Phase II of Golden Ratio

7/31/2019 Phase II of Golden Ratio

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Golden Ratio ProjectAlejandra ZepedaTC-Stats (Day)



Abstract:

The experiment was conducted in attempt to obtain information about distributions of human arm

lengths. The motivation behind this experiment is to find out if the a golden ratio is found in the arm

lengths of college and high school students. The golden ratio has been thought of a universal ratio found

in our daily lives. We collected data from male and female college and high school students to conduct

our experiment. Our experiment consisted of a population sample of 25 male and 25 female students.

The variable we measured was the ratios found in the arm segments for the males and females. The

study consists of ratio scale data and a continuous variable. We did several tests and assumptions to

make sure our data gave us the best results. From our results we found out that males consist of the

golden ratio in their arms but it was not found in the female population.

Introduction:

An Italian mathematician by the name Leonardo Fibonacci discovered a number sequence. The unique

sequence is called the Fibonacci sequence. The numbers in the sequence are obtained by getting the sumof the preceding two numbers. However, he found a special pattern within those numbers. He noticed

that by dividing one number by the number before it, the quotients were close to each other. There was

in fact a more interesting discovery; it was that after the 13th number in the sequence that a fixed

quotient, 1.618, appeared. That special number is known as the Golden Ratio or Golden mean. Scholars

agree that this proportion is found in the human body, design, architecture, and in nature. The golden

ratio found in the human body is based on the ideal human form that scientist and researchers have

agreed upon ("Golden ratio in," 2008). Research shows that things that have the golden ratio are

pleasing to the eye. In fact, Justin Kuepper says "nature relies on this innate proportion to maintain balance.” The equation used to find the golden ratio is a/b = (a+b)/a. The golden ratio is believed to be

the building block of nature. So can this golden ratio be found in parts of the human body? This

experiment was conducted to prove that the Golden ratio does exist in the human body in particular, the

arms.

Methods:

The data was collected from populations of males and females in college and high school. Each student

was approached in random times and random orders throughout a 3 day period. The random students

were asked information about their age, gender, and arm lengths. Each person was chosen according to

the age group range of 16-21. A sample 25 male and 25 female college and high school students was

taken from the population. The sampling technique implemented in this data collection used to represent

each population was through randomization because we want the best representation of the population.

The information about the students was laid on a spreadsheet then transferred unto a data set in an Apple

application called TC- stats. This application was used to generate all the data analysis such as summary



statistics, hypothesis test, histograms, t and f distributions, and normal plots. During each inquiry the

student was asked to get their arms measured in parts. A 60" plastic measuring tape was used to take the

measurements. The measurements were taken from fingertip to shoulder, fingertip to elbow, elbow to

wrist, elbow to shoulder, and fingertip to wrist. The fingertip is considered the middle finger tip. These

measurements will help obtain the variable needed. The variable of interest for each sample population

is the ratio of finger and shoulder to finger and elbow (Ratio A), ratio of finger and elbow to wrist and

elbow (Ratio B), ratio of elbow and shoulder to finger and wrist (Ratio C) of only the left hand of the

sample population. In the appendix there is a restatement of what Ratio A, B, and C represent and how

they were calculated. The measurement scale is classified as ratio. The devices used to calculate

numbers, information and to measure were all kept the same during the experiment. Consider that the

measuring procedure could cause error by chance because some people moved when they were being

measured.

Note: Measuring tapes and rulers are not recommended by scientist to get a theoretical measurement of

each person.

In TC-Stats we were able to identify if the distributions of each population were normal to begin with.

The normal plots for each sample population are found in the appendix section. The normal plots for

each male variable is found under Norm 1a, 1b, 1c and the normal plot for each female variable ratio is

found in under Norm 2a, 2b, and 2c. First, we checked the male population based on the sample. m

normal plots for ratio A, B, and C found in males are not normal distributions. The normal plots for ratio

A, B, and C found in females are also not normal distributions. There are potential outliers and some

gross violations along the reference line. With this information we were able to compare the ratios for males and females. To truly include the value of the population parameter we can estimate a confidence

interval for each variable measured. A sign test for the medians was used to calculate each confidence

interval since the median is the best point estimate for distributions that are not normal. The significance

level we used for all tests was a .05. Concerning the medians of the variables we did a Kruskal Wallis

hypothesis tests.

Results:

It is important we look at the individual variables for each sample so we can see if the ratio of the body

parts agree with the Golden ratio theory. Below are descriptive statistics and results are for each variable

concerning the male sample population.

