### Height Distributions & Taboos

```Alexa Curcio
•Original
Problem: Would a restriction on height, such as
prohibiting males from marrying taller females, affect the
height of the entire population?
•Height: Normally distribution, males are taller, same SD
•U.S.: Males: 69 inches, Females: 64 inches, SD: 2.8 inches
•No correlation between arranged marriages & height
•Parameters & Assumptions:
•Difference
in mean
•Standard Deviation
•Offspring calculation
•What creates a stable population
•100 males and females, each couple as one male and one female child
•Offspring
Calculation:
•Mean:
average mother and father height, add 2.5 inches
for son, subtract 2.5 inches for daughter
•Other mean: 1/3(opposite gender) + 2/3(same gender)
•SD: use variance of mother and father to find variance of
children [V(F+M/2)], SD = 2
•Stable
Population: Use U.S. means and standard
deviations. Generate children by choosing one male
and one female, then use the offspring calculation to
calculate the child’s height. Switch generations by
making son the father and daughter the mother.
•Taboos:
•Male
must be taller than female
•Difference in height between males and females remained constant.
•Means of both males and females remained relatively the same, with a
difference of about 1 or 2 inches.
•Why? Because of the difference in height and same standard
deviation. Odds of a female meeting a male shorter than she is are
very small
•Same
taboo, for more generations
•Same result
•Larger
taboo: males must be 10 inches taller
•Same result
•Larger
taboo, for more generations
•Same result
•Questions
•Clarifying assumptions & parameters
•Offspring calculation
•Comparing to normal curve
•Placing a cap on height
•Using more runs
•Graphs
of standard deviation and means over time
without taboo
•Graphs of standard deviation and means over time
with taboo
•Reversing Taboo
•Starting at Same Mean
•Conclusion
•Other thoughts…
•Accounting
for birth defects: The amount would be so
small, and the chances of this person procreating are not
likely – this would not be helpful for such a big model
•Children taller than parents: R code already allows this, it
has built in noise by using normal random number
generator, and the formula for offspring allows for sons to
be taller than their parents
•Code already takes into consideration the heights of
grandparents: new son and daughter based off of parents,
heights of parents are based on the parents of the parents,
or grandparents
•Based assumptions on what would create a stable
population (used U.S. as example, calculated offspring SD)
•Since
height depends on many other factors, not really
a widely accepted formula
•This formula was used by many pediatricians to
determine the height of a child before it’s birth.
•There is another formula (accepted by many) used to
calculate a child’s height but depends on child’s height
after a certain age.
•Added/subtracted 2.5 inches because it allowed for a
stable population (females shorter than males)
•Son:
(F+M)/2 + 2.5
•Daughter: (F+M)/2 - 2.5
•Normal
curve with
Mean=64 SD= 2.8
•Normal
curve with
Mean=69 SD=2.8
•Cap
on height: Code already excludes outliers
•Create a taboo that makes sure males can not
be taller than 7 feet
•Results: Did not affect height of population,
remained relatively the same.
•Why? – Code already takes this into
consideration when using a standard
deviation. The likelihood of producing many
males taller than 7 feet is very small.
•To
add more runs, place entire code into
another loop.
•Store the mean of each generation at the end
of each generation
•Print all of these in a histogram at the end of
the code to show how much variability there is
in the results.
•Mean
heights
varied.
•Difference
between the
mother and father
height remained
relatively constant,
ranging from 4.55.5 inches
•In
R: Store value of means and standard
deviations in separate vectors outside of the for
loop
•Inside loop, store the generation by generation
values
•At the end of the loop plot a vector against the
vector 1:gens.
•This will give the means of men over time on one
graph, the standard deviation of men over time in
one graph, means of women over time in one
graph, and standard deviation of women over
time in one graph.
•IN
R:
•Graph
of men and women mean before taboo
•Graph of men and women SD before taboo
•Graph of men and women mean after taboo
•Graph of men and women SD after taboo
•Their
graphs follow
a similar pattern
(keeping the
difference between
male and female
height relatively
constant.)
•Generally increases.
•The
standard
deviation for
both males
and females
fluctuates.
•So,
means of men
and women
fluctuate between
their mean and
below their mean.
Their graphs follow
a similar pattern
(keeping the
difference between
male and female
height relatively
constant.
•Similarly,
the
standard
deviation for
both males
and females
fluctuates.
•In
R, simply reverse the taboo: Have females only
marry men that are shorter than they are
•Result:
Height of population either increases by
•Depending on the height of the first population, if
females are generally shorter, then the males must
also be significantly shorter.
•If females are generally taller, then the males will
remain about the same height and then drastically
increase in height
•Height
tended
to decrease
most of the
time, but
sometimes just
stayed the
same.
•Started
at mean:65
•Used average of parents to calculate height
•Found that if the height went up for one
gender, it similarly went up for the other
gender and if it went down, it went down for
both.
•Inserting more runs…
•Inserting
more
runs showed that
usually the height
of the population
increased.
•The
difference in
height for men
and women
remained
relatively the
same (0 inches)
•Start
with the same mean, add taboo, but
calculate offspring height by using 1/3(other
gender) + 2/3(same gender)
•Result: A little taller…
more runs,
found that this taboo
led to a much taller
population.
•The
gap between
males and females
widened and males
were a little taller
than females.
more runs led to a more concrete answer.
Could easily find the means of each generation
and used this to determine what “type” of
population it often led to.
•Biggest change occurred when starting with the
same means, using the formula more dependent
on gender, and inserting the taboo and more runs.
•Using more runs showed that there was a lot of
variability for many of the taboos.
•The graphs of means and sd’s over time showed
that both tend to fluctuate.
•Taking
into account weight:
•A
little too difficult for this simulation.
•Weight depends on many more environmental
factors and genetics.
•No real correlation between height and weight.
•Likelihood
of a child to be taller than a parent:
•Research:
probability of a son being taller given his
father is taller than average: 71%
•Probability son being taller unconditionally: 50%
•Animal
•In
heights:
most mammals, the male is larger than the
female, no clear reason as to why
```