Post on 26-Apr-2023
DOI: 10.1142/S0129183110015324
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
International Journal of Modern Physics CVol. 21, No. 5 (2010) 567–581c© World Scientific Publishing Company
VISUAL AND AUDITORY CUES SIGNIFICANTLY REDUCE
HUMAN’S INTRINSIC BIASES WHEN TASKED TO GENERATE
A RANDOM NUMBER SEQUENCE
IRENE CRISOLOGO, RENE BATAC, ANTHONY LONGJAS,ERIKA FILLE LEGARA and CHRISTOPHER MONTEROLA∗
National Institute of Physics
University of the Philippines Diliman
Quezon City 1101, Philippines∗cmonterola@gmail.com
Received 2 December 2009Accepted 11 March 2010
Humans are deemed ineffective in generating a seemingly random number sequenceprimarily because of inherent biases and fatigue. Here, we establish statistically thathuman-generated number sequence in the presence of visual cues considerably reduceone’s tendency to be fixated to a certain group of numbers allowing the number dis-tribution to be statistically uniform. We also show that a stitching procedure utilizingauditory cues significantly minimizes human’s intrinsic biases towards doublet and se-quential ordering of numbers. The article provides extensive experimentation and com-prehensive pattern analysis of the sequences formed when humans are tasked to generatea random series using numbers “0” to “9.” In the process, we develop a statistical frame-work for analyzing the apparent randomness of finite discrete sequences via numericalmeasurements.
Keywords: Random number statistics; Linfoot’s criteria; brain models; pattern recogni-tion.
PACS Nos.: 05.40.-a, 02.70.Rr, 89.20.-a, 05.10.-a.
1. Introduction
Humans acquire an early sense of number by counting and arranging things. As
children, we were taught to count from a set of positive integers (1, 2, 3, . . . , 10) as
part of our language training. Formal education later on introduces a higher level
of mathematical abstraction, thereby affecting the way we perceive mathematical
processes.1 One example of a mathematical abstraction acquired through educa-
tion is the idea of random numbers. One encounters such sequences in statistical
mechanics, in a simple model called the random walk, and the very idea is used to
derive the principle of central limit theorem. Randomness is the ultimate absence
of predictability or order, achieved when successive events are independent of one
567
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568 I. Crisologo et al.
another. Examples of random events abound in physical phenomena; rain formation
and distribution, radioactive decay processes, even arrival times of cars in streets
all have exponentially-decaying inter-event occurrence time distributions.
Efforts to use humans for purposes of random number generation (RNG) are not
as successful as computer pseudo-RNG. In an earlier work, Persaud2 claimed that
by simply asking his subjects to generate and dictate numbers, humans can generate
sequences that are uniformly distributed, independent of one another and unpre-
dictable, a claim refuted by Bains3 and Figurska et al.4 The resulting sequences they
obtained were in fact substantially non-random, with an excess of sequential pairs
of numbers and a deficit of repeats of the same number. This failure is expected
because the human mind is exceptionally superb at recognizing patterns. Humans
are shown to have an intrinsic ability to carry out pattern recognition tasks even in
the absence of conscious knowledge of what must be learned as in the case of am-
nesic patients.5 In fact, non-declarative sequence learning is observed even in mice,
suggesting that generally, brains are primitively engineered to recognize patterns.6,7
Even for existing brain models, spontaneous activity of sensory neurons producing
background noise are deemed necessary so as to reinforce pattern identification of
temporal8 and spatial signals.9 Given the contrariness of the nature of the mind
(tasked to decipher patterns) and the idea of generating random sequences, it may
well be interesting to quantify the extent of how the former will fail at doing the
latter.
In this work, we detail the inherent biases when humans are tasked to generate
random sequences. We show that humans are intrinsically biased and in the absence
of any cues will generate a sequence that:
(1) is not uniformly represented, and
(2) favors sequential pairs while avoiding repeated pairs or doublets.
