analo-
gy, movement amplitudes are like signals and target tole-
rances or widths are like noise.
Fitts proposed an index of difficulty (ID) for a target
acquisition task using a log-term slightly rearranged from
Equation (1). Signal power (S) and noise power (N) are
replaced by movement amplitude (A) and target width
(W), respectively:
As with the log-term in Equation (1), the units are bits
because the ratio within the parentheses is unitless and
the log is taken to base 2. The source Fitts cited in intro-
ducing his formulation used a version of Shannon’s
Theorem 17, with the +1 removed [6, p. 388, 7, p. 157].
Although a convenient simplification, it was noted that
the formulation should only be used if the signal-to-noise
ratio is large. But, the A:W ratio in Fitts’ law experiments
is often as low as 1:1. Fitts prefixed A with 2 because it
“ensures the index will be greater than zero for all prac-
tical situations” [6, p. 388].
Several variations of ID have been proposed over the
years. Of relevance here are the Welford formulation [9,
p. 147]:
loglog 0.5
and the Shannon formulation [10]:
log 1.0
The Welford formulation was proposed on practical
terms: “[the subject] is called up on to choose a distance
W out of a total distance extending from his starting point
to the far edge of the target” [9, p. 147]. Going from the
center of the target to the far edge adds 0.5 W to A, which
is revealed in the first form of ID in Equation 3. Welford
dropped the 2 in Fitts’ formulation because, as he noted,
“the logarithm can never be negative, since in the ex-
treme case when the movement begins at the edge of the
target A = ½W” [9, p. 147].
The Shannon formulation (Equation (4)) was proposed
to create a direct analogy with Shannon’s Theorem 17
(Equation (1)). MacKenzie proffered that there was no
strong case given by Fitts or Welford to deviate from the
arrangement of terms in Shannon’s theorem. If the goal
is to measure “the information capacity of the human
motor system” (the title of Fitts’ 1954 paper), then it is
reasonable to arrange the terms in direct correspondence
with Shannon’s theorem.
The Shannon formulation is also appealing in that ID
smoothly approaches 0 bits as A approaches 0. This is
seen in Figure 1, contrasted with the Fitts and Welford
2This point is added for a reason. Hoffmann makes the peculiar claim
that Fitts’ law is not based on Shannon’s theorem and that “it was only
in the Fitts and Peterson paper of 1964 that the analogy with Shannon’s
17 theorem was introduced” [1, p. 207]. This view is narrow and selec-
tive. Certainly, the ID formulation used by Fitts differs in arrangement
from Shannon’s theorem, but the link is unquestionable. Where Fitts
actually introduces his formulation, he cites the resemblance to Gold-
man’s Equation 29 which itself is based on Shannon’s Theorem 17 [6, p
388, 7, p. 157]. Figure 1. With the Shannon formulation, ID approaches 0
as A approaches 0.
Copyright © 2013 SciRes. OJAppS
formulations which dip negative for small A. Although a
negative ID is unlikely in most situations, there are at
least four examples of ID < 0 in the Fitts’ law literature
[11-14]. With the Shannon formulation, ID < 0 is simply
not possible.
Note in Figure 1 that the lines are nearly parallel ex-
cept when ID is small. This is an important point. Quan-
titative analyses seeking to distinguish the three formula-
tions must attend to the range of ID s. Only where the
range includes low values of ID are differences likely to
emerge.3 We will return to this point later.
Fitts described three experiments in his 1954 paper.
The first involved reciprocal tapping of targets with ei-
ther a 1-oz or a 1-lb stylus. Four amplitudes and four
widths were used, yielding 16 target conditions. Fortu-
nately, Fitts published summary data tables so a re-exa-
mination of his results is possible. The data for the 1-oz
stylus condition are given in Table 1, and include target
amplitude (A), target width (W), error rate (ER), index of
difficulty (ID), and movement time (MT). The effective
target width (We) column was added, as discussed short-
Table 1. Data from Fitts’ tapping experiment with 1-oz
A (in) W (in) We (in) ER (%) ID (bits) MT (ms)
2 2.00 1.020 0.00 1 180
2 1.00 0.725 0.44 2 212
4 2.00 1.233 0.08 2 203
2 0.50 0.444 1.99 3 281
4 1.00 0.812 1.09 3 260
8 2.00 1.576 0.87 3 279
2 0.25 0.243 3.35 4 392
4 0.50 0.468 2.72 4 372
8 1.00 0.914 2.38 4 357
16 2.00 1.519 0.65 4 388
4 0.25 0.244 3.41 5 484
8 0.50 0.446 2.05 5 469
16 1.00 0.832 1.30 5 481
8 0.25 0.235 2.78 6 580
16 0.50 0.468 2.73 6 595
16 0.25 0.247 3.65 7 731
Fitts conjectured that the MT-ID relationship is ap-
proximately linear, implying a constant rate of informa-
tion processing. This is reasonably confirmed in the scat-
ter plot and linear regression analysis in Figure 2. With
R2 = 0.9664, the model explains 96.6% of the variance in
the data–a good fit, indeed. Nevertheless, there is a curv-
ing of data points away from the regression line, with the
most deviate point at ID = 1 bit (see block arrow).
