# 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

1. Have subject stand as in diagram 1.
2. 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.
3. Have subject take off shoes and stand upright as in diagram 1. Measure and record height.
4. 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.
5. 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.
6. Repeat steps 1-6 using 25 subjects.

## Data

The data collected can be found here in excel format.

Wingspan
(cm)
Height
(cm)
Wingspan-Height
proportion
Forearm+Hand
(cm)
Forearm
(cm)
ForearmHand-Forearm
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 t-test to determine whether the proportions found were significantly different to the established values.

One-Sample T-test
1. Individuals were independent.
Each individual's measurement was not dependent on another's.
2. Random Sample and <10% of the population.
We assume the students were selected randomly.
We can safely assume that 25 people is less than 10% of the human population.
3. Data is normally distributed.
HO: Body Proportion Height/Wingspan = 1
HA: Body Proportion Height/Wingspan 1
HO: Body Proportion Forearm Hand/Forearm = 1.618
HA: 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 = 1.076E-10
The p-value 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 p-value of 1.076E-10 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 one-sample T-test 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 one-sample T-test concluded that there is enough evidence to say the proportion is not 1.618, phi.

## Sources of Error

1. 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.
2. 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 -- 2005

Students Choice
1. Physics of the Body
2. Miscellaneous Topics