We then proceed by showing that the combinations of visual and auditory cues
significantly reduce these biases respectively. We also discuss how the stitching
procedure (digit stitching or based on auditory cues) can aid in weakening the in-
termittent and short transient biases (doublets and sequential pairs) during the
number generation. Our work provides a statistical framework in terms of digit
frequency histogram, inter-event occurrence distribution, relative frequencies of se-
quential pairing3 and Linfoot’s criteria, to comprehensively quantify the pattern of
the number sequences generated by humans.
2. Materials and Methods
Thirty-one college students from the University of the Philippines, with ages ranging
from 18 to 21 years, volunteered to participate in a three-part session where they
were asked to give a sequence of random numbers. Each part lasted two minutes,
the maximum length of time that the subjects found comfortable as determined
from pre-trials. Initially, the human subjects were tasked to generate a sequence
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Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 569
(a) (b)
(c)
Fig. 1. Numbered boards (a) with equal probability are used for Set 2, while (b) boards witha bias to the digit 2 are used for Set 3. (c) Histogram of digit occurrences showing statisticallysignificant increase in occurrence of digit 2 for Set 3.
of digits from 0 to 9 in random order [Set 1]. To investigate the effect of external
biases, we utilized visual and auditory cues. For the visual cues, the subjects were
shown a board with 20 numbers, containing digits from 0 to 9 as a guide. The first
board shown [Fig. 1(a)] contains the digits 0 to 9 in sequential order twice [Set
2], while the second board [Fig. 1(b)] is biased with four instances of the digit 2
compensated by less occurrences of the digits 0 and 9 [Set 3]. Each session was
recorded and transcribed for later processing.
We then conduct numerical experiments to determine the ordering characteris-
tics of the sequences generated. In particular, in a process we referred to as stitching,
we alternate in sequential manner, the digits of independently-generated sequences.
We then compared the inter-event properties of this ordering procedure with that
of merely appending the sequences.
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570 I. Crisologo et al.
For the auditory cues, seven subjects were asked to generate a random number
sequence of one participant after another for 14 minutes (equivalent to the length of
time for seven individuals to dictate random numbers at a rate of two minutes per
person). The first participant will call out one number, then the next participant
another number, then the third participant a third number and so on. They were
arranged in a semi-circle configuration and the direction of dictation was clockwise.
The results were analyzed in the absence [Set 4] or presence of the visual cues [Set
5 for unbiased visual cues; Set 6 for biased visual cues] described above.
The participants were only told to dictate random numbers and were not al-
lowed to ask questions and discuss with the proctor and other participants matters
concerning the activity before the experiment. This is to ensure that all subjects
had the same knowledge of the activity.
3. Results and Discussions
In the following, we present the results of 31 human-generated sequences and quan-
tify the intrinsic biases of human when tasked to generate a random sequence.
3.1. Digit statistics P (n)
Figure 1(c) shows the sample histogram of the digits generated by a group of
31 different individuals for the three experimental protocols conducted. The first
apparent effect of the visual cues is to lengthen the number sequence. In particular,
the presence of visual cues in Sets 2 and 3 produce, respectively, a sequence ∼20
and ∼24% longer as compared with Set 1, within the two minutes time limit (see
Materials and Methods). We also report that the total sequence by the group shows
uniform discrete digit distribution for Set 1 (without the digit 0) and Set 2 (all digits
included) based on the chi-square test for goodness of fit. However, the same test
rejects the discrete uniformity hypothesis for Set 1 (with digit 0) and Set 3 (with
or without digit 2). We elaborate below.
For Set 1 without the digit 0, and with the null hypothesis that the frequency of
the digits are equal, the chi-square statistic χ2 = 15.048 with df = 8, gives a p-value
of p = 0.058 > 0.001, hence observed discrete distribution are statistically uniform.
