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Mateo Solano
The Golden Ratio Project
Stats 1510 Night
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Abstract
This research paper covered the topic of the Golden Ratio and whether or not this ratio
appears to be derived from several measurements taken from specific areas of the
human body. Male and Female subjects were measured using a standard body tape in
order to obtain specific measurements and then calculate a total of two ratios for each
sex. Results indicate (p < .05) that the Golden Ratio of 1.6180399 is not equal to either
of the two ratios for male or female from the measurements taken from their bodies
Introduction
The Golden Ratio is a number that seems to appear everywhere in nature so much so
that it has also been called the Divine Proportion because many believe that this
number is all by which nature is governed. It is based on the Fibonacci numbers. These
are a series of numbers that begin with 0 and 1 after that you follow the rule of adding
the last two numbers to get the next 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,
987,... and so on. "A special value, closely related to the Fibonacci series, is called the
golden section. This value is obtained by taking the ratio of successive terms in the
Fibonacci series"(1). Phi is equal to approximately 1.61803399. "The Golden Section,
also known as Phi, is manifested in the structure of the human body. If the length of the
hand has the value of 1, for instance, then the combined length of hand + forearm has
the approximate value of Phi." (2) Similarly the proportion of navel to top of head to
shoulder line to top of head is in the same ratio of 1: Phi .The purpose of this research
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project was to determine if in fact these measurements would result in the ratio of 1:Phi
or the Golden Ratio. The measurements were taken from two populations the first was
adult males over the age of 18 and the second was adult females over the age of 18.
The Golden Section is yet another name for the Golden Ratio. The Golden Ratio is
denoted by the Greek symbol Phi and can be determined when the measurement of
A is added to the measurement of B and divided by the measurement of B giving you (A
+ B)/B. The variable A represents the measurements from the shoulder line to head in
Ratio # 1 and represents the measurements from the wrist to elbow in Ratio # 2 for both
males and females. The variable B represents the measurements from the navel to the
top of head for Ratio #1 and the measurements from the middle finger tip to elbow for
Ratio #2 for both males and females. This gives you the calculation equivalents for
Ratio #1 and Ratio #2 for both males and females as: {(A) shoulder line to top of head
+ (B) navel to top of head / (B) navel to top of head} and {(A) wrist to elbow + (B) middle
finger tip to elbow / (B) middle finger tip to elbow} respectively. In collecting the data 40
males and 40 females for a total of 80 subjects were measured for the length in inches
from the above mentioned areas to establish the two ratios and determine whether
those ratios were in fact calculated to be the Golden Ratio. These two sets of data were
then compared to each other.
Methods and Materials
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A simple random sample was taken from a population of adult males and adult females
over 18 years of age for a total 40 males and 40 females. Each were approached in
various locations and asked if they wouldn't mind being measured from their fingertips
to their elbows, from their wrist to their elbows, as well as from their shoulder line to the
top of their heads, and from their navels to the top of their heads. Each subject was
measured using a five foot body measuring tape. The results were recorded on paper
and transferred into tcstats on an ipad.
Results
Neither the males nor the female measurements when calculated using the formula (A +
B)/B resulted in the Golden Ratio and although the mean for both male and female ratio
# 2 did produce a number close to Phi it was in fact not Phi. Verifying this are the
results from the 1 sample t test performed for each of the ratios; two for the female and
two for the male. For female ratio #1 the p value from the one sample t test was 8.969
E-34 or approximately 0.0000. This p value indicates that the null hypothesis should be
rejected and therefore indicating that the mean for female ratio #1 is not equal to the
Golden Ratio of 1.6180399. The same holds true for female ratio # 2, male ratio # 1
and male ratio # 2 with their p values being 2.654 E-33 or approximately 0.0000, 2.654
E-33 or approximately 0.0000, and 0.0247 respectively. Table 1.1 lists the five number
summary for each ratio both female and male respectively along with the population
standard deviation and sample standard deviation, as well as the sum of all the data for
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the sample which was equivalent to 40 males and 40 females or n = 40; n being the
sample size. Included in this table is also the five number summary of female and male
respectively for each of the measurements, along with the population standard deviation
and sample standard deviation, as well as the sum of all the data for the sample which
was equivalent to 40 males and 40 females or n = 40; n being the sample size. The five
number summary consists of the minimum data value for each ratio, the Q1 value, such
that no more than 25% of all values are smaller and no more than 75% are larger, the
mean which is the average and the number that represents the center point of the data,
the Q3 value, such that no more than 75% of all values are smaller and no more than
25% are larger, and finally the maximum value of the data set. As you can see from
reading table 1.1 the average for female ratio #1 was 1.415585 which represented the
mean of the sum of the measurements from the shoulder line to the top of the head and
the navel to the top of the head divided by the length of measurement from the navel to
the top of the head. This number was not the Golden Ratio. The same calculation was
used to calculate female ratio #2 which was the sum of the measurements from the
middle fingertip to the elbow and the wrist to the elbow divided by the length of
measurement from the middle finger tip to the elbow; and as can be read in table 1.1
the average of the this data set came out to be 1.613727 which again comes close to
the numerical value of Phi but is not the Golden Ratio itself.