According to the summary statistics report the average ratio among men for Ratio A is 1.7015. The

standard deviation (SD) between individuals in the Ratio A is .0597 (Fig. 1). Figure 2 is a box-plot that

shows some leftward skewness that is why we consider the median as the point estimate, which is



1.7059. Based on the sign tests we are 95.7% sure that the true median of Ratio A is between 1.6857 and

1.7222.

Fig. 1

Fig. 2

Moving on to the second variable for males, our results report the average ratio among men for Ratio B

is 1.7461. The standard deviation (SD) between individuals in Ratio B is .0504 (Fig. 3). The box plot in

Figure 4 illustrates a mount shaped distribution and so we consider the mean as the best point estimate

which is 1.7461. Based on the one-sample T-test we are 95% confident that the true mean of Ratio B is

between 1.7253 and 1.7669.

Fig. 3



Fig. 4

The third variable for male Ratio C has a sample's data average ratio of 1.6453. The standard deviation

between these individuals in Ratio C . is .1486 (Fig. 5). The box-plot shows skewness to the left. Based

on the distribution the median was considered as the point estimate which is 1.6667. Based on the sign

test, we are 95.7% certain that the true median for Ratio C is between 1.5625 and 1.7333.

Fig. 5

Fig. 6

When we compare the statistical significance of these confidence intervals we know that Ratio C's

confidence intervals overlap with both Ratio A and B. Therefore the prevalence are deemed to not be

significantly different. In the other hand, Ratio A and B do not have overlapping confidence intervals.

Therefore the prevalence estimates are significantly different. Based on the sign tests for confidence

intervals of Ratio A and C we can say that there is enough evidence to say that the ratio of finger and



shoulder to finger and elbow (Ratio A) and ratio of elbow and shoulder to finger and wrist (Ratio C) are

within the golden ratio (1.618). However, Ratio B is not within the golden ratio. To confirm this a

Kruskal Wallis hypothesis test says that there is at least one of the ratios is different and by looking at

the data we can see that Ratio C is the different one.

The next data analysis was for the female population were we compared the same ratios (A, B, C).

Below are descriptive statistics and results are for each variable concerning the female sample

population. According to the summary statistics report, the average among females for Ratio A is

1.73778. The standard deviation (SD) between individuals in the Ratio A is .0469. (Fig. 7). Figure 8 is a

box-plot that shows a right skewed distribution and that is why we consider the median as the point

estimate, which is 1.7273. Based on the sign tests we are 95.7% sure that the true median of Ratio A is

between 1.7097 and 1.7500.

Fig. 7

Fig. 8

The descriptive statistics for the second variable said that the average among the female sample for ratio

B is 1.7722. The standard deviation between individuals for Ratio B is .1651 (Fig. 9). The box plot in

Figure 10 illustrates a very skewed right distribution and thats why we considered the median as the

point estimate. The median is 1.75 and based on the sign test we are 95.7% certain that it is between

1.7000 and 1.7714.



Fig. 9

Fig. 10

The third variable of interest is the ratio of elbow and shoulder to finger and wrist (Ratio C). Ratio C

data had an average of 1.7164. The standard deviation between individuals in Ratio C is .1469 (Fig.11).

Figure 12 is a box plot demonstrating that the data for Ratio C has some skewness to the left and for that

reason the median is the best point estimate. We are 95.7% certain that the true median is between 1.629

and 1.7857.

Fig. 11

Fig. 12



When we examine the statistical significance of these confidence intervals for the female sample

population we know that all the ratios are overlapping each other. Therefore the prevalence are deemed

to not be significantly different. Based on the sign tests for confidence intervals of Ratio A and B we can

say that there is enough evidence to say that the ratio of finger and shoulder to finger and elbow (Ratio

A) and ratio of elbow and shoulder to finger and wrist (Ratio C) are not within the golden ratio (1.618).

However, Ratio B has contains the golden ratio. The Kruskal Wallis hypothesis test there is insufficient

evidence to say that at least one of the ratio data was different.

Discussion:

According to scientists the golden ratio is found in the human body. Based on the experiment males

could have the ratio because only the left arm was measured for this experiment. Knowing that one arm

is proportional then it is possible that the rest of the body is made up of the golden ratio. The initial

hypothesis said the golden ratio did exist, however, the results from this experiment did not give us

sufficient evidence to make confident statements about the existence of the golden ratio. We need to testmore body parts and use a better measuring tool in order to make a better inference about the

populations. Clearly, the findings of this experiment were not sufficient to agree with what scientists say

about the Golden Ratio.

