Size of a Human: Body Proportions
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Abstract
The purpose of this experiment is to verify the proportionality of the human body.
Introduction
Nature has always been admired for its patterns and symmetry. The human body is an example of nature's proportion. Phi, the Golden Number 1.618, is a proportion found in many areas of the natural world as well as in the structure of the human body. Many of the bones that form our skeleton are thought to have a proportional relationship of 1:1.618. The human body itself is believed to be symmetrical as displayed in Leonardo DaVinci's Vitruvian Man in which a man with outstretched arms fits exactly into a square.
Diagrams
Procedure
 Have subject stand as in diagram 1.
 Use tape measure to the length of subject's wingspan [tip of middle finger of one hand to the top of the middle finger of the other hand]. Record length.
 Have subject take off shoes and stand upright as in diagram 1. Measure and record height.
 Have subject stretch out one arm and measure the length of his/her forearm [from elbow and wrist as shown in diagram 2]. Record forearm length.
 Have subject flatten their hand and measure the length of his/her forearm and hand [from the elbow to the tip of middle finger as shown in diagram 2]. Record hand length.
 Repeat steps 16 using 25 subjects.
Data
The data collected can be found here in excel format.
Wingspan (cm) 
Height (cm) 
WingspanHeight proportion 
Forearm+Hand (cm) 
Forearm (cm) 
ForearmHandForearm proportion 

172.3  172  1.002  44.5  26  1.712 
159.4  154.4  1.032  41.4  24.4  1.697 
182.1  178.2  1.022  46.1  26.4  1.746 
169.6  168.6  1.006  43.2  23.6  1.831 
164.5  167.6  0.982  43.1  26  1.658 
165.5  169.4  0.977  41.3  23  1.796 
172.4  167  1.032  46.8  27.2  1.721 
163.2  164.2  0.994  43.6  26  1.677 
164.5  162  1.015  42.5  25.5  1.667 
162.5  160.5  1.012  43.8  26  1.685 
180.5  173.5  1.040  47  27.5  1.709 
170.9  161.5  1.058  45.7  26  1.758 
150.5  150.5  1.000  39.1  23.8  1.643 
165.6  161.4  1.026  42.7  26  1.642 
175.1  165.9  1.055  46.6  27.6  1.688 
175  166  1.054  46.3  27.2  1.702 
192.5  187  1.029  46.7  27  1.730 
188.5  179.9  1.048  47.8  28  1.707 
184  176.5  1.042  47.5  27.5  1.727 
169  168.6  1.002  43.6  25.2  1.730 
178.4  174.1  1.025  47.7  28  1.704 
167  166.5  1.003  45.2  26  1.738 
158  157  1.006  42.4  24.6  1.724 
190.7  179  1.065  48.7  27.6  1.764 
48.8  28  1.743 
Analysis
It was previously believed that the wingspan of a person is equal to the height of that same person. It is also believed that the proportion of a person's forearm to their hand is in a ratio of 1.6.
The proportion of wingspan to height is found by dividing the wingspan of the person over their measured height. Graphically the proportion of height to wingspan is represented by coefficient A on the proportional fit equation, y=A*x, shown on the Wingspan vs Height graph. The proportion was found to be 1.023.
To find the proportion of forearm+hand to forearm, we divided the forearm+hand of the person over their measured forearm. Graphically the proportion of forearm+hand to forearm is represented by coefficient A on the proportional fit equation, y=A*x, shown on the forearm+hand vs forearm graph. The proportion was found to be 1.715
To further statistically analyze the calculated measurements, we performed a ttest to determine whether the proportions found were significantly different to the established values.



H_{O}: Body Proportion Height/Wingspan = 1 H_{A}: Body Proportion Height/Wingspan 1 
H_{O}: Body Proportion Forearm Hand/Forearm = 1.618 H_{A}: Body Proportion Forearm Hand/Forearm ≠ 1.618 
n = 24 people y = 1.044 s = 0.1166 t = 1.835 
n = 25 people y = 1.698 s = 0.0449 t = 10.797 
p = 0.079  p = 1.076E10 
The pvalue of 0.079 is not small enough for us to reject the hypothesis that the true mean body proportion is 1. We conclude that there is not enough evidence to say the proportion is not 1.  The pvalue of 1.076E10 is small enough for us to reject the hypothesis that the true mean body proportion is 1.618. We conclude that there is enough evidence to say the proportion is not 1.618 
Conclusion
We found the proportion of Height to Wingspan to be 1.023 which is within 2.3% error of the established value of 1. The onesample Ttest concluded that there is not enough evidence to say the proportion is not 1.
We found the proportion of Forearm+Hand to forearm to be 1.715 which is within 6% error of the established value of 1.618. The onesample Ttest concluded that there is enough evidence to say the proportion is not 1.618, phi.
Sources of Error
 When measuring the individuals forearm, we used our own judgment to determine where the forearm bone began. It is possible that these measurements were incorrect and affected the calculation of the proportion.
 To measure the wingspan of each subject we had the subject stretch his/her arms out horizontally or as horizontal as possible. We used no instruments to make sure each arm was straight, in alignment with each other, or horizontal; it was done by eye. It is possible that the wingspan was measured to be shorter than it should be.
Stacey Johnson, Kristine McPherson  2006
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