In this research age was not taken into consideration other than the fact that the
subjects were considered to be adults if over the age of 18. The only place where the
data might have been affected from this would be in the female measurements from the
navel to the top of the head as there was a deviation of over one and a half inches.
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Although even with this deviation the deviation in ratio #1 was very small, the sample
standard deviation being 0.03082269. Likewise the deviation in ratio #2 was also very
small, 0.0340273.
The same holds true for the male data listed in table 1.1. The standard deviation was
more than one and a half inches (1.713) from the navel to the top of the head but the
overall deviation for the ratio was very small (0.03323601). The male data also when
input into the calculation for the Golden Ratio failed to produce Phi as the result. Also
just as in the female data the mean for ratio #2 was numerically close to the Golden
Ratio but was in fact not Phi. All of the figures for the male data can be seen in table 1.1
A visual representation of the data can be seen in figures 1.1 for both of the female and
male ratios. Both sets of box and whisker plots show the distribution of data to be
symmetrical. You can see from these graphs that the offset of the data is similar in both
males and females. The graphs are based on the data from tables 1.1 for the female
and male five number summaries respectively.
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Table 1.1
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Figure 1.1
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Bibliography
1.)Jovanovic, 2001-2003, Fibonacci Numbers and the Pascal Triangle. GoldenSection.http://milan.milanovic.org/math/english/golden/golden2.html, April 2012
2.) R. Knott, D.A. Quinney and Pass Maths, 1997, The Life and Numbers of
Fibonacci. Plus MagazineLiving Mathematics.http://plus.maths.org/content/os/issue3/fibonacci/index, April 2012
http://milan.milanovic.org/math/english/golden/golden2.htmlhttp://milan.milanovic.org/math/english/golden/golden2.htmlhttp://milan.milanovic.org/math/english/golden/golden2.htmlhttp://plus.maths.org/content/os/issue3/fibonacci/indexhttp://plus.maths.org/content/os/issue3/fibonacci/indexhttp://plus.maths.org/content/os/issue3/fibonacci/indexhttp://milan.milanovic.org/math/english/golden/golden2.html7/31/2019 Stats Report1
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Appendix
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AppendixAdult female
data Head to
Shoulder Line
Navel to top of
Head
Female Ratio #1 Finger to Elbow Wrist to Elbow Female Ratio # 2
10.5 26 1.44 17.5 11.5 1.6571
10 25.5 1.3922 16.5 10 1.6061
11.5 27 1.4259 17 10.5 1.6176
11 30.5 1.3607 17.5 10 1.5714
11 26 1.4231 17 11 1.6471
12 27.5 1.4364 17.5 11 1.6286
11 25 1.44 17.5 11 1.6286
10.5 24 1.4375 16.5 10.5 1.6364
10 25.5 1.3922 17 11.5 1.6765
11 26 1.4231 17.5 10 1.5714
11 27 1.4074 15 9 1.6
10 26 1.3846 16 10.5 1.6563
12 27 1.4444 18 10.5 1.5833
11 23 1.4783 15 8.5 1.56679.5 25 1.38 16 9 1.5625
10 25 1.4 14.5 9 1.6207
10 24.5 1.4082 16.5 10.5 1.6364