Crossman first pointed this out in 1957 in an unpublished
report [cited in 9, p. 146].4 Similar observations and ana-
lyses were provided by Welford [16] shortly after. Both
Crossman and Welford sought to improve the model.
Welford’s approach was a new formulation for ID, as
given above in Equation (3). Crossman’s approach was
quite different.
Crossman sought to improve the information-theoretic
analogy in Fitts’ law by replacing the specified or set
target width (akin to noise) by an effective target width
that reflects the spatial variability in the human opera-
tor’s responses over repeated trials. Welford succinctly
paraphrases Crossman’s method thus:
“[The method] makes use of the fact that the informa-
tion in a normal distribution is log2((2 π e)½ σ), where σ
is the standard deviation in a normal distribution. Now (2
π e)½ = 4.133 and a range of ± half this, i.e., 2.062 σ,
includes about 96% of a normal distribution. We can
therefore argue that if about 4% of the shots fall outside
the target, log2W is an accurate representation of the in-
formation contained in the distribution of shots. We can
argue that if the errors exceed 4% the effective target
width is greater than W, and if the errors are less than 4%
the effective target width is less than W. How much
greater or less can be calculated from tables of the nor-
3One of the data sets Hoffmann analysed included an inappropriate
range of IDs. The lowest ID was rather high at 2.58 bits [1, p. 211-212,
15, p. 902]. Not surprisingly, the results were inconclusive, with no
consistent pattern emerging. Hoffmann made no mention of the rele-
vance of the ID range in comparing the formulations for ID.
Figure 2. Scatter plot and regression line for data in Table 1.
See text for discussion.
4It is worth mentioning that a chart showing scatter points and a regres-
sion line, as per Figure 2, was not included in Fitts’ original paper.
Copyright © 2013 SciRes. OJAppS
mal distribution. For example, suppose W = 2 and the
errors are 1%. Then the effective W = 2 4.133/5.152 =
1.604 in, since all but 1% of a normal distribution lie
within a range of ±2.576 (i.e., ½ 5.152) of the mean. [9,
pp. 147-148].5
Although Welford sought to improve the fit of the
model—bring the scatter points closer to the best-fitting
line—Crossman’s change has an even more important
consequence: If the model is built using the effective
target width (We), Fitts’ law truly embeds the speed-ac-
curacy trade-off.
The technique described above to determine the effec-
tive W is known as the discrete-error method since it uses
the error rate and z-scores from a unit-normal distribution
in transforming W. An alternative method is the standard-
deviation method. If the experimental apparatus records
the coordinates of selection for each trial, then the stan-
dard deviation (σ) is computed directly, with We = 4.133 σ.
Obviously, the standard-deviation method is preferred
since the transformation is more sensitive to the actual
spatial variability in responses.
The apparatus in Fitts’ experiment recorded “hits” and
“misses”, thus the error rate (ER) as a percentage was
easily obtained. The apparatus did not record selection
coordinates. The We column in Table 1 was developed
from the ER column using the discrete-error method, as
described by Welford. The first entry poses a problem,
however, since the task was easy (ID = 1 bit) and no
misses were recorded. The We value was developed using
a pragmatic approach. Fitts reported the error rate for the
top row as “0.00%”. This was converted to “0.0049%”,
which rounds to 0.00%, with the z-score obtained thus [3,
p. 108]. Although not explicitly stated, Welford likely
used a similar heuristic since the point corresponding to
ID = 1 bit appears in his reanalysis of Fitts’ data using
effective target widths. This is presented next.