A similar conclusion is arrived for Set 2 (χ2 = 26.784, df = 9, p = 0.00151 > 0.001)
strengthening the notion that a set generated by many individuals can be used to
create a longer sequence where each digit is equally represented. Consistent with
previous works in which visual cues were absent (Set 1), we have confirmed that
indeed humans have negative preference for the digit 0.3 This bias seems to be
inherent in the mind and is not a result of external stimulus. For instance, the
digits 1 to 9 appear on the average 414 ± 28 times in the sequence of Set 1, while
0 appears only 319 times, which is statistically far from the spread of this average
and differs by ∼23% from the mean. When the digit 0 is included in Set 1, chi-
square test results to χ2 = 35.482, df = 9, p = 0.00004 < 0.001, rejecting the
null hypothesis that the discrete frequency histogram is uniform. Note that the
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Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 571
statistical uniformity of the frequency histogram of Set 1 for all digits is recovered
when Set 2 is implemented as discussed above.
The effect of the biased board is apparent by looking at the digit statistics of
Set 3. While the rest of the numbers have an average occurrence of 471 ± 76, the
digit 2 (which appeared more often in the board) appeared 775 times, which is more
frequent by about ∼65% compared to the mean. The digit 9 which appeared only
once in the board has the smallest count of 331 and is ∼30% smaller than the mean.
These biases distorted the frequency distribution from being discretely uniform,
forming negative and positive preferences for different digits. The goodness-of-fit
test rejected our null hypothesis of uniformity of distribution for Set 3, in the
presence of digit 2 (χ2 = 257.494, df = 9, p = 10−10 < 0.001), in the absence of
digit 2 (χ2 = 98.805, df = 8, p = 10−9 < 0.001), or in the absence of digits 2 and 9
(χ2 = 49.63, df = 7, p = 2 × 10−8 < 0.001).
The above results indicate that the presentation of visual cues: (a) can signifi-
cantly reduce the bias with 0 (Set 2), and (b) allow generation of longer sequences
for a given time limit (Sets 2 and 3). Also, the response of human subjects to Set
2 and Set 3 establishes that visual cues can affect humans in generating a series
of numbers, a fact that can be harnessed to shape the number distribution to a
desired profile.
3.2. Inter-event occurrence distribution statistics P (∆n)
The randomness of a sequence is not only measured by the uniform distribution
of digits, but more so by the way they are arranged. To quantify how random
the sequence is, we plot the distribution of inter-event occurrence times (IOT) of a
digit — how long it takes for a digit to be repeated. Doing the IOT distribution for
the entire sequences of numbers connected end-to-end, we again find universality
in the distribution for any number across different individuals.
Random events are characterized by exponentially distributed IOT, signifying
independence of the events. For human-generated sequences, on the other hand,
Fig. 2(a) shows a universal trend of having a peak near seven contributing to about
8.6 ± 2.7% in the IOT histogram. This means that upon giving a particular digit,
an individual has a slightly higher tendency to give seven different values before
repeating it. As further proof of the bias against 0, the inset Fig. 2(b) shows that the
IOT distribution for 0 tend to have a lower peak (corresponding to less occurrences)
and a longer tail (which means that it is rarely mentioned by an individual). The
tail of the distribution for digit 0 may even reach 130, roughly the length of the
sequence generated by a single person.
Thus, we can conjecture the process of how an individual generates a sequence
of numbers he believes to be random: upon giving a particular digit, the individual
tends to exhaust all the numbers in his “natural set” before repeating any of the
digits already mentioned. Here, we used the term “natural set” to refer to the group
of numbers from “0 ” to “9” that an individual can subconsciously recall with high
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572 I. Crisologo et al.
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Fig. 2. Inter-event occurrence time distribution (IOT) for the appended sequences without board(in (a) and (b)) and with a biased board (in (c)). The distribution is universal for all samples andfor all digits, showing a peak at IOT near seven. In (b), the digit 0 shows an anomalously long tail,providing evidence to the idea that it is not part of the “natural set” of numbers of an individual.In (c), the IOT is altered when subjects look at a biased board. The possibility of successivelyrepeating the digit 2 increased upon biasing. (d) Shows the distribution of differences of adjacentnumbers in the generated sequence. Subjects avoid repeating the same number, as shown by thedip in zero.