Conclusion:

This experiment served as a initial process of a deeper experiment. A deeper experiment where various

body parts of the human body in both males and females can be measured. However, the data we

collected is important because it gave us an idea of the possibility of the existence of the golden ratio inthe human body. This implies that there is a great chance that the golden ratio does exist since the

average ratios found in the left arm were near each other.



Bibliography:

Duke University. (2009, December 21). Researchers explain mystery of golden ratio. Retrieved from

http://www.physor g.com/news180531747.html

Golden ratio in human body [Web]. (2008). Retrieved from http://youtube.com/watch?v=085KSyQVb-

U

Kuepper, J. (2004, March 31). Fibonacci and the golden ratio. Retrieved from

http://www.investopedia.com/articles/technical/04/033104.asp#axzz1nzcXaVgP

Lahanas, M. The Golden Section and the Golden Rectangel Retrieved from www.mlahanas.de/Greeks/

GoldenSection.htm

Sage, E. The golden ratio~fingerprint of "God" Retrieved from http://worldtruth.tv/the-golden-

ratiofingerprint-of-god-2/

Appendix:

F= Female

M= Male

Ratio A= ratio of finger and shoulder to finger and elbow

Ratio B= ratio of finger and elbow to wrist and elbow

Ratio C= ratio of elbow and shoulder to finger and wrist

Gender Ratio A Ratio B Ratio C

F 1.7419 1.8235 1.6429

F 1.6562 1.7778 1.5

F 1.7419 1.7222 1.7692

M 1.6176 1.7895 1.4

F 1.7273 1.7368 1.7143

M 1.7647 1.7895 1.7333

F 1.746 1.6579 1.88

M 1.6176 1.7 1.5



M 1.6944 1.8 1.5625

M 1.7778 1.7143 1.8667

M 1.697 1.7368 1.6429

M 1.5821 1.8108 1.3

F 1.7097 1.7714 1.6296

M 1.7222 1.7143 1.7333

F 1.7742 1.6757 1.92

F 1.7231 1.8056 1.6207

M 1.7941 1.7895 1.8

M 1.7941 1.7895 1.8

M 1.7273 1.7838 1.6552

M 1.6471 1.7436 1.5172

F 1.6562 1.6842 1.6154

F 1.7273 1.7368 1.7143

F 1.7188 1.7778 1.6429

F 1.7576 1.7368 1.7857

F 1.8621 1.8125 1.9231

M 1.7143 1.8421 1.5625

F 1.7097 1.7222 1.6923

M 1.7143 1.75 1.6667

M 1.7105 1.6522 1.8

F 1.7647 1.7 1.8571

F 1.7143 1.8065 1.6

F 1.7143 1.75 1.6667

M 1.7353 1.7895 1.6667

M 1.7222 1.6744 1.7931



Calculations of obtaining the ratios:

male ratios:

Divide the measurement of finger to shoulder by finger to elbow to get a Ratio A.

Divide the measurement for finger to elbow by wrist to elbow to get a Ratio B.

Divide the measurement for elbow to shoulder by finger to wrist to get a Ratio C.

female Ratios:

Divide the measurement of finger to shoulder by finger to elbow to get a Ratio A.

Divide the measurement for finger to elbow by wrist to elbow to get a Ratio B.

M 1.7027 1.6818 1.7333

M 1.6812 1.7692 1.5667

F 1.8 1.7647 1.8462

F 1.75 1.6842 1.8462

F 1.7 1.7647 1.6154

F 1.8 1.8182 1.7778

M 1.7059 1.7 1.7143

M 1.7576 1.7368 1.7857

M 1.6857 1.7949 1.5484

F 1.8 1.7647 1.8462

F 1.7647 2.5185 1.2683

F 1.6875 1.6 1.8333

M 1.7059 1.7 1.7143

M 1.7059 1.7 1.7143

M 1.5588 1.7 1.3571

F 1.697 1.6923 1.7037



Divide the measurement for elbow to shoulder by finger to wrist to get a Ratio C.

Normal plot for male sample:

Ratio A Ratio B Ratio C

Normal plot for female sample:

Ratio A Ratio B Ratio C

Kruskal Wallis hypothesis test for male sample:

null hypothesis: θra = θrb = θrc

alternative hypothesis: at least one is ≠

α=.05 p-value: .0211

(subscript ra is ratio A, rc is ratio B, and rc is ratio

C)

Reject the null hypothesis.

Kruskal Wallis hypothesis test for female sample:

null hypothesis: θra = θrb = θrc

alternative hypothesis: at least one is ≠

α=.05 p-value: .6662

(subscript ra is ratio A, rc is ratio B, and rc is ratio

C)

Fail to reject the null hypothesis

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Phase II of Golden Ratio