If the data in Tabl e 1 are plotted as in Figure 2, except
using the Welford formulation or using the effective tar-
get width, the fit of the model is indeed improved. This is
evident in Figure 3(a) in which both Welford’s ID for-
mulation and Crossman’s adjustment for accuracy are ap-
plied. Welford presented a chart that is essentially the
same, with following observation: “the results lie close to
a straight line which passes through the origin” [9, pp.
148-149]. Indeed, the correlation is very good (R2 =
0.9885) and the intercept is very small (1.22 ms).6
Using the Shannon formulation (see Figure 3(b)),
there is a slight improvement in the fit (R2 = 0.9877),
although the intercept is larger (31.43 ms). Importantly,
the charts in Figure 3 contain 16 scatter points. Note that
Figure 3. Scatter plot and regression analysis using data
from Fitts’ tapping experiment with a 1-oz stylus. Both
charts use the effective target width. (a): Welford formula-
tion; (b): Shannon formulation. Source data: Table 1.
the point identified by the block arrow is now much
closer to the best-fitting line. Including this condition is
important, since (a) it was the most deviate point in the
original analysis, and (b) low values of ID are needed to
distinguish the different formulations of ID, as demon-
strated earlier (see Figure 1).7
3. Analysis
Hoffmann’s analysis leading to the conclusion that the
7This point is given particular emphasis for a reason. Hoffmann in-
cluded a reanalysis of Fitts’ data [1, p. 211] using the Fitts and Shannon
formulations and using the effective target width. But, he used only
15 points. The condition with ID = 1 bit was excluded. This is unfortu-
nate, particularly in view of prior research demonstrating similar
analyses with all 16 data points [3, Figures 7, 9, Figures 5.4]. Thus,
Hoffmann’s analysis is incomplete.
5A variation of this method was originally described by Crossman [17,
p. 75-77].
6The source and interpretation of the intercept is hotly debated in the
Fitts’ law research community. For the most part, the debate is avoided
here. A detailed discussion is provided by Soukoreff and MacKenzie
Copyright © 2013 SciRes. OJAppS
Shannon formulation for Fitts’ index of difficulty is inva-
lid hinges on two points: “movements are not a conti-
nuous signal” [1, p. 210] and there is “as incorrect sub-
stitution of an amplitude in place of a signal power” [1, p.
213]. It is certainly true that movement amplitude in
Fitts’ law is substituted for signal power in Shannon’s
Theorem 17. Whether this is incorrect is a matter for de-
bate, which we get to shortly. Hoffmann expounds on the
possibility and mechanisms for using a more power-like
variation of movement amplitude in Fitts’ law. The dis-
cussion is interesting and might very well suggest a new
formulation for Fitts’ index of difficulty. But that is a
separate issue (and, we might add, an issue in need of
empirical evidence).
Hoffmann’s invalidity claim is deficient in at least two
ways. We preface the first with observations on Fitts’
law and modeling in general. Most models are developed
from within a discipline: Low-level established princi-
ples are used to explain higher-level phenomena. In hu-
man-computer interaction, the best-known example is the
keystroke-level model (KLM) introduced more than 30
years ago [18], and still widely used today. With the
KLM, the low-level principles are primitive actions such
as the key stroking time for commands, mouse-to-key-
board homing time, and so on. High-level phenomena are
actions like search-and-replace, file copy, delete a para-
graph, etc. The KLM is a model developed from within
the discipline. Most models can be characterised simi-
larly. No so, with Fitts’ law. Fitts’ law is a model by
analogy, with no basis in human motor control. The mo-
del uses low-level established principles in electronic
communications. But, the phenomena of electronic sig-
nals exist in far-off world from the phenomena of human
movements. Because of this, the correctness of Fitts’
law—or any such model by analogy—cannot be establi-
shed through analysis. One might postulate that move-
ment amplitude is like a signal or that target width is like
a noise distribution, but there is no mathematical or ana-
lytic basis to deem the is-like-a link between the two
worlds correct, incorrect, valid, invalid, or whatever. The
only choice is to the test the model empirically—to
weigh observations against predictions.
The first deficiency in Hoffmann’s analysis is the pre-
sumed equivalence of the phenomena of electronic
communications systems with those of human movement.