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 573
and equal probability during number generation. This process produces the peak
in the distribution observed. With such procedure, individuals also tend to avoid
successive repetitions of numbers, resulting in the low value of the IOT statistics
for small intervals.
The effect of biasing the board in Set 3 is also apparent from the IOT distri-
bution, and is characterized by an increase in the peak for the occurrence of the
digit 2 as seen in Fig. 2(c). The subjects tend to repeat the digit 2 more often in
the sequence they generated because it is represented more often in the board they
look at. The resulting IOT in Fig. 2(c) shows that the IOT statistics of the digit 2
is significantly different from the general trend observed in Fig. 2(a), with higher
tendency for repetition as shown by the peak near low IOT values.
3.3. Statistics of consecutive number differences P (ni+1 − ni)
These biases are also captured upon looking at the distribution of the difference be-
tween successive digits. For uniformly distributed random numbers, the histogram
of differences will look like a triangular distribution with base from −9 to 9. This
is clearly not the case upon appending the data as shown in Fig. 2(d). The dip in
a difference of zero is a clear indication that individuals avoid repetitions of the
same number. Another deviation from the expected triangular shape is seen for
near half-period differences; this fact tells us the manner by which humans exhaust
his “natural set.” While having the numbers 1 to 9 in mind, the individual thinks
it is random enough to choose a digit half a period away from it. The individual
repeats the process until all the numbers in the “natural set” get mentioned.
Alternatively, we can look at how often a number pair AB appears in the se-
quence to reveal inherent biases. By definition, AB pairs where A and B are equal
are called doublets, and pairs where B is one digit greater or smaller than A (i.e.
A = B ± 1) are termed sequential pairs.3 For a truly random sequence, the prob-
ability of a certain number pair AB being mentioned is one out of the 100 total
possible number pairings, or 0.01. Figure 3 shows the Normalized Number Pairing
Matrix (NNPM) where each element of the matrix corresponds to a specific num-
ber pair AB, for both the appended and stitched sequences for all the sets. For a
random sequence, NNPM is expected to be flat (visually, it has only a single color)
since all digit pairings are equally probable. For the set with no cues, the low val-
ues of the diagonal, which represents the doublets or the pair with the same digit,
supports the result that humans avoid repeating the same number consecutively in
their sequences. High values of the sequential pairs suggest that humans tend to
give numbers that are in succession even though they were asked to give a random
sequence, which agrees with the result of Bains.3 However, when the uniform board
was presented (Set 2), the bias towards sequential pairs is no longer evident, while
still disinclined to doublets. The biased board (Set 3) showed a positively correlated
influence on the generated sequences, as seen from the increased frequencies of the
2-2 doublet and the A-2 and 2-B pairs.
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
574 I. Crisologo et al.
Fig. 3. Normalized Number Pairing Matrix (NNPM) indicates that humans tend to avoid dou-blets for all cases, and seem to prefer number pairs when not aided with a number board (Set1). When shown a board with numbers of equal probabilities (Set 2), the number pair preferencevanishes but the doublets are still avoided. When shown a board with bias to digit 2 (Set 3) thesubjects become biased to digit 2.
Therefore, using the statistics for (a) the occurrence of each digit P (n); (b) the
inter-event occurrence time (IOT) P (∆n); and (c) the difference between adjacent
digits P (ni+1 − ni), we quantify the following biases in human random number
generation:
(i) The existence of a “natural set” of numbers in the human mind of which the
digit 0 is not included;
(ii) The inability to generate longer sequences due to fatigue; and
(iii) The bias against successive pairs of the same digit.
The first bias (i) can be altered by external influencing via a visual stimulus.
The second bias (ii) can be addressed by appending or stitching independently
generated sequences. In the succeeding discussions, we discuss a possible solution
to considerably weaken the third bias (iii).
