VISUAL AND AUDITORY CUES SIGNIFICANTLY REDUCE HUMAN'S INTRINSIC BIASES WHEN TASKED TO GENERATE A...

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∗[email protected]

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

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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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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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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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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.

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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.

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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

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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

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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

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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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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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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.

May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324


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

May 20, 2010 9:4 WSPC/141-IJMPC S0129183110015324


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.

References

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VISUAL AND AUDITORY CUES SIGNIFICANTLY REDUCE HUMAN'S INTRINSIC BIASES WHEN TASKED TO GENERATE A...

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