HEIGHT DISTRIBUTION

PART 2

Alexa Curcio

REVIEW•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

REVIEW•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.

REVIEW•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

OUTLINE•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…

QUESTIONSPARAMETERS & ASSUMPTIONS

•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)

QUESTIONSOFFSPRING FORMULA

•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

QUESTIONSNORMAL CURVE COMPARISON

•Normal curve with Mean=64 SD= 2.8

•Normal curve withMean=69 SD=2.8

QUESTIONSCAP ON HEIGHT

•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.

QUESTIONSMORE RUNS

QUESTIONSMORE RUNS

•Mean heights varied.

•Difference between the mother and father height remained relatively constant, ranging from 4.5-5.5 inches

GRAPH OF MEANS AND STANDARD DEVIATIONS

OVER TIME•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.

GRAPH OF MEANS AND STANDARD DEVIATIONS

OVER TIME

•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

GRAPH OF MEANS OVER TIME BEFORE TABOO

•Their graphs follow a similar pattern (keeping the difference between male and female height relatively constant.)•Generally increases.

GRAPH OF SD OVER TIME BEFORE TABOO

•The standard deviation for both males and females fluctuates.

GRAPH OF MEANS OVER TIME AFTER TABOO

•So, means of men and women fluctuate between their mean and about 5 inches below their mean. Their graphs follow a similar pattern (keeping the difference between male and female height relatively constant.

GRAPH OF SD OVER TIME AFTER TABOO

•Similarly, the standard deviation for both males and females fluctuates.

REVERSING TABOO•In R, simply reverse the taboo: Have females only marry men that are shorter than they are

•Result: Height of population either increases by about 5 inches, or decreases by about 5 inches. •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

REVERSING TABOOWITH MORE RUNS

•Height tended to decrease most of the time, but sometimes just stayed the same.

STARTING WITH SAME MEANUSING AVERAGE CALCULATION

•Started at mean:65•Used average of parents to calculate height•Added taboo:•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…

STARTING WITH SAME MEANUSING AVERAGE CALCULATION

•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)

STARTING WITH SAME MEAN USING 1/3(OTHER GENDER) + 2/3(SAME GENDER)

CALCULATION

•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…

STARTING WITH SAME MEAN USING 1/3(OTHER GENDER) + 2/3(SAME GENDER)

CALCULATION

•Adding 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.

CONCLUSIONS•Adding 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.

SOME OTHER THOUGHTS…•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 heights:•In most mammals, the male is larger than the female, no clear reason as to why