3.4. Stitching procedure
3.4.1. Digit stitching
We demonstrate here that the biases in the generated sequence resulting from dou-
blets and sequential pairing can be weakened using a stitching procedure, i.e. stitch-
ing the first digit by individual 1 followed by first of individual 2, etc. [see Fig. 4(a)
for visualization of the procedure]. Whereas peaks occur for the IOT of the raw
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 575
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Fig. 4. (a) Stitching procedure showing that the “new” sequence is similar to generating a se-quence with one person giving one number per round. (b)–(d) shows the exponential IOT distri-butions for post-processed data from the samples.
samples that are appended, the stitched sequence seems to decay exponentially as
shown in Fig. 4. It should be noted that similar distributions are obtained for the
data from Sets 1 and 2, as shown in Figs. 4(b) and 4(c). The only apparent differ-
ence in the data for Set 3 (biased board), on the other hand, is in the maximum
number of occurrence, which is expected given that the digit 2 is mentioned by the
subject almost twice more than the other numbers as shown in Fig. 4(d).
The fact that stitched sequence seems to have exponential IOT indicates that the
correlated period valued at “∆n = 7” is reduced by stitching. This is expected since
the strength of correlation of period s · ∆n becomes statistically insignificant after
a few stitching of s sequences decaying to the power of s, as verified numerically in
Fig. 6. As an illustration, for s = 7, the correlation strength of sequence with period
s · ∆n = 49 becomes ∼(0.086)7 ∼ 2−11, and is ∼zero for practical purposes. Here,
0.086 ± 0.027 is the average correlation strength for “∆n = 7” of the unstitched
generated sequences in Fig. 2.
The stitching procedure discussed here to some extent resembles the idea of card
shuffling that was shown to be truly random after seven shuffles by mathematicians
Diaconis, Graham and Kantor.12 One shuffling procedure in their mathematical
proof assumed that after a cut, or a process of dividing the entire deck of cards
into two, the piling order of one half of the deck of cards will be made to alternate
with the ordering of the other half. Note that such process is essentially stitching
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
576 I. Crisologo et al.
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Fig. 5. (Color online) Fidelity (F ), Structural Content (C) and Correlation Quality (Q) of theNNPM of appended sequence {ΨAS} (blue circles) and stitched sequence {ΨSS} (red triangles) ascompared to that of a randomly generated sequence {ΦRS} (green squares). The dashed verticalline denotes N = 7.
the first set of cards with the remaining set. A work by Trefethen and Trefethen
two decades later suggests that six shuffles are sufficient.13
By simply appending the generated sequences [fourth row of Fig. 4(a)], the
individual’s tendency to avoid digit repetition within succession is seen to be still
present. Moreover, the bias towards sequential pairing is still evident. Both these
biases are not observed when independently generated sequences are stitched [fifth
row of Fig. 4(a)], and the resulting NNPM are generally flat except for the pairings
of digit 0 in Set 1 and for the pairings of digits 2 and 9 in Set 3. For a biased
random number generator of Set 1, a lower value is expected for the frequency of
the pairs sequential with 0. While for a biased random number generator of Set 3,
a higher (lower) value is expected for the frequency of the 2-2 doublet and the pairs
sequential with digit 2 (digit 9). The stitched sequence of Set 2 provides the best
random number generated sequence, that is, it has the smallest possible bias both
in the frequency histogram distribution and the number pairing frequencies.
To further compare the NNPM of the appended {ΨAS} and stitched {ΨSS} se-
quences with the NNPM of a randomly generated sequence {ΦRS}, we use Linfoot’s
May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324
Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 577
Fig. 6. The correlation strength of period ∆n · s decays to the power of s. Inset: In a semi-logplot the decay is a straight line.