Of course, no such equivalence exists: Human move-
ments are not electronic signals—in any form. Because
the link is by analogy, it is irrelevant whether the signal
in Shannon’s theorem is peak or power, discrete or con-
tinuous, filtered or unfiltered, etc. Validity or invalidity is
simply the wrong construct. The issue is utility, not va-
lidity. Aside from that, the Fitts and Welford formula-
tions use the same measure of movement amplitude and
in exactly the same way—in the numerator of the log-
term. This point is examined next.
The second deficiency in Hoffmann’s argument is his
opposing and incompatible positions on the Shannon
formulation (invalid) and the Welford formulation (valid).
His claim is perplexing since the two formulations differ
only in the use of +1.0 (Shannon) vs. +0.5 (Welford) in
the log-term. Why would one version be valid, the other
invalid? Welford’s rationale for +0.5 was to add the dis-
tance from the center of the target to the far edge, which
is 0.5 W. MacKenzie’s rationale for +1.0 was simply
that this is the arrangement in Shannon’s Theorem 17. If
the rationale for the Shannon formulation was different,
would that matter? What if MacKenzie said nothing
about the Shannon formulation, but simply argued to
change Welford’s +0.5 to +1.0 because of the desirable
property that +1.0 yields ID = 0 bits when A = 0? Let’s
call this the Plus-one formulation. It is identical to the
Shannon formulation. So we ask: Is the Plus-one formu-
lation valid? On what basis would Hoffmann deem the
Plus-one formulation invalid? Clearly, there is no argu-
ment on the basis of signal power or continuous signals.
Once again, we see that validity vs. invalidity is an ill-
conceived construct. The only issue is utility: Do the
formulations work? Which one provides a better descrip-
tive or predictive ability to explain human responses for
rapid-aimed movements?
4. Evidence
Data from three sources were analysed by Hoffmann and
offered as evidence for his position that the Fitts and
Welford formulations for ID are valid and that the Shan-
non formulation is invalid. Although we already noted
the irrelevance of empirical evidence in view of an ana-
lytic determination of invalidity, let’s examine Hoff-
mann’s evidence to see what insights are offered. Once
again, we find deficiencies. Two data sets analysed by
Hoffmann have already been dealt with, and are not dis-
cussed further (see footnotes 3 and 7). The third data set
is from a paper published by MacKenzie in 1995. Let
revisit Hoffmann’s reanalysis.
First, it is worth noting that the 1995 paper cited is not
a research paper. It is a review paper with a pedagogical
intent. An example data table was used to illustrate ap-
plications of Fitts’ law. The table is a subset of a table
from MacKenzie [2], which is cited in the 1995 paper
and which has been available online since the mid-1990s.
By using a partial data set, Hoffmann’s analysis is in-
complete.8 The results of Hoffmann’s analysis are given
in Figure 4. There are six Fitts’ law models. The top
three use set target widths (W), the bottom three use ef-
8Hoffmann mistakenly cites the data as from an experiment for “mouse
movement on a computer screen” [1, p. 212]. In fact, the data are not
for a mouse, As stated in the paper he cites, the data are for a stylus on
a tablet [19, p. 485].
Copyright © 2013 SciRes. OJAppS
Figure 4. Hoffmann’s reanalysis of data from MacKenzie
fective target widths (We). Within each group, there are
models for the Fitts, Welford, and Shannon formulations.
Reflecting on Figure 4, Hoffmann notes, “there is a
marked reduction in the correlation when both the effec-
tive target width and the Shannon formulation are used in
the regression” [1, p. 212].
There are at least three problems in Hoffmann’s analy-
sis. First, his observation is simply wrong. Yes, there is a
reduction in the three correlations using the effective
target widths (“eff” in the figure). This effect is well
known [e.g., 20, p. 479]. However, with respect to the
Shannon formulation, his observation is wrong. In fact,
the opposite is true. The correlations within each group
are highest using the Shannon formulation.
Second, Hoffmann did not bring the same standard of
rigor to the analysis as used in the paper he sought to
criticize. In MacKenzie’s [10] comparison of the Fitts,
Welford, and Shannon formulations, correlations were
computed and a statistical significance test was used to
determine if the differences were significant. Hoffmann
included no such test. An appropriate test is Hotelling’s
t-test for the correlations of correlated samples [e.g., 21,
p. 164].