11 29 1.3793 15.5 9 1.5806
11 28 1.3929 16.5 10 1.6061
11.5 25 1.46 15.5 9 1.5806
10.5 25.5 1.4118 14.5 9.5 1.6552
11 25 1.44 14.5 9 1.6207
10.5 27 1.3889 17.5 11 1.6286
11.5 28.5 1.4035 17 10.5 1.6176
11.5 27 1.4259 16.5 10.5 1.636410.5 27 1.3889 17 11 1.6471
10.5 23 1.4565 16 9.5 1.5938
9 27 1.3333 15 10 1.6667
10.5 24 1.4375 16 10 1.625
10 25 1.4 16.5 9 1.5455
11 25 1.44 16 9.5 1.5938
11.5 25.5 1.451 16.5 10 1.6563
10 22.5 1.4444 16 9.5 1.5938
10 25 1.4 16 10 1.625
11 25 1.44 16 9 1.5625
10 24.5 1.4082 16 9 1.5625
9.5 26.5 1.3585 18 10.5 1.5833
10 24 1.4167 18 11.5 1.6389
11 24.5 1.449 16 10.5 1.6563
11 26 1.4231 16.5 10 1.6061
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Mateo Solano
The Golden Ratio & the Human Body
Stats 1510 Night
Introduction
The Golden Ratio is a number that seems to appear everywhere in nature so much so that it has also been
called the Divine Proportion because many believe that this number is all by which nature is governed. It
is based on the Fibonacci numbers. These are a series of numbers that begin with 0 and 1 after that you
follow the rule of adding the last two numbers to get the next 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610,987,... and so on. "A special value, closely related to the Fibonacci series, is called the golden
section; this value is obtained by taking the ratio of successive terms in the Fibonacci series"(1). The
Golden Section is yet another name for the Golden Ratio. The Golden Ratio is denoted by the Greek
symbol Phi and can be determined when the measurement of A is added to the measurement of B and
divided by the measurement of A giving you (A + B)/A. Phi is equal to approximately 1.61803399."The
Golden Section, also known as Phi, is manifested in the structure of the human body. If the length of the
hand has the value of 1, for instance, then the combined length of hand + forearm has the approximate
value of Phi." (2) Similarly the proportion of navel to top of head to + shoulder line to top of head is in
the same ratio of 1: Phi .The purpose of this research project was to determine if in fact these
measurements would result in the ratio of 1:Phi or the Golden Ratio. The measurements were taken from
two populations the first was adult males over the age of 18 and the second was adult females over the
age of 18. In collecting the data 40 males and 40 females for a total of80 subjects were measured for the
length in inches from the above mentioned areas to establish the two ratios and determine whether those
ratios were in fact calculated to bathe Golden Ratio. These two sets of data were then compared to each
other.
Methods and Materials
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In collecting the data 40 random males and 40 random females were approached in various locations and
asked if they wouldn't mind being measured from their fingertips to the their elbows from their wrist to
their elbows and from their shoulder line to the top of their heads and from their navels to the top of their
heads. Each subject was measured using a five foot body measuring tape. The results were recorded on
paper and transferred into tcstats on an ipad.