Criteria of Fidelity F , Structural Content C, and Correlation Quality Q that can
be expressed as10:
FAS/SS = 1 −〈Ψi,j
AS/SS− Φi,j
RS〉2
〈Ψi,jAS/SS
〉2; CAS/SS =
〈Ψi,jAS/SS
〉2
〈Φi,jRS〉
2;
QAS/SS =〈|Ψi,j
AS/SS| |Φi,j
RS|〉
〈Φi,jRS〉
2
(1)
where FAS/SS measures the general similarity, CAS/SS describes the relative sharp-
ness of the peaks, and QAS/SS measures the alignment of peaks and troughs of
{ΨAS} or {ΨSS} as compared to {ΦRS}. The indices i and j run from 0 to 9 cor-
responding to the elements of the NNPM in Fig. 3. The experimental NNPM of
the actual data {Ψ} are similar to that of a random sequence {ΦRS} if exactly
F = C = Q = 1.
For Set 1, the FCQ values of {ΨAS} (blue circles), approach 0.93, 2.0, and 1.4,
respectively (Fig. 5). This means that a collection of people generating sequences
of numbers one series after another do not produce a random sequence even as the
number of persons N , which is proportional to the length of the combined sequences,
is increased. On the other hand, the FCQ values for {ΨSS} (red triangles) of Set
1 to Set 3 approach 1 (deviation is <3%) for N ≥ 7. This result indicates that by
stitching a minimum of seven independently generated number series, humans can
produce a longer and close to random sequence. Compared to Set 1, the FCQ values
of Sets 2 and 3 approach the value of 1 faster, however they are still near N ∼ 7
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578 I. Crisologo et al.
when the deviation of FCQ from 1 is consistently <3%. This implies that with the
aid of visual boards, alternating sequence of numbers generated by seven individuals
can better approximate a random number series, partly due to generation of longer
sequences. One visual limitation of using FCQ for comparing NNPM is its failure
to highlight the bias of Set 1 to digit 0 and Set 3 to digit 2. This is traceable from
the fact that F or C or Q is a single valued number that represents the average
deviation of the ensemble of observations.11 Since the relative deviation of the
relative frequencies of digit 0 in Set 1 in NNPM (or digit 2 in Set 3) deviates by less
than 3% in total, the fluctuation of F or C or Q from 1 as a result of this bias is only
within 3% as observed in Fig. 5, and is visually small compared to the deviation
resulting from the biases due to human’s avoidance of repetition and/or human’s
leaning towards sequential pairs. Thus, while Linfoot’s criteria provide an intuitive
guide (since each term is single valued) for comparing the randomness of sequences
of different lengths, the use of frequency histogram [Fig. 1(c)] and visualization
of NNPM (Fig. 3) are still necessary to ascertain the degree of randomness of a
number sequence.
3.4.2. Auditory stitching
We now check if the results of digit stitching above can be obtained by stitching
human responses using auditory cues. As discussed in the methodology, seven sub-
jects were asked to dictate random numbers that is heard by all participants, one
at a time. They were arranged in a semi-circle configuration and the direction of
dictation was clockwise.
Figure 7 top row shows the sample histogram of the dictated digits and the
bottom row shows the digit differences IOT histogram generated by the participants
for the next three experimental protocols conducted [Sets 4–6].
Fig. 7. Histogram of digit occurrences (top row) and digit differences IOT histogram (bottom
row) of generated sequence in the presence of auditory cues for Set 4 (no visual cues), Set 5 (withunbiased visual cues), and Set 6 (biased visual cues).