Third, Hoffmann did not exercise due diligence to ob-
tain and use the original and complete data set for his
analysis, even though the source is cited and the data are
readily available online.9 Hoffmann excluded the data
point for ID = 1 bit because the example data table only
included error rates and the error rate was 0.0% at ID = 1
bit (E. R. Hoffmann, personal communication, June 19,
2013). As noted earlier (see Figure 1), it is with low
values of ID that the distinction between the three for-
mulations emerges. So, to needlessly exclude this data
point in a critical analysis that seeks to compare the three
formulations falls short of the standards of rigor de-
manded in the analysis. The complete data set is given in
Table 2 and includes a column labeled We(SD) for the
effective target width as computed using the standard
deviation in the selection coordinates. With this, it is easy
to compute the effective index of difficulty for all 16 data
points. For convenience, six columns are included show-
ing ID computed using the Fitts, Welford, and Shannon
formulations using set target widths (W) and effective
target widths (We).
Hoffmann’s analysis is repeated in Table 3, using the
data in Table 2. The ranking of correlations within each
target width is Fitts (lowest), Welford (middle), Shannon
(highest). So, the results are favourable to the Shannon
formulation. However, the differences in correlations are
modest. Hotelling’s t-test deemed the difference between
the Fitts and Shannon correlations not significant both
using set targets widths (t16 = 1.29, p > 0.05) and using
effective target widths (t16 = 0.34, p > 0.05).10
The results in Table 3 are not dramatically different
from those in Figure 4. The purpose here is the do the
analysis correctly: using the full data set, employing an
acceptable standard of rigor, and drawing correct conclu-
One final point about the analysis in Table 3 will be
made. The correlations are lower for the models using
the effective target width (We) compared to those using
the set target width (W). This is a natural consequence of
the reduced range of IDs when computed using the effec-
tive target width. Note, for example, that the ID range in
the Fitts-W column in Table 2 is 7 1 = 6 bits, whereas
the ID range in the Fitts-We column is 6.620 – 1.988 =
4.633 bits. The lower correlations in the latter case are
much like the statistical effect known as “regression to-
ward the mean.” It is important to remember that the
benefit in using the effective target width is not because
it produces a model with higher correlations (although
this sometimes occurs, see Figures 2 and 3), but, rather,
it brings accuracy into Fitts’ law, and makes it a true
speed-accuracy model of human motor behavior. Further
discussion on this is provided by Soukoreff and Mac-
Kenzie [4, section 3.2].
5. A Modern Example
Since the Shannon formulation was introduced in 1989
[10], it has been generally accepted as the preferred for-
mulation for Fitts’ law. This is particularly the case in
human-computer interaction (HCI), where there is an
active community of researchers exploring and pushing
the limits of Fitts’ law. For the most part, there is no de-
bate on which formulation to use. Other issues are con-
sidered more interesting, such as applying Fitts’ law in
3D virtual environments [22], using Fitts’ law for touch
screen input where fingers select small targets [23], or
examining if input control using device tilt can be model-
ed by Fitts’ law [24]. The Shannon formulation is gene-
ally the formulation of choice. r
10In other analyses, the improvement with the Shannon formulation is
statistically significant [2, Table 3, Table 11, 10, Table 3]. No exam-
les have been reported with statistical significance that favor the Fitts
or Welford formulation.
9The data are in the table labeled “Tablet-Pointing”, available at
Copyright © 2013 SciRes. OJAppS
Copyright © 2013 SciRes. OJAppS
Table 2. Data from MacKenzie (1995) with an additional column for We (SD).
Index of Difficulty (ID or IDe) (bits) MT (ms)
Set Target Width (W) Effective Target Width (We)
We (SD)
Fitts Welford ShannonFitts Welford Shannon
8 8 4.034 0.00 1 0.585 1.000 1.988 1.312 1.577 254
8 4 2.845 1.88 2 1.322 1.585 2.492 1.728 1.931 353
16 8 4.690 0.83 2 1.322 1.585 2.770 1.968 2.141 344
8 2 1.560 1.67 3 2.170 2.322 3.358 2.493 2.615 481
16 4 3.231 2.08 3 2.170 2.322 3.308 2.447 2.573 472
32 8 5.562 0.63 3 2.170 2.322 3.524 2.645 2.756 501
8 1 1.149 8.84 4 3.087 3.170 3.800 2.900 2.993 649
16 2 1.629 2.14 4 3.087 3.170 4.296 3.368 3.436 603
32 4 3.252 2.71 4 3.087 3.170 4.299 3.370 3.438 605
64 8 6.624 2.51 4 3.087 3.170 4.272 3.345 3.414 694
16 1 1.053 7.01 5 4.044 4.087 4.925 3.972 4.017 778
32 2 1.795 3.42 5 4.044 4.087 5.156 4.196 4.235 763
64 4 3.464 2.34 5 4.044 4.087 5.208 4.246 4.284 804
32 1
1.165 8.50 6 5.022 5.044 5.780 4.806 4.831 921
64 2 1.867 3.33 6 5.022 5.044 6.099 5.120 5.141 963
64 1 1.301 9.88 7 6.011 6.022 6.620 5.635 5.649 1137
Table 3. Fitts’ law models and correl ations using the Fitts, Welford, and Shannon formulations for ID using se t and effective
target width s.