Results
Neither the males nor the female measurements when calculated using the formula (A +B)/A resulted in
the Golden Ratio and although the mean for both male and female ratio# 2 did produce a number close to
Phi it was in fact not Phi. Tables 1.1 and 2.1 list the five number summary for each ratio both female and
male respectively along with the population standard deviation and sample standard deviation, as well as
the sum of all the data for the sample which was equivalent to 40 males and 40 females or n = 40; being
the sample size. Tables 1.2 and 2.2 list the five number summary of female and male respectively for each
of the measurements, along with the population standard deviation and sample standard deviation, as well
as the sum of all the data for the sample which was equivalent to 40 males and 40 females or n = 40; n
being the sample size. The five number summary consists of the minimum data value for each ratio,
theQ1 value, such that no more than 25% of all values are smaller and no more than 75%are larger, the
mean which is the average and the number that represents the center
point of the data, the Q3 value, such that no more than 75% of all values are smaller and no more than
25% are larger, and finally the maximum value of the data set. As you can see from reading table 1.1 the
average for female ratio #1 was 1.415585 which represented the mean of the sum of the measurements
from the shoulder line to the top of the head and the navel to the top of the head divided by the length of
measurement from the navel to the top of the head. This number was not the Golden Ratio. The same
calculation was used to calculate female ratio #2 which was the sum of the measurements from the middle
fingertip to the elbow and the wrist to the elbow divided by the length of measurement from the middle
finger to the elbow; and as can be reading Table 1.1 the average of the this data set came out to be
1.613727 which again comes close to the numerical value of Phi but is not the Golden Ratio itself. In this
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research age was not taken into consideration other than the fact that the subjects were considered to be
adults if over the age of 18. The only place where the data might have been affected from this would be in
the female measurements from the navel to the top of the head as there was a deviation of over one and a
half inches. Although even with this deviation the deviation in ratio #1 was very small, the sample
standard deviation being 0.03082269. Likewise the deviation in ratio #2 was also very small,
0.0340273.The same holds true for the male data listed in Tables 2.1 and 2.2. The standard deviation was
more than one and a half inches (1.713) from the navel to the top of the head but the overall deviation for
the ratio was very small (0.03323601). The male data also when input into the calculation for the Golden
Ratio failed to produce Phi as the
result. Also just as in the female data the mean for ratio #2 was numerically close to the Golden Ratio
but was in fact not Phi. All of the figures for the male data can be seen in Tables 2.1 and 2.2A visual
representation of the data can be seen in figures 1.1 for and 1.2 for both of the female ratios and 2.1 and
2.3 for the male ratios. Both sets of box and whisker plots show the distribution of data to be symmetrical.
The graphs are based on the data from Tables 1.1 and 2.1 for the female and male five number summaries
respectively.
Table 1.1
Table 2.1
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Table 1.2
Table 2.2
Figure 1.1
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Figure 2.1
Appendix
Table 1.1
Table 2.1
Table 1.2
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Table 2.2
Figure 1.1
Figure 2.1
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AppendixAdult female
data Head to
Shoulder Line
Navel to top of
Head
Female Ratio #1 Finger to Elbow Wrist to Elbow Female Ratio # 2
10.5 26 1.44 17.5 11.5 1.6571
10 25.5 1.3922 16.5 10 1.6061
11.5 27 1.4259 17 10.5 1.6176
11 30.5 1.3607 17.5 10 1.5714
11 26 1.4231 17 11 1.6471
12 27.5 1.4364 17.5 11 1.6286
11 25 1.44 17.5 11 1.6286
10.5 24 1.4375 16.5 10.5 1.6364
10 25.5 1.3922 17 11.5 1.6765
11 26 1.4231 17.5 10 1.5714
11 27 1.4074 15 9 1.6
10 26 1.3846 16 10.5 1.6563
12 27 1.4444 18 10.5 1.5833
11 23 1.4783 15 8.5 1.56679.5 25 1.38 16 9 1.5625
10 25 1.4 14.5 9 1.6207
10 24.5 1.4082 16.5 10.5 1.6364
11 29 1.3793 15.5 9 1.5806
11 28 1.3929 16.5 10 1.6061
11.5 25 1.46 15.5 9 1.5806
10.5 25.5 1.4118 14.5 9.5 1.6552
11 25 1.44 14.5 9 1.6207
10.5 27 1.3889 17.5 11 1.6286
11.5 28.5 1.4035 17 10.5 1.6176
11.5 27 1.4259 16.5 10.5 1.636410.5 27 1.3889 17 11 1.6471
10.5 23 1.4565 16 9.5 1.5938
9 27 1.3333 15 10 1.6667
10.5 24 1.4375 16 10 1.625
10 25 1.4 16.5 9 1.5455
11 25 1.44 16 9.5 1.5938
11.5 25.5 1.451 16.5 10 1.6563
10 22.5 1.4444 16 9.5 1.5938
10 25 1.4 16 10 1.625
11 25 1.44 16 9 1.5625
10 24.5 1.4082 16 9 1.5625
9.5 26.5 1.3585 18 10.5 1.5833
10 24 1.4167 18 11.5 1.6389
11 24.5 1.449 16 10.5 1.6563
11 26 1.4231 16.5 10 1.6061