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Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 579
For Set 4 the subjects were guided by auditory cues only, rendered by hearing
the numbers dictated by the other participants in the group. The discrete digit
distribution of the total sequence by the group is found to be statistically uniform
based on the chi-square test with χ2 = 11.471, df = 9 and a p-value of p =
0.2448 > 0.001. The digit differences IOT histogram, however, failed to make a
marked improvement in the preference for sequential pairs, in which the frequency
of the digit difference of 1 and −1 deviate from the ideal triangular shape (Fig. 7)
and a deficit of doublets as seen in the dip in the difference of zero. Figure 8 visually
confirms this showing in a distinct region of high frequency for sequential pairs and
low frequency for doublets. Comparison of the resulting NNPM matrix from the
NNPM of a random sequence using Linfoot’s criteria results to the following values
F = 0.96; C = 1.23; Q = 1.52, which is significantly different from the ideal value
of 1. The auditory cues alone can also make the distribution of dictated numbers
statistically uniform but there is still the presence of excess sequential pairs and
deficits of doublets.
For Set 5, the digit histogram also passed the test for uniformity with χ2 =
19.284, df = 9 and p = 0.0229 > 0.001. However, in contrast to Set 4, the effect of
using both the visual and auditory cues simultaneously [Set 5] significantly reduce
the bias against doublets and that of the sequential pairs. Moreover, the digit
difference histogram for Set 5 seems to resemble the ideal triangular distribution,
which is a signature of uniformly distributed numbers. We quantify the resemblance
of the stitched sequence for Set 5 by comparing its NNPM to that of a pseudo-
random sequence and note that they are indeed close to 1 (F = 1.00; C = 0.96;
Q = 0.91) and are within the variance of the results obtained using digit stitching
discussed in the last subsection for seven individuals.
For Set 6, where auditory cues are combined with visual cues that are biased
towards the digit 2, the histogram reflects this bias. While the rest of the numbers
have an average occurrence of 95 ± 17, the digit 2 (which appeared more often
in the board) appeared 153 times, which is more frequent by about ∼61% com-
pared to the mean. Despite the bias towards the digit 2 seen in the histogram,
the digit difference histogram shows no sequential pair preference or doublet aver-
sion, showing a triangular distribution. Comparison of the NNPM to that of a
pseudo-random sequence proves its similarity, with Linfoot’s criteria being close
to 1 (F = 1.00; C = 1.03; Q = 1.06). Note that the peak at n = 7 in IOT for
individually generated sequence no longer exists in both Sets 5 and 6, instead a
monotonically decreasing trend that is close to an exponential decay is observed
(Fig. 8).
Similar to the digit stitching procedure discussed in Sec. 3.4.1, the audi-
tory cues imposed here significantly weakens the intermittent and short tran-
sient biases (doublets and sequential pairs) during the number generation, while
at the same time, preserving the effect of visual cues in influencing the number
distribution.
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580 I. Crisologo et al.
Fig. 8. NNPM and IOT of generated sequence in the presence of auditory cues for Set 4 (novisual cues), Set 5 (with unbiased visual cues), and Set 6 (biased visual cues). The average IOTplot of each set fits well with an exponential decay (R2 = 0.94 for Set 4; R2 = 0.92 for Set 5;R2 = 0.96 for Set 6).
4. Conclusion
We quantified in this work the inherent biases reported when humans are tasked to
generate random numbers. We realized and verified the following limitations that
make human random number generation not directly feasible:
(i) the existence of a “natural set” numbers in human’s mind of which the digit
0 is not included,
(ii) the inability to generate longer sequences due to fatigue, and
(iii) the bias against doublets and sequential pairs.
The first one was lessened with the help of visual stimuli. The second human limita-
tion was remedied by repeated sampling. The third bias was extensively decreased
by either stitching the digits of independently-generated number sequences or using
auditory cues to stitch several individuals’ sequences in succession. We also demon-
strated that the IOT histogram of human generated sequence in the average, peaks
at ∆n = 7, a bias not reported in previous literatures. We showed here that this
bias was weakened by digit stitching or experimental stitching via auditory cues.
In summary, the use of unbiased visual cues (to make the frequency of occurrence
histogram flat) and stitching of the individually generated series both via digit
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Visual and Auditory Cues Significantly Reduce Human’s Intrinsic Biases 581
stitching and via auditory cues (to correct the bias on sequential pairs and doublets)
significantly reduce the main biases of humans during random number generation.
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