Target Width ID Formulation Equation r R2
Fitts MT = 54 + 148 ID 0.9921 0.9843
Welford MT = 138 + 161 ID 0.9946 0.9893
Shannon MT = 81 + 173 ID 0.9951 0.9901
Fitts MT = 123 + 181 IDe 0.9869 0.9739
Welford MT = 0 + 193 IDe 0.9875 0.9752
Shannon MT = 55 + 203 IDe 0.9876 0.9753
Of course, the analysis above can be pursued with
other data sets, provided summary data are published or
are available first-hand. One recent example is a data set
for an experiment comparing a mouse and a gyroscope-
based remote pointer [25, p. 253]. The data set is in the
same format as in Fitts’ original publication, with an ad-
ditional column for the effective target width (We). With
such data, it is easy to compare ID formulations, as a
demonstrated above. See Table 4. There are four tests: 2
devices 2 methods of calculating target widths. In all
four cases, the rank of correlations is Fitts (lowest),
Welford (middle), Shannon (highest), although the dif-
ferences are modest.
Comparing by target width in Table 4, the correlations
are consistently lower with the effective target width (We)
vs. the set target width (W). To help illustrate why, an
extra column is added showing the ID range for each mo-
del. The range varies due the inherent differences in the
ID formulations and to the method of calculating target
widths, as noted above. For all 6 formulation device
comparisons, the range is less using We compared to W.
For example, the ID range in the top row is 5.00 1.00 =
4.00 bits, corresponding to the Mouse-W-Fitts model.
The range for the Mouse-We-Fitts model (three rows
Table 4. Comparison of ID formulations in a Fitts’ law experiment comparing a mouse with a remote pointer.
Device Target Width Formulation Intercept Slope ID Range r R2
Fitts 184.0 153.2 1.00 - 5.00 0.9890 0.9781
Welford 252.9 176.2 0.58 - 4.04 0.9920 0.9840 W
Shannon 172.7 196.5 1.00 - 4.09 0.9924 0.9849
Fitts 172.2 158.0 0.99 - 4.90 0.9805 0.9613
Welford 242.6 182.3 0.58 - 3.95 0.9842 0.9686
Shannon 159.0 203.8 1.00 - 3.99 0.9850 0.9702
Fitts 580.0 324.9 1.00 - 5.00 0.9750 0.9506
Welford 722.4 375.3 0.58 - 4.04 0.9822 0.9648 W
Shannon 548.2 420.1 1.00 - 4.09 0.9859 0.9719
Fitts 615.3 319.9 1.08 - 4.95 0.9603 0.9222
Welford 762.8 366.4 0.64 - 4.00 0.9642 0.9297
Shannon 596.9 408.4 1.04 - 4.04 0.9660 0.9332
down) is 4.90 0.99 = 3.91 bits. The lower ID using We
is simply an artefact of one’s choice to include accuracy
in the Fitts’ law model.
6. Conclusion
We have examined Hoffmann’s claim that the Shannon
formulation for Fitts’ index of difficulty is invalid. Seve-
ral deficiencies in his analysis were noted. We have ar-
gued that because Fitts’ law is a model by analogy, there
is no analytic basis on which to deem the Shannon for-
mulation (or any other formulation) valid or invalid. The
only test is utility, which demands empirical evidence.
Hoffmann’s empirical evidence (although irrelevant to
the question of invalidity) was also examined. Again,
deficiencies were noted, such an erroneous observation,
the use of an incomplete data set (when the full data set
is available), and the failure to exercise the same stan-
dard of rigor as used in the research where the Shannon
formulation was originally introduced. In a proper reana-
lysis using the full data set and in an analysis of a re-
cently published data set, the Shannon formulation is re-
affirmed to provide better predictions than the Fitts or
Welford formulation.
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